September 1990 M90-79

}. R. Wait

<,

Antenna PerformanceInfluenced by theFinite Extent andConductivity ofGround Planes:A Collection ofReprints byJ. R. Wait, et ale

....,

o

DTICElECTE "MARO 41991 .

Approved for public release;distribution unlimited.

MITRE

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September 1990

J. R. Wait

CONTRACT SPONSOR MSRCONTRACT NO. N/APROJECT NO. 91260DEPT. 085

Approvt'd for public release;distribution unlimited.

MITREThe MITRE CorporationBedford, MauaChuMttI

Antenna PerformanceInfluenced by theFinite Extent andConductivity ofGround Planes:A Collection ofReprints byJ. R. Wait, et ale

L

ABSTRACT

Project 91260 "High-Frequency (HF) Antenna Element Modeling" is a MITRE sponsoredresearch project to develop computer programs and accurate models for predicting the elementpattern, radiation efficiency, and input impedance of HF monopole elements on ground planes ofvarioUs forms that rest on or are in close proximity to earth. A unique aspect of this project hasbeen the use of a technical advisor group composed of outside interested investiyators to assistMITRE personnel in the development and evaluation of analytical models.

As a mermer of this group, Dr. Jame. R. Wait has assembled a collection of twenty-tworeprints, authored by himself and his colleagues, on antenna performance influenced by the finiteextent and conductivity of ground planes primarily in the presence of earth. The objective of thiscollection is to provide a focus and filter of relevant papers from among the more than 750refereed papers by Wait and his associates; this paper presents this collection. Also included areDr. Wait's discussions of U.S. Navy Electronics Laboratory reports related to reprint 1.11 (Wait andWalters, 1963) and an annotated listing of selected related publications. A biography of Wait anda complete listing of his refereed papers through April 1990 are given in appendices.

Accesion For..-----f--~

NTIS CRA.&IOTIC TAlJU"CI:::'O~ :;t.:dJldlil:::dlio"

- ~_. -.. -.. .. - .. .. _.."""------~---._----

ByOi, t';0' ,'tia :T ..

. .---.-'.-----4iW.:iii.3 I):;:ty Co:ics

au

A-I

ii

AUTHOR'S PREFACE

The perfonnance of an antenna is dependent on the environment. In some cases theeffect is minor but very often we need to worry about the structure upon which the antenna ismounted. It becomes part of the radiating structure! For example, if the system consists of avertical wire fed from a perfectly conducting plane sheet of infinite extent, simple image theorywould suffice to predict the radiation pattem and the input impedance. But the ground plane maybe imperfectly conducting and it would always be of limited extent. Over the years, I have had anabiding interest in this problem, particularly in connection with the design of very low frequency(VLF), low frequency (LF), medium frequency (MF), and HF antennas in situations where theinfluence of the finitely conducting earth is significant. In such cases, radial-wire or mesh groundsystems are often employed to mitigate the adverse effect of poorly conducting earth in the lowangle radiation and to improve the overall power efficiency.

My publications, including those coauthored with my colleagues dealing with the topic, arescattered over many journals, reports and symposia proceedings. Some of these are no longeraccessible. For this reason and because I have received numerous requests to supply copies, Ihave felt it is desirable to assemble some of the relevant material in a reprint collection. I havetaken the opportunity to correct typographical errors in of the papers. Also I have added someextracts from related reports, particularly those from the U.S. Navy"s Ocean System Center in SanDiego where I often visited in the 19605. Finally, I have included a listing of related papers,pubUshed in the open literature, along with specific comments.

I wish to thank Melvin M. Weiner of The MITRE Corporation In Bedford, MA for getting me toassemble this collection. I am al$O grateful for his initial support. By way of introduction to thissubject, the reader should consult Melvin Weiner'S monograph, -Monopole Elements on CircularGround Planes: (coauthored by S. Cruze, C-C U. and W. Wilson) published by Artech House,1987. Their analyses deal with the case where the monopole and circular disc are located in freespace.

James R. WaitConsultant in Electromagnetics

and Electrical Geophysics2210 East WaverlyTucson,Arizona 85719

Iv

ACKNOWLEDGMENT

M. M. Weiner, Principal Investigator of Project 91260, wrote the abstract, edited the

manuscript, and motivated the author to assemble the enclosed reprints.

v

SECTION

TABLE OF CONTENTS

PAGE

1 Reprints 1-1

1.1 Wait, J. R., and W. J. SUrtees, May 1954, "Impedance of a 1-3Top-loaded Antenna of Arbitrary length Over a Circular GroundedSCreen, "Journal ofApplied Physics, Vol. 25, pp. 553-556,(Reprint No. 41)

1.2 Wait, J. R., and W. A. Pope, 954, "The Characteristics of a Vertical 1-7Antenna with a Radial Conductor Ground System," Applied ScientificResearch, Vol. 4, Sec. B, pp. 1n-195, (The Hague), (Reprint No. 43)

1.3 Wait, J. R., and W. A. Pope, , May 1955, "Input Resistance of lF Unipole 1-27Aerials," Wireless Engineer, Vol. 32, pp. 131-138, (Reprint No. 59)

1.4 Wait, J. R., April 1956, "Effect of the Ground Screen on the Field 1-37Radiated from a Monopole," Institute ofRadio Engineers TransactionsonAntennas and Propagation, Vol. AP-4, pp. 179-182, (Reprint No. 70)

1.5 Frood, D. G., and J. R. Wait, Jan. 1956, "An Investigation of Slot 1-41Radiators in Rectangular Metal Plates," Proceedings ofthe Institution ofElectrical Engineers, Vol. 103, pI. B, pp. 103-109, (Reprint No. 72)

1.6 Wait, J. R., and D. G. Frood, October 1956, "The Radiation Patterns 1-49and Conductances of Slots Cut in Rectangular Metal Plates," Proceedingsof the Institute ofRadio Engineers,Vol. 44,1467, p.1467, (Reprint No. 85)

1.7 Wait, J. R., October 1957, "Pattern of a Flush-Mounted Microwave 1-51Antenna," Journal ofResearch ofthe (U.S.) National Bureau ofStandards,Vol. 59, pp. 255-259, (Reprint No. 107)

1.8 Wait, J. R., August 1958, "A Study of Earth Currents Near a VLF 1-57Monopole Antenna with a Radial Wire Ground System," Proceedings ofthe Institute ofRadio Engineers, Vol. 46, pp. 1539-1541, (Reprint No. 121)

1.9 Wait, J. R., and A. M. Conda, 1958, "The Patterns of a Slot-Array 1-61Antenna on a Finite and Imperfect Ground Plane," L'Onde Electrique,Supplement #376, pp. 21-30, (also 1957 Proceedings of the CongresInti. Antenna, Vol. 1, pp. 21-29, Paris,) (Reprint No. 128)

1.10 Wait, J. R., 1963, "The Theory of an Antenna Over an Inhomogeneous 1-71Ground Plane," extracted from Electromagnetic Walle Theory, editedby E. C. Jordan, pp. 1079-1097, Pergamon Press, Oxford (ReprintNo. 208)

vii

SECTION PAGE

1.11 Wait, J. R., and L. C. Walters, April 1963, "Influence of a Sector 1-91Ground SCreen on the Field of a Vertical Antenna", U. S. NationalBureauofStandords, Monograph 60, pp. 1-11, (Reprint No. 215)

1.12 Wait, J. R., 1967, "Pattern of a Unear Antenna Erected Over a 1-103Tapered Ground Screen," Canadian Journal ofPhysics, Vol. 45,pp. 3091-3101, (Reprint No. 327)

1.13 Wait, J. R., Sept. 1967. "On the Theolj of Radiation from a 1-115Raised Electric Dipole Over an Inhomogeneous Ground Plane."Radio Science. Vol. 2 (new series), pp. 997-1004, (ReprintNo. 330)

1.14 Wait, J. R., October 1969, "Impedance Characteristics of 1-125Electric Dipoles Over a Conducting Half-Space, Radio Science,Vol. 4, NO.1 O. pp. 971-975, (Reprint No. 388)

1.15 Wait, J. R., 30 October 1969, "Surface Wave Effects with Large 1-131AntennaEarth Screens," Electronics Letters. Vol. 5, pp. 552-553,(Reprint No. 389)

1.16 Wait. J. R., and K. P. Spies, 2 October 1969, "Fields of Electric 1-135Dipole on Radially Inhomogeneous Ground Surface," ElectronicLetters, Vol. 5, pp, 478-479, (Reprint No. 394)

1.17 Wait, J. R., November 1969, "Transient Response of a Dipole 1-139Over a Circular Ground Screen," IEEE Transactions on Antennasand Propagation, Vol. AP-17. pp. 806-809, (Reprint No. 396)

1.18 Wait, J. R.. and K. P. Spies, January 1970, "Integral Equation 1-145Approach to the Radiation from a Vertical Antenna Over anInhomogenous Ground Plane." Radio Science, Vol. 5, pp. 73-79,(Reprint No. 397)

1.19 Wait, J. R., and K. P. Spies, July 1970, "On the Radiation from a 1-153Vertical Dipole with an Inductive Wire-Grid Ground System," IEEETransactions on Antennas and Propagation, Vol. AP-18, pp. 558-560,(and extended report version) AFCRL-69-0404, (Reprint No. 411)

1.20 Hill, D. A., and J. R. Wait, Jan. 1973, "Calculated Pattern of a Vertical 1-177Antenna with a Finite Radial-Wire Ground System," Radio Science,Vol. 8, pp. 81-86, (Preprint No. 488)

1.21 Hill, D. A., and J. R. Wait, March 1973, "Effect of Edge Reflections on 1-183the Performance of Antenna Ground Screens," IEEE Transactions onAntennas and Propagation,Vol. AP-21 , pp. 230-231, (Reprint No. 503)

viii

SECTION PAGE

1.22 Wait, J. R, September/OCtober 1980, "Low Angle Radiat:on of an 1-187Antenna Over an Irregular Ground Plane," extracted from Aliidella Fondazione Giorgio Ronchi. Vol. 35, pp. 576-583, (Florence,Italy), (Reprint No. 689)

2

3

Discussion of U.S. Navy Electronics Laboratory Reports Based onReprint 1.11

Annotated Usting of Selected Related Publications (1967-1989)

2-1

3-1

APPENDIX A Biography of James R Wait

APPENDIX B Complete Usting (to April 1990) of Refereed Papers oflleS R Wait

ix

A-1

B-1

SECTION 1

Reprints

Note: The reprint number in parentheses corresponds to that in the listing of refereed papers ofappendix B. The reprints in this section are assembled in the order of increasing reprintnumber.

'-1

1-2

1.1 Wait, J. R., and W. J. Surtees, May 1954, "Impedance of a Top-Loaded Antenna ofArbitrary Length Over a Circular Grounded Screen," Journal ofApplied Physics, Vol. 25,pp.553-556

(Reprint No. 43)

Reprinted with the permission of the American Institute of Physics

1·3

Impedance of a Top-Loaded Antenna of Arbitrary Lengthover a Circular Grounded Screen·

WALna J. SVUU" V~".,.i',,,r.,....'•. T_. c.......(Recrivcd s.pl~m~r 21. 195.1)

The probl~m 01 a verlical monnpnle .ilualed o\·,r " circular pnleclly CftftfluetinglCrem Irinl "" a finilel)'Clln<luctin~ .rnun,1 i. con.i<le,ed. An approsimllie ",~lhr.<1 _plo"ed oriIEinan)· 11)' Ahhott tn ,aleubte theKIf.imped:Ulce i. discussed. Using this Inrmub e",,'icil ~""rfS5iens are drri\-ed ler th~ ",'.impedance "' athin. top-loaded v~,tical monopol~. Sinll'SOidal curmll distrihution is--.l.

and

THE APPROXIMATE SOLUTION

(2)

(3)

(4<:)

(4b)

" all.E.-----,. azaE, aE.----;""'/11 ,ap az

"'here ,. and , art the intrinsic propaption constantand characteristic impedance of the medium, respec­lively, and defined by

.,- [i,.w(,,+iwf»1

,- [i,.w/(,,+iwf»l.

A subscript zero is affi.'lIed to these qu:anl it its in orderthat the,' should pertain to the air. That is,

.,.-i(f.,.)""- i2..t>.,'II'hert A is the free space wavelength in meters and

.'It- (../ ••)~J77 ohms.

Following the method of Storer,' the difference he­tween the impedance Z of the antenna over a finitescreen and the impedance Z, for an infinite screen isdenoted by 4Z and is given by

I1Z-Z-Z.- _-.!.-f' (~!..PIlI')'(I)tl:. (4a).,,'1.'. p ap

In this equation HI' is lhe change of the magnetic fieldn. for a finite screen from the mllJtl1etic field HI- for aninfinite screen. Thr current on the antenna is 1(a) andthe lJ.,se current is 1•. It then follows th:at

1 -!It.Z= -- f. BI-(P,O)E.(P,O)2..pdp,

1.' ,,..here

I iJ fa exp[-,.,(::+;)IJH.-(P,O)- -- - 1(z)II:.

2.. iJp. (::+;)1

INTRODUCTION

U SUALLY the most important factor contributingto the inefficiency of low.frequency antennas is

to be found in the ground s}"Stem. The ruin for groundsystem design are usually empirical and b:astd on re·suits of ellJlt!riments on existing installations. Recently.the general problem has been investigated analytic:allyby Abbott' who has developed a design procedure toselect the optimum number of radi:al conductors speci.fied given the values of the electrical constants of theground.

An important related problem is the actual change ofinput impedance of the antenna for different sizes ofground systems. This analysis has been carried out fora vertical antenna situ:zted centrally O1~r a ~r/eclly

conducting disk by Leitner and Spence! and morerecentl}' by Storer.' They, hO\\'ever, only considered thecase when the surroundinlt medium was free space. Amore appropri:ate situation is when the disk is lying onthe surface of a homogeneous conducting half.space'II'hieh corresponds to the ground. It is not surprisingthat the solution of this problem is in general very'difficult.

FORMULATION OF PROBLEM

With reference to a cylindrical pol:u coonlin:II"system (p,41,:) the antenna of heiRht " is coincident wilhthe positive z axis. The circular screen of radius a liesin the plane:= 0 which is also the surface of the ground.The conductivity and dielectric constnnt of the Itfoun,1nrc denoted by" nntl f. re51'C<"tively, "nd the dielectricconstant of the air by fo. The perme:tlJility of the wholespace is ,. whieh is t:aken to be the S3me as free space.

Due to the ob,'ious pol:ar symmetry in this problem,Muwell's equations t:ake the following form for timedependence according to exp(u.:/):

" iJpE,---(PHI ) (J).., iJp

(5)

Tht m:agnetic field in the ground outside the screenis a solution of the wave equation

(A-r)81 (p,:)-O,

...",... • (;) ': )....!... L ~bf f }f

• P"per p,...,nted "I Joint In'.,national Scitnlific Radio Cnion-Institute of Ra<lin Engin.... m"linK io Ottln. October 61h, 1953

1 F. R. Abboll, rroc. Insl Radio Enlf'" 40,846 (1952).t A. L.itn.r Ind R. : .. Spfnct. ]. Appl. Ph}·s. 21. 1001 (1950)•J. E. Storer. J Appl Ph)", 22. IOS8 (1951)

55J

1-4

554 J. R. WAIT ..'Nil W. J. SURTEES

so that

Now the differential equation for J is liven by

where .,- (A'+.,.') I and the electric field is liven b)'

(14)

O~,~h.

where II-,.';i- 2,.j).. The muimum currenl ampli·tude is 1_, and the conditions at the upper end of theanlenna specify a. For unterminated ends l(h)-O sothat a-If/r and then

1(:)-1, sin8(h-I:)/sin8h.

for

fnr 47. in F.q. (II) ..as fir.ol Riven in thi~ form byAbbott' ..ho quoted it wilhout I,mur hilt IlOinted outthaI it ..:as a plau~ible mlUlt. )..,Ier lIInnleath' lll~

arrh'ed at this same formula from an eXlension or thecompens.'tion tMorem well known in cirnlit theory.

" better approximation to Eq. (III) might be to set

1I.CP,G)-lI.-CP.O)F{p). (12)

"'here F(p) is Sommerfeld's "surface \\,ave allenuationfactor" which is a function of the numerical distance p"'hich for a good conducting ground (o'»ftoI) is given by

p- (,.p/A)(,..../O').

When p«1 it can be shown or seen from tabulatednumerical values' that F(p) is very close to unity.Jf p> 1 the magnitude of F(p) drops appreciably belo'\"unity. HO\\"ever. since H.- varies essentially as c-I •

and since F(p) is a slowly varying function the contri·bution from the int~nd in Eq. (II) for the cases"'here p> 1 are negligible. At .low \':ldia-rrequencies« 1000 kc) and for moderate ground conductivities(0'> II)-' mhos per meter) the IIp!lroximo'lle fomlulal:i,"C'n by Eq. (II) is sullicienlly close to the formula inEq. (10) to justify its use.

Up to Ihis point no restriction has been placed on thecurrent c1i~lribulion on the antenna. For thin antennasit can usually be assumed that

1(,)-1. sin(a-~%) (13)

In some instances at low frequencies it is customary toIo.,d the anlenna near the 101' so that elTecli"el)'l(h)P'O. It is convenient then to set

a-II(lI+h'),

..here h' specifies the del:reC of top Io.,din~.

Employinx the value of 1(,) gil'en by Eq. (13) antiinserting it into Eq. (ok) the inlegntion with respectto h can be carried au t to give

ij·l. c'" ro#,H.-CP.O)-- -cos(ph-a)--cOStt

2,. p p

ihc'" '~~~1---SID ,p ,

(i)

(10)

(II)" f-AZ-:::.- [ll.-CP,O)]2,.pdp.I,' _.

"1-AZ - -: II.-(P,O)H.CP,O)2,.pdp.I,' _.

(~~~P-A')JI(Ap)-Oap p ap

and so it readily rollows that, for .-0

E.- -;1[- .,.J I (Ap)e·'p(A)M>.

+ [- (I/2oy)J I (Ap)e·'p(A)A'dA+·· ·1. (8)

The binomial expansion or II is of the form

A'11-,.(1+"':"+'" ).

2-y'

and therefore

n.cp,,)- f- JI(Ap)e·'p('A)NlA, :~I), (6),

" a 1 aE.- -"H.-- - --(PH,l

2-y' ap p ap

± terms containing ,...... (9)

The second term is neglil:ible ir the propagation constantin the ground is sufficiently large and if pH. is notvarying too rapidly "'ith p. That is plI. should notchange appreciabl)' in a distance equal to 1,.-11 in theradial direction ror points outside the screen. Thisapproximation is alwa)'s "'ell justified under practicalconditions at low radio.rrequencies.

Employing this result for E. the equation for thesell-impedance is now given by

To apply this formula to an actual situation it can beassumed that H• (p,O) is not very differeDt from lI.-CP,O)in the region or the Rround plane where the losses aresignificant, and thererore

An equivalent statement or this approximation is thatthe electrical radio''' current density at the surrace ofthe grountl is not apl.rccio',lIly affected II)' the (mileconductivity of the soil. The approximate expression

, F. R. Abboll, 51andard Radial Ground Sr'lrm, 'or 1.1. andm.f ~10fI0I101. Transmillinc Anlrnnu. U. S. /IInr EIKlronics1....or3'''''· (San 1MJo), Rrpnrl ~n. 219, Orl. 19.;lI

• G. n. ~1""1r.llh,I'roc.ln'" F:1...1. .:n.... (I...n<l"n) 9~. rl.IV,23 (I9.~I).

• E. A. Notion, Prot. Insl. Radio Enen. 24, 13(,7 (1936)

1-5

IMI'EUANCE OF A TOr-LO,'I)F.O ANTENNA sss

'.If'''· 611"0·FIG. 1. The incremental IeIC·impedance 4Z for a "uartCf·...~e

monopole OYCf • circllbr ,rounded lCfeen.

where r- (p'+h')'. Using this value for 11.- the inte·gration indicated in Eq. (11) can now be carried outto yield .,~-----lcos·Ulh-a)ll+cos·al.

211' sin'a (-r -13#i~-It' sin'Ul/I-a)11+2ila cososin~ •

/~tJ!r-2 cosUlh-a) cosol.- 2ilasii\1

XcosCIJh-a)l,J. (IS)

where I .. I •• I.·· ·1. are integrals which can be ell­pressed in terms of the exponential integral

-r··Ei(-mN>--! -tis

• sas follows:

f-rIO'. I

/ 1- -;-dp",,-;;E.(-2i/J,,)_. rep II'

I--lr""Ei[ - li/J(,,-la»

2h'+e""Ei(-2~(,,+h)]1

• r li,,, 1I,. r -tlp--{E{-ZiP(,,+lrfY'"

J_.,P 2h-£i(-2i/J(,,-Jr)]r""1

• ri't...,) 1I,-f -dp--Ic'''/:i[-itle,,+I)Jr]_,p II-r'''Ei(-ille,,-I)"]/

f-r·Il....·'1.- -dp-Ei(-itfhf)

- p-Ei(-ifje,,+ l)hJe;f'-Ei(-iae,,-I)h]r'"

f -e--1.- -tlp- -41£i( -2i/J(,,+h»)eiU._. p

+Ei( - 2i/J(,,-Ir)]rU'1

1-6

"'hereI

'.- (0:+11')' and f- [.+ (.'+11')1]-.II

The exponential integrals occurrinlt in Ihe abnve ex­Il~~innsare all of im.'ll:inaf)· a~ment !Ill therefore they("311 he eSllrcs..qo,I in terms of the sine and cMine in­tejtrals, Si(z) and Ci(z). respectively, as follows

f:i( -iXl -Ci(Z)+lli-Si(X)1When the antenna is not terminated such that

Jr' - 0 or a- air the expression for 6Z simplifics to

~7.---""-[I-:i[ -lid('.+lrly····4" sin~IJ"

+F.I{ - 2~(,.-la»)e-··,.+2cns·d"/:i(- 2i~a)

+4 cMIJItIF.i(-i8ht)-I-:i[-idh(r+ I)y'"

-1':;(-itJlaCe-I)]r""l (16)

It has Men found that the results for .17. can be Jlutin a very corl\"enicnt form b}' writing

4Z-.1R+iAX

and then I"otting M as a function of AX for \':l1uesof the ratio "lla. The cun'cs take the form of spirals"'hich arc solrle\\"hat similar to the "impedance spirals"of S,orerl lor rM anrmll4 and disk .itu:uea in I~

space.:\5 a specific example thc impcdance '1liml for a

quarter "'ave \"ertkal monopole (i.e. Ia- AI,,) withouttop loading is shown in Fig. I. Since the displacementcurrents in the sround can be ncslectcd at low radio­frequencies li.e. ..«_/,) the curves can he convenientlynormalmd by multipl)'ing M and A.X hy (-If)I. It isinteresting to note that M approaches infinity as,,11r al",roaches zero. This is not surprising .ince ther;round loss would be infinite if the lower end of theantenna were in direct contact ,,·ith a homogeneoussemi-infinite conductor.

Further calculations from Eqs. (IS) and (16) ha"elJ«n cllrrit<1 OIIt in this laboratory by !\Ir. D. A.Trvmlilcr. In addition the effect of the Gruun,1 screenon the radiation field has abo been .tudied. Theseresults are the IUbjcct of subsequent papen.'-'

ACEIIOWUDGMEIfT

The authon "'ould like to thank Mr. H. Page. ~lr.

G. D. Monteith. and Dr. F. R. Abbau lor their in­terestinlt and helpful COmment. in connection withthis "·ork.

• W J. lOurtftS. ~fmft Rnearch Board Rrport on Grant :-".6:. DfJ>artnmlt of Electrical £osi_rinc. UniVCfsil)" of Toronlo(Octolwr. 19521.

•J R. Wail and W. A. r..,.... rarer lI:... 42.1. Instilut, of RadioF.n~Mft'S COlI,,"tion Rccord (~tarch. t9S4).

1.2 Wait, J. R., and W. A. Pope, 1954, "The Characteristics of a Vertical Antenna with aRadial Conductor Ground System," Applied Scientific Research, Vol. 4, Sec. B,pp. 177-195, (The Hague)

(Reprint No. 43)

Reprinted with the author's permission

1-7

Appl. sci. Res. Section B, Vol" 177

THE CHARACTERISTICS OF A VERTICAL ANTENNAWITH A RADIAL CONDUCTOR GROUND SYSTEM "')

by JAMES R. WAIT and W. A. POPE

Radio Phyaics Laboratory. Defence Researcb Board. Ottawa. Onto Canada

SummaryEmploying an approximate method the input impedance of a ground

based venical radiator is calculated. The ground system consists of anumber of radial conductors buried just below the surface of the soil. Theintegrals involved in the solution are evaluated. in part. by graphicalmethods. The final results are plotted in a convenient form to illustratethe dependence of the impedance on number and leDltth of radial conduc­tors for a specified frequency. antenna height. and Kround conducth·ity.It is finall~' shown that under usual conditions the radiated field~ arc mO<Ji­fied by only a few percent due to the ptCllCncc: IIf the: Itround ll~'stem.

§ I. Introduction. Antenna systems for low radio frequency aredesigned. usually, to work in conjunction \\;th a radial wire groundsystem buried just below the surface of the earth. The purpose ofthis wire grid is to provide a low-loss return path for the antennabase current and consequently to improve the efficiency of the trans­mission.

The rules for ground system design are usually empirical and basedon the results of experiments on existing installations. The firstsystematic study of this problem was carried out by B row n 1) I)and his associates who were mainly concerned with the operation ofhalf-wave antennas for the broadcast band. Sometime later A b­bot t·) developed a procedure to select the optimum number ofradial conductors. gi\'cn the values of the electrical constants of theground. An important related problem is the actual change of inputimpedance of the antenna for different sizes and types of groundsystems. This analysis has been carried out by Lei t n e rand

-I Work earned out under ProJect ~o. 048·95·55·07. Pap<'r p....vnlC~d at th~ '·.S.A.~ationalClln"enlion 01 the Institute 01 RadiO £nllilne~n III ~CW York. ~Iarc:h \954.

1·8

III

178 JAMES R. WAIT AND W. A. POPE

S pen c e ') and more recently by S tor e r i) for a vertical an­tenna situated over a perfectly conducting disc. However, they onlyconsidered the case where the surrounding medium was free space.

The purpose of this paper is to consider, in some detail, the cha­racteristics of a vertical antenna of any length situated over a cir­cular ground screen composed of N radial wires of equal spacingsituated at the interface between the air and a semi-infinite homo­geneousground. An approximate method to calculate the input impe­dance will bl' employed similar to that described by M 0 n tea t h ')who developed an extension to the compensation theorem of electric­circuit theory. The current distribution on the antenna is consideredto be sinusoidal.

With reference to a cylindrical polar coordinate system (e. tp, :)

the antenna of height Is is coincident with the positive z-axis as in­<licat('d in fiJ.:. I. The ground screen is of radius a and lies in the plane

Fig. I. The schematic representation of a vertical antenna situated over aradial conductor ground system.

== 0 which is also the surface of the ground. The conductivity anddielectric constant of the ground are denoted by (I and E respectivelyand the dielectric constant of the air by EO' The permeability of thewhole space is taken as fl which is taken to be that of free space. Theintrinsic propagation constant" and characteristic impedance '7 ofthe ('arth medium are defined by

y = [iilCt/(a + iWE)JIand

'] = [ijiw.(a + ;wE):I,

whefe til is the an~ular frequency. The propagation constant ;'0 and

1-9

THE CHARACTERISTICS OF A VERTICAL ANTENNA 179

characteristic impedance '70 of the air are then defined by

Yo - i(EoP)t w = i2n/). = ifJand

7Jo = (jjjEO)t ~ 377 Q.

where). is the wavelength in air.The self-impedance at the terminals of the antenna is now denoted

by Zr and can be broken into two parts by setting. Zr = Zo + LlZrwhere Zo is the self-impedance of the same antenna if the groundplane were perfectly conducting and infinite in extent. On the otherhand LIZr is the difference between the self-impedance of the antennaover the imperfect and the perfect ground plane. It is called the self­impedance increment and can be written in terms of a real andimaginary part as follows:

JZr = JRr + iJXT (I)

whc:n: LI/~T and JXr represent the resistance and reactance incfl'­ment. If the current at the tcrminal£ of the antenna is 10• the powerfl'qllired to maintain this current is J~UT' If the ground w('re per­f('ctly conducting the input power would be nUo where Uo is thereal part of ZOo The additional power required to maintain the samecurrent 10 at the terminals is J~ JRT *).

It is seen, therefore. that the quantity LlUr represents an impor­tant parameter of a radio frequency antenna.

§ 2. Tile impeda1tcc calclIlation. It is shown in appendix I that theimpedance increment is given by

(2)

(3)

where H';'(I]. 0) is the magnetic field of the antenna tangential to aperfectly conducting ground plane and E,(!!. 0) is the tangentialelectric field on the imperfect ground. If the current on the antennais / (:). it follows that A

J C (e-~("""·Il:.H';' = - 2;- -=- (:2 + 2)" 1(:) d:.

:t c!! w !!'o

.) It IS und"~lood 'h"l 10 "Ih,' rool1ll('3n ,qua... \'alul' ollhe currenI i1lh(' I'0\\('r ,­tu I", c:cpr("ss...d in "'atb.

1-10

180 JAMES R. WAIT AND W. A. POPE

For thin antennas it can be usually assumed that the current distri­bution is sinusoidal. that is

l(z) = 10 sin (a - fJZ)/Sitl a (4)

(5)

if the antenna is fed at the base (i.e. a monopole). The quantity a isdetermined by the height of the antenna and the top-loading, thatis

a = fJ(], + II').

The quantity],' specifies the amount of top loading and is usuallyobtained from experiment. For a thin antenna without top-loading(i.e. unterminated case) a = fJh. The integrals of the type indicatedin (3) can be expressed in closed form for a sinusoidal current distri­bution. The result is given by

i/ [e- itJ,fl";' (~, 0) = - .0 - cos ({lII-I1) -

2n san (1 (!

e-·It ill c-'!l' ]--COsu.---sin(<<.,-PJr) •e e r .

where r = (!'2 + ],2) Ii•.

Since the electric field E,((}. 0) is an unknown quantity, it is neceS­sary to make several simplifications at this stage. Since 1,,1 ~ fJ, anapproximate boundary condition (see appendix II) is employed.expressed by

(6)

where H!J (!! , 0) is the tangential magnetic field for the imperfect~round system and 'Ie is the surface impedance of the air-groundinterface. If !! is greater than a. the radius of the ground screen, 'I.can be replaced by '7. If (} is less than a, '7. is the intrinsic impedance'7, of the ground screen which is in parallel with the impedance '7 of .the ground. In a previous iD\'estigation 7) ') of this problem it wasassumed that 'I, was zero so the ground system was equivalent to aperfectly conductin~ disc of radius a. The limits of the integrationin (2) arc then from!! = a to fJ = 00. A more general case is when '7.is comparable in magnitude to 'I. in which it follows that

'7'7, f 0'1, = -,- or <: fJ <: a,'I -:- 'I.

1-11

(7)

THE CHARACTERISTICS OF A VERTICAL ANTENNA 181

where

11. = i1104 In~

A 2"tc

and where 4 is the spacing between the radial conductors and c is theradius of the wire. The expression for 11. has been derived') for awire grid in free space where it was necessary to assume thatiratil ~ 1. Since the grid is l};ng on the ground plane, this restrictionmust be replaced by 17'.41 ~ 1 where 7'. is the effective propagationconstant for propagation along a thin wire in the interface and isgiven by 10)

= (,,~ + ,,~1:.". 2·J .

If there are N radial conductors. it can be seen that 4 can be replacedby 2"t!}IN since N is usually of the order of 100. It is assumed alsothat H" (!}. 0) is not very different from H';'(y. 0) in the region of theground plane where the losses arc significant. This approximationhas also been discussed previously') I) and it certainly appears tobe valid if 1,,1 ~ (1.

The impedance increment -dZr is then written in the followingform:

(9)

(8)

(10)

•I~.1Z =:: 11 f[H';'(f!, 0)]2 2."t!,1 <1Q

•

-dZr = -dZ + -dZ,.when'

and •I~ .JZ, =::(~ [H;" (fl, 0)]2 2.!} dg.

• 11 + 11.o

The first expression L1Z corresponds to the self-impedance of themonopole over a perfectly conducting discoid. whereas the secondexpression dZ, accounts for the finite surface impedance of the radialconductor system.

It is instructive to consider L1Z. first in some detail. since theintegrations can be carried out and the result expressed in terms ofthe exponential integral defined by

•

'

'"e- iIllIEi (- ipa) = - - dg.

• g(11)

•

1-12

182 JAMES R. WAIT AND W. A. POPE

The procedure ~\'as outlined previously 7) 8) where an expression forLJZ was given for any value of the height h and top loading h'. Thespecial case where the antenna is unterminated (h' == 0) is given by

LJZ == \ {Ei[- 2iP(TO+ h)] ei~4+4n: SID tJh+ Ei [- 2iP(To- h)] e-2i,!4 + 2 cos2 Ph Ei (- 2ipa)

+ 4 cos tJll [Ei (- iPhg) - Ei [- iPIl(g + I)] eiil4

- Ei [- iPh(g - I)] e-i,!4]}, (12)where

Ig == h [a + (a2 + h2)1/2] and TO == (a2 + h2)1/2•

This equation may be put in a suitable form for computation byemploying the relation

Ei (- ilia) == Ci (pa) + i [~ - Si (pa)] , (13)

whl'reCi (pa) and Si (pa) are the cosine and sine integrals respectivelyas ell'fined and tabulated by J a h n k e and Em dell). The resultsof the calculations are presented in a most general form by plotting471LJZ'1l as a function of a/A for various values of h/A as shown in figs.2 and 3. It can be seen that the magnitude of LJZ increases withoutlimit as a approaches zero..This formulation is not actually valid inthis limiting case since one terminal of the generator would then beconnected directly to the earth medium (or to a disc of vanishingradius) and would lead to an infinitely resistive path for the antennabase current.

A slightly more convenient way to i11ustrate these calculations isto plot (a:/)1 2 JR. wherc iJR is the real part of LJZ, as a function ofa,A as shown in fig. 4 where I is the frequency in Hz. This is onlypermissible if displacement currents in the ground are negligible(note if EW <{ (1, '1::::: ((iiO),a)"'. It is interesting to note that JRactually assumes negath'e values under certain conditions. In thiscase, the input power to obtain a given current /0 at the antennaterminals is actually less for an imperfect ground than for a perfectground. This fact can be reconciled by sho\\ing that the radiatedpower is actually reduced if iJR < O. A physical explanation for theoscillating nature of the cun'c is that a wave is reflccted from thediscontinuity in the surface impedance at f} == a. As the radius tl

1-13

THE CHARACTERISTIC.S OF A VERTIC.-\L ANTENNA 183

increases, the phase lag ofthe reflected wave will continually increase.This viewpoint is substantiated when it is noted that the period ofthe oscillations is nearly equal to twice the diameter of the groundscreen.

10t•

-

II \

I N.8. VALUES OF "/~ ARE SHOWN,\ 1\ ON THE CURVES

\\\ 0

.~ ~~0' ~,o~""..t.

o~~...., -r-- Ir Of ''':':'~'--''-- "..........,;;:., ~.::--~..::;.:::~~------- .._- --0.0

°

I.

02

~:: 0.'10

.x.0-

0.'

1.0

1.2

2 3 _ IJ 6 7

RELATIVE SCREE,. RADIuS 0'"Fig. 2. Thl: incremcntal ~c1f.imlll:dancC"for a \'crtical antl:nna situated ,,\"(Oran idcaliz"d perfectly conductinl( discoid Iyin/: on a homQr;:cncous r;:rounll.

SlO'

315'

270'

".....225'N

<I~..

110'...Q

ISS'IUIn..:t '0'lL

_5'

0' L.-_"--_,;,.J._.l..-~~_~--l~_:--"-_--l'--"".......l.o 2 3 5 10

RELATIVE SCREEN

FIC::. J. Thl' phase of the Incremental self-impedancc correspondinr;: to th~

conditions of fig. 2.

1·14

184 JA~IES R. WAIT AND W. A. POPE

When the ground screen of finite surface impedance is consideredthe integrations indicated by (10) must be evaluated. This appearsto be a formidable task for the case when the antenna is of arbitrarylength. However, if the antenna is a quarter-wave monopole, withouttop-loading. the integrations can be carried out fairly readily bygraphical means. In this case

(14)

which is a special case of (5) with h' = 0 and h = ),14, and hencethe integral in (10) can then be expressed in the following form:

I~ pq cos [:r(1 - 4R) + 3: - tan-I p + qJ dP

LJR = P (15), ~ y'2"! [p2 + (I' + q~~lil P

o

and. similarly. LJX, with cos (-) replaced by sin (-). The realdimensionless quantities p. q••': and R arc defined by

p = 120 :rcS/y'2 with 6 = (EoW/a(l.24O:r2 P P

q =: N In NC with C =: cl),•

.4 = al), and R =: v'p2 + (t)2.

•2.II:CI

toi

0

-Io z s •

RELATIVE SCllEEN IlADIUS ./ft

Fil.(. 4. The rea) part of the incremental self·impedance corresponding tothe conditions of fig. 2.

It has been assumed in writing (15) in this form that displacementcurrents in the ground arc negligible, that is (ewlC1) ~ 1. Csing the

1-15

THE CHARACTERISTICS OF A VERTICAL .o\NTENNA 185

results of the numerical integration for LJR•• values of LJRT (= LJR ++ LJR.) are plotted as a function of A in figs. 5, 6 and 7 for variousvalues of d and N with C = 0.1 X 10-5• and in fig. 8 for variousvalues of C for N = 100 and d = 0.1. The value of d can be readilyobtained from fig. 9 when the ground conductivity a and the fre­quency in kHz are specified.

o

\ 1.'010

1'\'00--

"10 --0

!\'~

-=V r-..... -t"'-....

\~ V •• 10 f"""oo,.. -r-~

0 .!!=.

r\ /' '"\ J "- .-'

-000 o-t 0·1 o·~os 0,' 0,' 0,' 0" I'

0-0

1 •j

A.~

IllUTM IUllUI f1II SCIlEEII

FiK. 5. The incremental self-resistance of a vertical quarter-wave monopoleon a radial conductor ground system for a wir.e radius equal to 10-0 of a

free-space wavelenRth.

It is immediately e\'ident that the oscillations in the cun'es forLJRT have been damped if .v is finite. This can be expected sincethere is a smaller change of the surface impedance at !! = a if theground conductivity is reasonably high (15 < 0.1). The limiting casewhere N = 00 corresponds to the perfectly conducting disc discussedpreviously. It is quite apparent from these curves that a groundscreen radius greater than about 1 of a wavelength is wasteful. Onthe other hand it would be quitc feasible to choose a large number ofradials to reduce the resistance increment to a low value. Although.in practice, it is usual to employ ~o. 8 wire (e = 0.5 :< 10-5 at 1~lHz); some improvement could be obtained by using larger wirediameters. From a theoretical standpoint, huwevcr, it would appearthat. for a gi\'en total wcight of wire, it is prefcrable to us\:' a cun-

\ ppJ. ,<'" /<:.,. II 4

1-16

186 JAMES R. WAIT ....ND W. A. POPE

ductor of smaller diameter, say No. 22 wire (C = 0.1 X 10-5 at 1MHz) and to employ 150 or more radial conductors.

0.1 o.s 0.4 0.' 0.' 0.7 . 0.' 0.9 . 1.0

0

1\\ ".100

\ 1\~.IO".

.0

\ \~\

100..

IV' r........... -"GJ

.0\ ..o~~ l[

.0

I

o0.0 0.1

14.

II.

RELATIVE RADIUS 0' SCREEN

Fil(. 6. Th~ incremental self-resistanCe of a vcrtical quartcr-wavc monopClleon a radial conductor ground system for a wire radius equal to 10-1> of a.

free-space wavclength.

0·'01~5 06 010402 oa

I.oos

~ fe''''''1'-.. .....

'\~

.·100I ,

~••••0

.................. ./

~__--L-I.Q

00 01

54

I.. 24<I

Ao~l

lIIUITIV! IIID11S OF SCREEN

Fil;. 7. The incremental self-resistance of a vertical quarter-wave monopoleon a radi.11 conductor ground system for a wire radius equal to 10-1> uf a

free-space wavelength.

1-17

THE CHARACTERISTICS OF A VERTICAL ANTENNA 187

§ 3. The earth "men/s. It is interesting to examine how the an­tenna base current is shared by the radial conductcrs and the grounditself. If the current flowing in the ground is denoted by I, and thetotal current in the radial wires by I •. then the ratio 1,/1. is equal

\\ ..lOCI

\ ao_0

,~ c.......

\\ LJ; .......-, ~

ItCo-wr'.

\ i":: :::;:....-- ICOIQ1"~

!\. ./' r---..

" ./f-cc••-1.0~ w ~ ~ ~ ~ U M U ~ ~

Aa~

IlIL.AT1'II UllIUS f11 seMEN

l;ig. 8. The inert-mental seU·resistance as a function of the wire radius witha fixed number of radial conductors and ground conducti\·ity.

to the ratio of the surface impedance of the grating of the wires com­posing the earth system to the surface impedance of the ground.Therefore it follows that

(16)I, 'I, iqIe = -;J ~ (1 + i) p'

where p and q have been defined previously. Since the total currentis given by I, = I, + I •. it follows that

I~~ I= II +11/,/'1I= [p2 + ~2: q)~':I' (17)

I t has been assumed here that displacement currents in the groundare negligible (i.e. EW U ~ I).

Equation (17) is not a function of the height of the antenna andtherefore it would apply also to top-loaded antennas as long as the

1-18

188 JAMES R. WAIT AND W. A. POPE

circular symmetry is essentially retained. Employing this equation,curves are plotted in figS'. 10 and 11 to show the dependence of thecurrent in the radial \\;res on the various parameters. The abcissaeare the lengths of the radial wires in wavelengths measured from thebase of the antenna. It is noted that if the radial wires are increasedbeyond a certain length. nearly all the current flows in the ground.When the ratio ile/I,; is equal to !. the current in the ground is equalto the total current carried by the radial wires.

~ .• 1---1-~d--';::'"k:::--::~~:...,j~...

.: .001--l--+-.,..,..-~~--:r~:w~"'lo~,j,...,;:.....::~::o.."",...,j

"11---1--+-+-.p~...

"'I,~-~--I~~,-~--~--,----""'~---".... ..I •.,'-'10 c_""n .,._,.,

Fig. 9. Ground conductivity.

§ 4. The radialtd power. While the main subject of this paper hasbeen to e"aluate the input impedance of the antenna. it is also ofsome interest to know if the presence of the ground screen appreciablychanges the radiated power. A simple and approximate analysis isnow carried out which indicates that this change is small.

The power dissipated in an elemental area of the ground is equalto the real part of '1;H~;. It is then evident that the change of powerlost .lIJL in the ground due to the presence of the ground screen isgi"en by

•LJPL=Ref.J"I!H,,:22~!Jd!!. (18)

o

where JI} is the difference between the surface impedance '1, of theradial conductor earth system and the surface impedance 'TJ of theground. Howe,·cr. the change of input power at the antenna termi­nals is gi"en by

•JP = R,·.r J'IH; 2:T!! d!J.

o

1-19

(19)

THE CHARACTERISTICS OF A VERTICAL ANTENNA 189

Now, since there is conservation of power, the change of the totalradiated power .dP, beyond the edge of the earth system is equalto.dP - .dPL or

•.dP, = Ref .d7J [H; -IH;IJ 27r!> dp. (20)

oI.Or--------------,

0.9

0.'

0.7

0.'

..-\'0.5•--

0.4

O.S

0.2

0.1

0.0'.....__a.--'-....I1.o...........I-""'-.....-""'-~0.0 0.2 0.4 0.6 0.' 1.0

.y~

Fig. 10. The ratio of the current carried by the radial conductors to th~

total earth current as a function of the distance from the base of theantenna.

For a good ground screen, J'1 =:: - '1 and, if the antenna is a quarterwa\'e monopole (iJ = ;.;4), the integration can be carried out in

1·20

190 JAMES R. WAIT AND W. A. POPE

closed form to yield

iJ~. = Real part of 4: {[2ln.~+ 0.5773

n _/---::-~+ In"2-Ci (v~ + (2,ia)2- n ) + Ci (2"t)

- Ci (\/;r'l + (2,ia)2 -+- :E)] + i [Si (V~+ (2,ia)2- n)

- Si (2"f) + Si (V~ + (2,ia)2 + ;or)]}. (21)

NolOO

8. r;:;;;-

o V7 00.100.1

0.7

0.6---\0,5-- 0.4

0.3

0.2

0.1

0.0.0 0.2 0.4 0.6 0.1 1.0

">'Fig. II. The ratio of the current ·~;,."l'Ied by the radial conductors to thetotal earth current as a function 01 the distance from the base of the

antenna.

If the ground screen radius a is small compared with a wavelength,the change of power radiated is given approximately by

LJP. 2 ' 1 2~ Re i'7P a· 2 ~ -'7 -, (fJal. (22)

l~ 2\/2

1-21

THE CH.O\RACTERISTICS.OF A VERTICAL ANTENNA 191

This is usually a small quantity with the total radiated power so it isof minor significance at low radio frequencies.

§ 5. The radiation paUern. The effect of the ground screen on theradiation pattern is also of some interest. For convenience, it isassumed here that the ground screen is equivalent to a thin, per­fectly conducting. circular disc laid on a homogeneous ground. Themagnetic field H.(e. z) in the air can be written as the sum of thefield H:"(e. z) for an infinite screen and a secondary field H; (!!. =),

In appe!1dix I it is shown that .--H~ - Yo ffl. (~) I. (le') e-"" JAo' E,«(1', 0) e' de' .i dA, (23)

~ .,'._1.0. '

where"0 - (.i

2 + ~tl.

The change of the magnetic field AH" due to the presence of thescreen is now given by

.JH~ - H~ -IH~:_.o -

--- - YOf)"£,«(1'.0) Q' dQ' I. (1.1/) I. (IJ,!) e-""',,;I.i cli.. (24)'10 -,·.01-0

The integration with respect to .i can be carried out by the saddlepoint method of integration since pf! ~ 1 in the radiation zone, Theresult is

where

-pe-i6R,'" ,AH" = ---,r E,('!. 0) I.

7J~' •o

(25)

(26)

R = ve2 + :? P= 2-ril = - iyo. and fJ - tan-I =,f!.

The approximate boundary condition. E,(f!'. 0) ~ -7J H.(r,l'. 0). canno\\' be p~p'oyed so that

-AH - P'7 e- oIIR ,"'H (0' 0)• " -' I Ii? ' 0)' d '-- = -- --,r- I \#'0 cos 0 o.H" 7Jo .n:. H,,(f!. =) - _.

oAn approximate expression for .JH<; is now obtained by replacing

1·22

192 JAMES R. WAIT AND W..... POPE

H.,(e, z) on the right hand side of (26) by H';'(e, z). If the antenna isa quarter wave monopole,

-i1 .• _H-(e' 0) = -- e-"" ,"+(AI4)' (27).' 2-r(l' '

and for Pe ~ I

- i1. (:r )H-(o -) ,.... -- e -i6R cos - sin 8.. -' - - 2-r(! 2' (28)

so that

(30)

LJH..--=-H.

2IM,Ar )!!...J- e-'V~'+ ~ I. (1' cos 0) dp. (29):r. '10

cos 2'sm8 0

The right hand of this equation is of the order of I'1i'1oI which is smallcompared with unity. For small screens where a <A the relationsimplifies to

~. ~-I~ I·-;~C·~O)' (>-;) ,cos ism

which is of second order magnitude.

§ 6. Conell/siml. The results of this analysis. while not exhaustivc,are sufficiently developed to be useful in the design of vertical an­tennae with radial conductor ground systems. The work has shownthat the input impedance of the type of antenna discussed is depen­dent mainly on the number and the length of the radial ground con­ductors and on the conductivity of the ground in which the wiresare buried. The dependence of antenna impedance on ground wiresize is shown to be very slight. It may be concluded, from this dis­cussion. that a sensible design criterion for an optimum groundsystem is attained by a suitable choice of number and length ofground wire radials so that they \\ill always carry an appreciablefraction of the total earth current.

ApPEXDlX I

Fo,.naulation 0/ the input impedance. An expression is here for­mulated for the input impedance at the terminals of an antennasituated over a circular screen. The total flux F of the vector E ;< H

1-23

THE CH.-\RACTERISTICS OF A VERTICAL ANTENNA 193

over a surface surrounding the antenna is given byF =/E X H.nds (31)

swhere n is the unit outward vector nonnal to S. As is customary inother problems of this type. S is chosen to be a slender cylindricalsurface of vanishing radius!! concentric with the antenna so that

•F == -lim melE,H. dz. (32)

,..0 0

It then follows that•

Z == l~ [ - ~fE. [(:) dzJ.o

(33)

(34)

It is now convenient to let E, == E:' + E:. where E:' is the corre­sponding value of the electric field for a perfectly conducting groundplane and E; is the change of the field to account for the finite con­ductivity !n the soil and the ground s~'Stem. The impedance incn'­m'~nt JZT is then given by

•JZT - [ __[12 rE; 1(:) d:J ...

o. II.o ,..0•

['10 ,10(1 0 J

- ---2 - -f!H~) 1(::) d: .)'0/ 0 • f! Of! lim

o ,..0

Since H~ is a solution of the wave ~uation.- (35)

-H~(g. z) == I I. (~) e-.... /().) ldi..

ofor z ;> 0, where JAo == ().2 + ~)I/•• From Maxwells equations it isseen that

EII(!!.O) == '10 , I. (i{J) I(l) /Iol dl. (36))'0 •

oand by applying the Fourier-Bessel theorem it follows that-

I(l) == )'0 f I I (Ag') Ell (g'. 0) fl' d!?'. (37)'1oJl o

o

This equation for I(i.) can then be substituted back into {35) toobtain an expression for H~(,!. ::) in tenns of EII (!? 0). It is also noted

1-24

194 JAMES R. WAIT AND W•. A. POPE

that II(}.e) can be replaced by Ae/2 as e tends to zero so that-- .~~ H~(e, z) = ;;0 ffell (Ae') e-"" p.;1 A2 <U E,(e', 0) e' de', (38)

o 0

and introducing Sommerfeld's Integral

J~-""Pol Alo (~) dA = (z2 + (i~)-I!I e-Y'("+")''', (39)o

the integration in (38) with respect to A can now be carried out to~ve _

fa -,..("+,,,)'1,

lim H;(e,:) = 2"1oe ':'""7 e(z2 '2)'1, E,(e', 0) e' de'. (40)~o PIa ee + (!

o

Inserting this expression into (34) leads directly to (2) for the inputimpedance ;'lc·ment.

ApPENDIX II

The approximate bOl4ndary condition. The magnetic 'fieid in theground outside the screen is a solution of the wave equation

(.::I - y2) H.(e. z) = 0

(41)-H.(e. :) = JI. (Ae) cP' peA) i.dAo

for: "' 0 and where II = (A2 + y2)11l. The electric field is given by

and therefore

(42)-

E, = _!!- ( J.l II(~) eJ" pel) l<U.y-o

The binomial expansion of J.l is of the form

J.l = "1(1 + ;;2 + .... )so that _ OlD

E g~ _!!. { (i' I'<~) eJ" Pll) ldi. +(~ II (.'.e) eJ" P(l) l3 di.+.. }. (~y . .,; 2y

o 0

~ow the differential equation for II is given by

( c 1 C 2);;:--;:-!!-l I,().e) = 0,C!! f] Cf]

(44)

1-25

THE CHARACTERISTICS OF A VERTICAL A~TENNA \95

so it readily follows that

11 0 \0 . 4E = -flH -----f!H ± termslny-. (45)

• ~ 2y2 of! !! af! "

The second and remaining terms are negligible if the propagationconstant of the ground is sufficiently large and if H" is not varyingtoo rapidly with fl. That is. !!H., should not change appreciably in adistance equal to Iy-I! in the radial direction.

Acknou·ledgement. We would like to express our appreciation tollr. G. D. M 0 n tea t h of the British Broadcasting Corporationand to Dr. F. R. A b bot t of the U.S. Navy Electronics labora­tory who communicated to us unpublished reports of their work. Weare also indebted to Dr. W. J. Sur tee s and Mr. D. A. T rum p­I e r who assisted us in the earlier phase of the investigation and to}Ir. P. A. Fie I d and Dr. W. Pet r i e who read the manuscript.

Note added in prooj. Further calculations for short top-loadedantennas have been carried out. The results are available in RadioPhysics Lab. Report No. \9-0-7 April. \954 by j. R. W a i t andW. A. P ope. Good a~lJ'eement has been obtained with experimentfor a 250 foot mast at \00 kHz.

RfC~ived 171h ~an:h. 19$4.

RHFEIU~SCES

II B r 0 ... n. G. B .• Proc. of the I.R.E. =:1 (193$) 31 \.2) Brown. G.H.. R. f. l.ewls andJ. Epstein. Proc.ofthe I.R.E.:4119J7)

7S3.J) A b bot t. F. R•• Proc. of the I.R.E.• 40 (19521846.4) l. e ito e r. A. aDd R. D. S pen c e. Appl. Pb),.:U (19$01 1001.5) Storer. J. F., J.Appl. Pbys.:::l (1951) 1058.l>\ ~ 0 n tea t b. G. D.. Proc. of the lost. Elect. EOI".••• Pt. IV (1951) 23.7) Surtees. W.J.andJ.R. Wail. J.AppI.Pbys.:::;(I954)55J.81 W a i I. J. R.. Radio Pbysics l.ab. Report 19·0.4, 1953. (Call. Defence Research

Board Project D 48.95·55·07).9) ~I a c Far I a n e. G. G.• Jour. Insl. Elect. EOl!". 83. Pt. III A (1946) 1523.

101 Col e maD, B. L.• Pbil. ~lag. "I (1950) 276.II) Jab n k e. E. and F. Em d e. "Tables of Fuoctions". Dover Publications. !'lew

York,1945.

1·26

1IIII

1.3 Wait, J. R.. and W. A. Pope, May 1955. "Input Resistance of LF Unipole Aerials."Wireless Engineer. Vol. 32, pp. 131·138

(Reprint No. 59)

Reprinted with the author's permission

1·27

INPUT RESISTANCE OF L.F. UNIPOLEAERIALS

If/ith. Rf;'dial "--ire Earth Systems

By J. R. Wail, B.A~ I\I.A-Se.. Ph.D. and W. A. Pope(Ru", PIJ,.... ~.....,. D,{nec, Rftle,,1J Bo.J. Ott.rc. 0 ......... c.....,.)

SUMMARY.-The input resistance of a low-frequency unipole aerial is calculated. The earthsystem consists of a number of radial conductors buried just Ix'lo\\' the surface of the soil. The integralsinvolved in the solution are evaluated. in part. by ~ra.phical methods. The final results are plotted ina convenient form to illustrate the dependence of the input resistance on numlx'r and lenl;th of radialconductors for a specified frequency and earth conducti\·ity. The curves should he USl'ful in thedesign of earth systems for low.frequency transmitting aerials. It is pointed out that increasing theradius of the earth system beyond a certain limit gi\'cs only a slight impro\'ement in radiation efficlt'ncy,

d

b

N

AZ

,- anti 8

I'H,F and '"

Si (Jr)

Ei (- x)

01.1')

CII.

Ci IJrI

= - r~lll)dt• Jr

= a quantity whos.· onlt'r "f mal(lIilud.. ise'lual tl' ,"

mol:.:nitudr and phase' of thr complexnumbE'r l't J8d"tin"d in equation [15(h) ~

\'ariabl" of intt'::rat ill" (= pl:l)= Jlt'i~ht "f a"rial in wa\'"I"nl:ths (~II;}.)

= ma,::nitudr and phas" Ilr th.· comp!<'xnumber F,J'" .It·fined in "quation (ISa)

1'. q. ~. ,.,. 8. C1 = dim"nsionl,"" quantities tlcfinc·<\in th" trxt f"lI"win!: equation 115(a)]

= thr input resi ..tancf' "f II... a"rial f"r 11

prrf"clly-condnclill/: "arth planr= EIJI.·r', numh"r (,- (/'5772 , . ,)= al'tual height of the unl":III..d arrial

f~in t d--, I

In

= - f;'is~ dl

• r

= (':"')1= ground conc1l1t'ti\'ily l'arallll't .. r

Introduction

x:I~ I.-\L systems for 10w-frequl'IKies aredesigned. usually, to work in conjunctionwith a radial-wire earth s\'stern buried

just lx·low the surface of the ('arth. - The purpose(If this wire grid is to pro\'ide a low-loss path forth(' aerial base current and consequently to im­pro\'e the radiation efficienc~·.

The rules for earth-system design are usually('Itlpirical and based on the results of experimentson existing installations. The first systematicst\llh' 01 this problem was carried out by Brownu

a1111 his associates who were main'" concernedwilh th(' operation of half-\\'a\'e aerials for thehwadea,;t band, Sometime later Abhott' develop­(,J a procedure for selecting the optimum numherof radial conductors. ~i\'{~n the values of the(''''elrical constants of the ~round. An importantr('blrt! problem is the actual change of input1llll'I.'dance of the aerial with diflerent sizes and

E p (P. 0)

AZ,AR,AX,1.

H':(p.O,

".

AZ,

t

Y,

LIST OF SYMBOLS("..k.s. ",.ils art mll'l",.,d IhrO"/:hollt,

(P. ~. z) = cylindrical polar co-ordinatrsh height of an ideally top·loa,hl al'fial

or the equivalent height of anunloaded aerial

lenl;th of radial wires= conductivity of ground= permittivity of ground= permitti\'ity of air (= 8'85 X 10''''"" permeability of free space (= 4.. " 10 ')= intrinsic propallation constant "f I:rollnd= charactt'ristic impedance of ground= propagation con'ltant of air= charactt'ri~tic impc.lance of air 1= I:?H.. ,= wa\'e number in air (= - lY.)= angular frequency= wavelenl!:th in air= self-impedance at terminals of aerial"" self·impedance 01 the aerial for a

perfectly-conduct in!: earth plan(= self-impedance increment (= Z, - Z.l= real part of AZ,= imaginaf)' part of AZ,= current at terminals of at'rial

(= ,/2 ... r.ms currrnt)= the talll("ntial maJ:l\l·tic 'i"I,1 .. r II...

aerial on a p"rf"ctly-conduct inl: ,';art itplant· of intinitl' extent

the tanJ:ential electric field of the arrlaJon the imperfect ground

I(z) >= current .lIon,;: the aerialH. (P. 0) -= the tanJ:cntial ma,::netic fi ..lcI of thr

aerial on the impl'rft'ct ~mund>= the surface impedance of the air-ground

interface.., the surface Impedance of the ra.holl

wire grid.. the spacinJ: between the radial wi"".., radius of the wires of the grid= the effective propagation constant or a

wire in the air-~round intt'rface"" number of radial wires in tht' earth

s,"stem.. self:impedance increment f"r an illt-al

circular ground screen of ratllU' II

.. correction to 4Z to account for till'losses within the ~round scrren

-= limit of integration for equations (I '.!Jand (13)

".

UI).

Z,Z.

aaaa.P.y'IY.'I.fJ

---_._--- - '-_.- ----MS accepted by the Editor. May HIS-l

WIRELESS ENGtNEER. ~IA\' 1955 Vol.32, pages 131-138

'-28

(2)

p

t.,. (AIR)

/// /. ///;'

UDlAl tDNDUCTOREARTH SYSTEM

Fig. I. The I.trtical atrialWIth a radinl'flIlldllctor ,arth

5.1'5''''',

z

h1

real part of Zo0 The additional power requiredto maintain the same r.m.s. current 10 ,,- ,12 at theterminals is I otiJR,i2.

General TheoryIt was shown in the pre\'ious paper' that the

self-impedance increment iJZ/ could be writtenin the following form:

I r'"'iJZt = - I t H; (p,O) E" (Po U) 2rrp dp• , 0

whl're H; (p, U) is the magnetic field of the 3l'rialtangential to a perfectly conducting- earth planeof infinite extent and E" (Po 0) is the tangentialelectric field on the impl'rfect l'arth- Thisformula also follows immediatl'h' frolll the workor ~Jollteath7. If the current 'on the aerial is1(:) amps it follows that

I ~ r' _c:.-'\:l~- j~!5.t .-1:. p!)I") 1(-) d-:!" fJp (:1 + pl)I" "

, II

'I'll!' ell'ctric field E p (p, 0) is l'ssentialh' anunknown quantity. How('ver. since )' ~ Yo'an approximate boundary condition call beel1lplo~'ed expressed b~·.

E" (p. 0) ~ - "'1e H. (Po 0) (-I)

Whl'Cl' H 6 (p. 0) is thl' tang<'ntial J1la~lIeti(' fipldfoc thl' imperfel't earth and TJr is the surfaceimpedance of the air-ground interface.

If p is ~reater than a (the radius of thl' earthserel'n) "'1e can bl' replaced b~'"'1' If p is less thana. "I, is the intrinsic impedance "'1' of the earthsen'clI in parallel with the impedancl' 7J of theearth. In a pre"iolls investigationS" of thispwbll'm it was assumed that "'1' wac; zero, so thatthe I'arth s~'stem was equi"alent to a perfectl~'-

types of earth systems. This analysis has beencarried out by Leitner and Spence4 and morerecently by Storer', for a vertical aerial situatedover a perfectly-conducting disc. Howe\'er, theyonly considered the case where the surroundingmedium was free Sp'lct:.

In a previous paper' the electrical charac­teristics of a vertical aerial wiih a radial conductorsystem was studied. Employing an approximatemethod the input impedance was calculated.To illustrate the nature of the problem onl~' aquarter-wave unipole was considered in detailsince it was the case most easily computed.Curves were plotted showing- the dependence ofthe input base resistance on number and length ofradial conductors for a specified frequenc~' andground conductivity. It is the purpose of thispaper to extend the solution and calculations forshorter aerials with top-loading.

With reference to a cylindrical polar co­ordinate system (p, .". z) the aerial of height" iscoincident with the posith'e % axis as indicated inFig. I. The earth screen is of radius n and lies inthe plane % = 0 which is also the surface of theground. The conducti\'ity and pemlitth'ity ofthe ground are denoted by C1 and ~ respecth·e1y.and the permitth'ity of the air is denoted by (0'

The permeability of the whole space is taken asP-o which is taken to be that of free space. Theintrinsic propagation constant)' and characteristicimpedance "'1 of the earth medium are defined h~'

)' = UP-ow (a + jw~)J1 t

and TJ = UP-OW! (a + jW()]l't

where w is the angular fre(luel1l'~·. The pfopaga­tion constant )'0 and the characteristic impedann'"'10 of the air are then defined by

)'0 = j ((oP-o) I 2w = j2rr" = j{Jand "'10 = (P-o' (0)11 ~ :477 ohms :H; (p.O) =

where " is the W3"elength in air.The self-impedance at the terminals of the

aerial is now denoted h\" Z/ which can be brnk<,ninto two parts by setting. Z/ = Zo ~ .JZ, wheceZo is the self-impedance of the same aerial if thl'earth plane were pcrfectly conducting and infinitein extent. Thus LlZ, is the difference Ilt'tweenthe self-impedance of thl' aerial over thl' impt'ffef'(anG the perfect earth plane. It is called tlie sell·impedance increment and can be written in tefmsof a real and imaginary part as follow:'>

LlZ/ = LlR/ + jLl.':t (I)

where iJRt and iJ Xt represent the resistalll'l' andreactance increments. If the current amplitndeat the terminals of the aerial is 10 , thl' J!nWl'frequired to maintain this current is 10:U, :! wa"~If the ground were perfectly conductin~, tllf'input power would be I 02Ro'2 where Uo is the

2 \\'IRELEo;S ENGINf.ER, )I.\\' 1!'l55

1-29

(6)

(8)

(5)

Impedance CalculationFrom equation (3) it follows that

The impedance increment ~Z, is thl'1I wriUt'nin the following form

AZ, = AZ + AZs

,'ar:

where 101 AZ ~." {H; (P. O)JI :!.1TP lip

wO

'and 101 AZs R:S fO -:!l+."Is [H; (p, 0)]1 21Tp dp. (7)

.'" "IsThe first term AZ corresponds to the self­impedance of the unipole over a perfectly con­ducting discoid. whereas the second term AZsaccounts for the finite surface impedance of theradial-conductor earth system.

Assuming a sinusoidal current distribution for1(:) the magnetic fidd H: (p, 0) can be ('xpr('sscdin clO!lCd form. The integration indicateu byequation (6) can then be carried out and theresult expressed in terms of sine and cosineintegrals', Curves of the function AZ fur un­tcnninated aerials ha\'e been computl.'<! from thisformula'. The integrations indicated in equation(7). howe\'er, cannot be carried out allal,\·tically.It is necessary to resort to a numerical procedurefor this case.

conducting disc of radius a. The limits of theintegration in equation (2) are then from p = ato p =00.

A more general case is when '1$ is comparablein magnitude to '1. in which if follows that

'1'1$ f'Ie = -_. or 0< p < a"I + "Is, did

where "Is = J"Io A og~ 21TC

and where d is the spacing between the radialconductors and c is the radius of the wire. Theexpression for '1$ has been derived10 for a wiregrid in free space where it was necessary toassume that I 'Yod ' ~ I. Since the grid is lyingon the earth plane, this restriction must bereplaced by I 'Ytd :~ I where 'Y~ is the effecth'epropagation constant for propagation along a thinwire in the interface and is gil'en byll

'Ye = ('Yol

~ t.Y'I

If there are N radial conductors it can be seenthat d can be replaced by 21Tp!N since N is usuallyof the order of 100. It is assumed also thatH. (p,O) is not \'ery different from H': (p,O) in theregion of the ground plane where the losses aresignificant. This approximation has also beendiscussed previously'·' and it certainly appearsto be valid if : 'Y I > i 'Yo·'......................... -.- -.- .

H 2O( 0) = .p- J'. rL~XP:-C j~ (pI + ~:llll] + jj3 e.:xp, [ -=- jfJ (~~±- ::~J J lz) dz.

• p, 21T (pI + ZI)U pI + Zl,=u .......................

( a) ( b)

O'l4

Fig. 2. Thr illr,rmr,,' oj II,eanial ba.'r rr.'i5Ia"rr a.' a j"'''',,,,,of Ihr ,·adl1l~ of Ihr rarth 5.\"."..,11(n) j,., ,'a,·icJIIS ,'al"r., ''1 S ./", III'",'al"f., "1 & (/,,,1 Ihl Jv, ,.", i ..".,

m/llrs oJ & for.\" = 7_.

0'0'oo 2400'200'160'12

a/)..0'0'0'04

2 1

\\ h/).._o·on

•\ a-o·1

I 8-0'os,~ ~

N- ~o

1\ ~:~~-~

~-~)~ ~-IO~ .::::----uo-.

o

I"

1'4

"6

0'6

0'4

0'2

WIRaLass ENGINEJ!R, MAY 1955 3

1-30

For top-loaded aerials that are short compared \\;th a wavelength a reasonable approximation is1(:) AI 10 , over the length of the aerial. Also. exp_ [- jfJ (pi + %1)111] is given to sufficient accuracyby expo (- jfJp) so that

H a:> (0) e -jfJp PIOfli[( I I- 1)0311 + 'Q ( I + I)-IJ d e -jb plo[ 11 jfJ - hJ~ p. lll:l~- 0 P - % 111 P % % AI--2;,-- p(pi +lIi )i,! + p tan I p

(9)This can be expanded in a power series in (II/p) as foUows:

H: (p.O) AI ~-;;:0 [(~)(l + jflp) - (~)' (1 +~ flp) + --..J (10)

and correspondingly.

[H:(p.O)JAI !-o:::;l(~)' (1 + jflp)' - (~y (I + jfJp)(1+ 2jflpJ3) + ....J (II)

Ca> (b)

0'24020

FiR, 3. Tilt iltrrtllltnl of liltlurial bast rtsislnlt" as a funrlionof lilt radifls of lilt tarlh s.VSltm(al for t'QrioltS l'QllltS ('If N for Iwovalues 01 Ii alld (bl for various

vailUs of Ii for X = 00.

0·0.0'04oO'l4O'lO0'010'04o 0'12 0-1.

a/A

The self-impedance ~Z is then gi\'en by a series of integrals as follows:

fb lilf'"'~Z = -'\ [H; (p, 0)JI211'pdp + _'L~ e°'li'. (I + 2jfJp - (JIpl) p'ldp10 • 211' b

-~~ f~ e'2jfJp (I + 5jfJpJ3 - 2fIZpl,'3) p'$ dp + ... . (12)

The integration of p has been broken conn:nientl~' into ~wo ranges, from a to band b to 00, Thedistance b is chosen sufficiently large so that the series of intl'gTals convergl's rapidly. The integralwith limits a to b is integrated by graphical means, lJsing the exponential integral defined by,

E; (- 2jflb) = - f~ e- 2". p-I dp

the impedance ~Z is written

~Z = .!!.. fb [H: (p 0))1 2.".p dp + ,-~2fJ: [ I ( .. - + j) e'2"b - Ei (- 2jflb)J10

1 . ' 211' {3b '1.fJh

'1hlfJI (h2){[( I I ) . (7 fJb)] 2 .( .)} ,.,h

2fJ2 (ht

)- 211' hI 4iPbi + is + J i8jjb -9 r '21'~ + 9 fJlb lEI - 2JfJb + 211' x 0 ht

(13)

,I---H---+---+---r----+--+~

O's

'·sl---t-1I----J---1----L..---I---~

4,....--,r-'1r---r---r---r----r--~

_ z-st---I\---f---+---+--+---Hona:i-zt--r--f--ltt--1I---+---f---+---Hct."

""J

WIRELESS El'GIl'EER. MAY 1955

1·31

The computed values of JJR, for fixed aerialheights are plotted as a function of a/A for variousvalues of 8 and N in Figs. 2 to 4. The wire radiusto wavelength ratio, ciA, is taken to be 10-1 whichcorresponds to No.8 B. & S. wire at 183 kc/s. Thecurves in Figs. 2(a), 3(a) and 4(a) are for two fixedvalues of the ground conductivity parameter 8,whereas the curves in 2(b), 3(b) and 4(b), for aperfect ground screen N =00, show a wider range of8. With the results plotted in this form. values ofJJUt for intermediate \"alues of Nand 8 can beestimated quickly by interpolation. For purposes

and

When hl/bl « I, the terms containing higher powers of hllbl can be neglected.The increment of the input resistance JJR, for purposes of calculation is now written

JJR, == Real part of (JJZs + JJZ)

v'2fA pq FI (H I.P) ( 3Ir _ P+ q)== 2", 0 [pI + (p + g)'I1I1' cos 21/1 - 4",P +.. - tan I p tiP

.+ ~2J: P (HI. P) p-l cos (21/1 - 4",P + "'/4) tiP + v'2" PHil Y COS (8 + ",/4) ..

where F (HI' P) ejlj == "[1 +~iJ~P)~111 + j 2",P tan-I (H ./P),

- 24Orr1P PP == 12OIr8/v2, 9 = -W-loge-NC1' 8 == (c....la)lIl,

HI = hlA, P == pIA, A == a/A, B. == blA and CJ == cIA,

Vej· == 2 [tiJ(1 - ~b)e-j2.. - Ei(- 2ill/)] ..

Summarizing, this formula for JJR, should beaccurate to within a few per cent under therestrictions that HI < 0·1 (electrically shortaerial), cwla « 1 (negligible displacement current~ soil), and (BI/H1)1 == (b/h)1 == 25.

Presentation of ResultsIt hardly needs to be mentioned that the major

part of this work has to do "ith the e\'aluationof the integrals in equation (14). A graphicalprocedure was adopted employing a conventionalarea planimeter. The resulting values of theintegrals, so obtained, are believed to be accurateto within 1%.

(14)

(15a)

(ISb)

O'ZO ' O'Z4

Fig. 4. Thl' illrrl'II/I'III of'he al'rial bast rrsisllUlUas a Ilmciion of Ihe MdillSollh, tarlh 5>,s/l'lII (a) lorl'arious 1'Q11lts 01 N lor 1"'0valurs 01 6 allti (b)lor varimlS l'allltS 01 6

for N = co.

O'IZ 0'16

ai'll.0'01

(b)

0'04o0'010'.

7r--r-T-----.----r------,.------r--""T""'"'"'>(4)

61----Jj---+--+-----I>-.---+--+-I

o

s

-;;:;-z: 4:z:0-r; \

"'J , \

z

WIRELESS ENGINEER, MAY 1955 5

1-32

O'ZO0'16

I:ilt. 5. 1'1" inert"""'tlf tlrt base rtsistanu for.. quartn-W/JlIe uJlipole.

O'IZ

0./>..0'0'0'04o

that the radiation resistance 1<0 can be computedby cnnsid('ring onl~' the curr<'nt on th(' \'<'rticalportion of the aerial. Th(' contributions fllr thecurrents flowing on the loading 'umbrdla' or'cone' are usually negligible if th(' top-loadinghas been adjusted for maximum radiationresistance. This radiation resistance is ginn by U

Ro~ 160,,1 (hl>.)1It is possible now to apph' the aOOn r('sults toa('rials \\;thout top-Iuading h~' d('finin~ an'('lluh'al('nt' or 'effecth'(" h<,ight II, Thi~ ""hl<' isobtained by assuming that the loaded and

It is usually justified in these cases to assumethat the current distribution is constant from thebase of the aerial to its upper end. It has beenshown also by Brownl.I , Smebyl:, and othersl3

(b)

Fig. 6.

001 ~-4---+---+---+----+--~__ j...:::!l ~__.......::::iI

o·oosl.-_-'-__.L-_..J..._....L..__....L.._--l.._....J._--=:::.L._....:J• Z S I Z 1

10-4 10-' 10-1

'.OUIID COIIDUCTIVITY rT (mllo'/'Ile".)Tire /HJrflmdtr 8 as a fllllclioll of frtqllency ill kc/s alld

tround condllCtil:i'y.

O'ZO0"10'0'Q

.I---+-Jb-L.--~I_-__r--_+---+_l

• I---ft---I--~I_---'--_+---+_i

ll---+-

10,..---r,---.__---.,...----,--....,.---.,.--,

7t---t-~~.__-~i---t--_+---+_i

";;' It---t---'Ik-~--ii----'--_+---+_i

Ezo

- st-t--t---I-~IM~<J

of comparison, the values of iJR, for a lluarter­wave unipole (II = >'/4) are shown plotted in Fig.5(a) and 5(b). The data for these cun'es aretaken from a previous paper'.

(a)

O'IZ

~/'A

It is apparent immediately thatthe increase of iJU, with diminishingearth-screen radius is much more rapidfor the short aerials than for thequarter-wave unipole. This beha\'iour o'osr---+--..;::o",~-~~~o::-~~is connected with the fact that the 81induction and static fields of short 1~lb o·os ~-+--~I_-p...o::--~~-aerials are more significant than those ~ HZ 1---~---lI--I--,-~""""---l~~~~p."",,,""-i~"'C""'Ifor a higher aerial, such as a quarter- ~wave unipolc.

The appropriate valll<' uf 8 til use inconnection with these cun'es isobtained conveniently from Fig. 6,when the ground conductivity andthe frequency are specified.

The effect of changes in wire radiusis slight. This fact is illustrated inFig. 7 where AR, is shown plotted asa function of at>. for various values of the wireradius/wavelength ratio for a quart('r-wave unipnll·.

For earth screens which are small comparedwith the wavelength iJR, varies in a linearmanner with 8. That is, it varies directh' as thesquare root of the frequency and inverst'1~· as thesquare root of the ground conducti\·ity. Howev('r.for larger values of at>. as is illustrated in Fig. S,the values of AR, become somewhat less sensiti\"Cto changes in 8.

Up to this point. the discussion has been limit('lImainly to vertical aerials with ideal top-loading.

6 WIRELESS ENCINEER, MAY 1"'55

1-33

0'120'100'010'010'04

hIA_o'lcIA-lo-·N-IOO

4

Q:."

"::::J Z1---+--7t------::JIIC---I----±=--""1

.......; Jt-----r--'+-zo-

unloaded unipoles are electrically equh'alent if abo\'e mentioned approximate proccdure andtheir radiation resistances are equal. For thin are also shown in Fig. 9. The encircled points onunloaded unipoles the current distribution is the dotted curves are obtaint'd from calculationsapproximately sinusoidal and if its actual height carried out from e«luation (16) of refert'nce 6,is denoted by ho' the radiation resistance. R.. The points indicated by x for h>' = 0·2.:; canis given by be obtained either from equation (16) of reference

sin l fJho R o = ao (C + log 2fJho - Ci 2fJh.) + 15 (Si':;Jih~"~"2 si2.6h~j·~i~··2,6h~···'···················""'···'······

+ 15 (C + log fJh. - 2 Ci 2fJh. + Ci o4/Jh.) cos 2fJh. . • • • • • (16)where C = 0·5772. Si and Ci are the sine and cosine················································ ,....... . - _ .integral functions. Ro is just one-half of the self- 5

resistance of a thin, centre-dri'·en. aerial oflength 211 0 situated in free space·'. When 1,.'Ais small compared with unity, Ro ~ 40,,2 (ho'A)I.

Using th($e formulae to calculate the ('(Iuh'alentheight h of the unloaded aerials it follows thatfor h/A = 0'0'25. 0·050 and 0·10 h.IA = 0'050.0,095, and 0·175 respectively. In other 'Words.it is probable that the curves in Figs. 2.3. and 4for loaded aerials also apply to unloaded aerialsof heights O'050A, 0·095A and 0·175A respecti\·ely.The accuracy of this procedure can be checkt'd bycomparing the results with more exact pre\'iouscalculations for an unloaded unipole situatedover a circular. perfectly conducting, disc laid onthe ground. Employing the data in Figs. 2, 3and 4 the function iJR, for the loaded unipole andN =00 is plotted in Fig. 9 as a function of h'A forselected value!' of alA. The corresponding cun'es,for the unloaded unipole are obtained by the

Fig. 7. Tilt t/f,cl n/ chal/.lIl11g u'i" Tat/illS 0" lilt illc""",,'0/ in/'III "s;sla"ce IJRt.

FiR. tl. A" ill,ulrnliol/ I" sll"u' ht';l' AU, I'Qrics u'ilha,o,,"d cond"clivily alld /r(qll(IIc." .

/

10 r---r---.---"-T""--"T'""1

00~~OZ:-:-0·.l:-0):---::0.L:·OC:-5---:-0'':-1--"o~'z~

hI>. AND h. o/ >.

-­ttlEzo--

FiR.9. Th,so/i,/ r;("".,~ """5'''' <:;:I/JU, for all id,al·'-" /nad,d "ni/,ol,(Ifh,it"l". u'lI,r,·as III, dasll,delln"s eorr,sf'OndIn III, ,sl"nal,d"alll'~'t" an 'III·laminnl,d 1111;'

pnl, of 1I1'1~'" ".'I'll, '''dlCaltd,....""5 ar, l"oII,dlrn", mo". , Ifact

jorm,lia,.

O'Z4

8-((::) '/'_0'\

hlA - 0'25

N-Ioo.

0'04 0'01 O'IZ 0'16

alAo

,

•7

•.......~z:z0-Q:."

4"::::J

WIR£UlSS ENGIXEEIl, ;\1.,,' 1955 7

1-34

10,......--..---r---.....--...........

6 or directly from Fig. SIb). The good agreementbetween the two methods of calculation for /JR,is reassuring.

It is also instructive to plot /JR, as a function

OJ hi>" for both loaded and unloaded unipoles fora finite value of N. The results are shown inFig. 10 for N = lUO, 8 = 0'1, CI). = WI em·ploying data from Figs. 2, 3. 4 and 5. Tht'encircled points correspond to results, comnlllni·cated to us privately by Mr. G. D. Monteath of tht'B.B.C. for unloaded unipoles with heights of o· W,\and 0·167>". Again the agreement is ver~' lk"ltis­factory.

Hoth the curves in Fif.(s. 9 ilnd 10 illustrate tht'near linear log-log rdationship between iJR, amihi>". This beha\'iour is also pre\'alent for otll(.'rvalues of Nand 8 and pro...i~es a cOIl\'enientmeans of interpolating and extrapolating forvalues of hi). other than those shown in Figs.2 to 6.

ConclusionNo attempt has been made in this paper to

consider the economic factors but ratht'r tht'emphasis has been on showing the manner inwhich the impedance varies with the numbt'r andlength of radial wires. aerial height. and ~rollnd

conductivity. The curves should be useful in thedesign of earth systems for low-frequencytransmitting aerials. It would appear that man)"earth systems are probabl}' more extensh'e thannecessary since the benefits gained by employinglarge radius screens are lost if there are not a

sufficient number of radial conductors. This isparticularl)' so if the ground conductivity isrelatively high.

The calculations for the base input-resistanceare deri\'ed on the assumption of uniform currentdistribution along the vertical portion of theaerial. It has been indicated, however. that theresults are also applicable to aerials with non­uniform current distribution if the quantity h isregarded as an effective height.

AcknowledimentsThe need for the curves presented in this paper

was pointed out by Dr. T. W. Straker. Furtheradvice and suggestions were also given byDr. F. R. Abbott. Dr. W. J. Surtees. Mr. R S.Thain and Mr. J. S. Helrose. Appreciation mustalso be expressed to Mr. G. D. Monteath of theBritish Broadcasting Corporation who com­municated to us unpublished reports of hisresearch.

APPENDIX~Il'a~urcml'nts on a 250·ft umbrl'lIa top.load('d unipole

show good agrf'f'm~nt with thl' thl'Ory. Th~ follow in.:ob~l'n'l-d "alul's were supplied to U!i by 1\Ir. ){. S. Thainof this lahoraton':

l.c.-ngth of Radials, .... 81KI ft.~umhl'r of Hadials. X = 120Ground Conducth·it)". f1 ... 2·0 X 10- 1 mho!i/metreI'rN!u('ncy, 97 kc/sHadiation H«'!'i!itance 3!i calculat('d frolll fil'hl str('lIl:th

m('a..~ureml'nts, R. = 0·50 ohmInput rl'~istance m('a!iurl'd on a "ridgl', U, = 0·75 ohmOI)Sl'n·('() rt'Sistancl' incrl'm('nt, AR, = ()·25 ohmThlorl'tical "alue of resistancl' incr('mcnt (for

h;A -= 0·025•• /A = 0·(18. 6 = 0·07). AR, = 0·23 ohm.

REFERENCESI G. H. Bro-n. "Geaf"r~1 Cou,.idcrationl of l"o,,·rr Antl'unas for

IIrOoM...... l·...... hoc. huf. RUlo f. ..ps. 1935. Vol. 23. p. :111.• Ci. H. bro.-n. R. t~. 1....;1./. Erslt°in. ·'t'found S\·"'r...~ a".l I:,,('tor

in Anl"",n.. rftiri.....r)··. l'rflt . ..." HUltJ ,..",. •. 191:17, '"ul. :!~. I'. ;~~I.-.- N. Abht.It. ··I~i ftr lIuri,... M. I: .• ;,n",..1S'.trm.··.'·,tw.'"".

II..., tt5::. '· 411, I'. '1411.• .0\ nnrr .lad R. U. S,N·nl"f'. ··I~n."f't o••1 l i,.·uta, t~r(llll1t' l'l.uu- 011

.4nlrnn.. lC..di..'i....... } ••,.,.'.I·..... IlSU. '''1.21, r. I'M'I.• J. t·, StOffOf. "Impnbll('p 0' an ."ntelm;], O\'rr a Lire" lurul.tr

Srr........ } .•,.,.,. ""·s., .951 Vol. 72. p. '0511.• J. R" \'".ie and \\'..~. I'opf'. "CharactrrlSlic:s of ~I '-."fli, ~II .\nl"IIII.l

... ,th ~ kad,.. , londuUOT tilound S.,.·'t~m·· '-"}'"" ~o. ":!.~. \' ,S.A.~ .. I'UI,..ll""",,"lion. In"..nle RadIO Enllrs. Mar<:h 195. lin"" rnb·..'h.... i .. A,.,.,. SCI. Rrr.• Vol. U·••.

'Ci. V. ~IUI'.f'... lh. "Arrliraeioll of Ihf' l"omprn!fo.ltlOn Thf'('lrrm to(, rrUm f(.-.di",'ion and Pror-".tion I)rob~m"'t P'fK. /"1,,.,1'1".1-."('1.!!lSt. '·0198. I'.n 1\'. p. 23.

oW. j. SUfi........d j. R. W.i......n.........'e 01 a Top·lo.'d.... An'enn,01 .',hll'."· Unc'h 0'·... Cirrular l>round«l Sc:reen",) .•,.,.,. l'Ar<.,)I.. liS•. \·ol. 25. p. 553. .

• J" k. "·.,t. "lu,ped.,,(C" of an Anteonna O~.,., a Cir~"t~r (~r(\\",dSnl.n R i '.ab. Ie ' II·U·•••9S:l (l.lU. II · .k_ud. 'Io.ord I·ro' P411·95·!l5·111I.

I. c,;. (i. ~bct.ul.anf'. ·'Surf.act' In1pt"d3f1ct' 01 :11I1 In(mitt' \\'Ir,' (,.;,rld OItOb....u. Ana'os of 'nridm«". ). h .., .. ,IN,. E...... 19.~. ,.,,1. 93, ".orlII". p. 1523.

II U. L. (oIC"ntan. "Prnr.1.aliofl 0' F.1f'C'frOf1l,llt"rtac Ill!olurh;wr",alu..c • Thin "'ur In • Hor.IOI,lalh· Strollll;M Mt''Chum''. "lui . •\,.,.,1950'. \·..1 41, p. 276. .

II I.. l" Sntf'b,· "Sbort AnlfOnn.1 fh.u.1Ctf"ri'tlrs- Thf'Ol'('h~.1I··.I',ot. 1.. 01 1161.0 .....:-•••9.9. "01. 37. r. 11115.

II <.. L. Sn..'b and E. ~•. john!lOlI. "I'~rtnrlllanr~ of Shorl Antenn....I',ot. t .... H6I,o ....p •. It." Vol 35, p. 11126.

II S..,. Sr.....kunoll. "£I«.'On,.,n.'" W.vo.... \·.n SOI'r.nd, ItU.

',"'r III. r..-m' ".""'. G. I\.kof>.lf." J....,. of '·A\S,"~~. VIII. 32.r ~...s, hoi' arrhrd " \-"I.Juon.11 I('('hlllqu(' 10 obuln ~ lolulIOI\ _hlch...'"'" 1,.\ ..."abh· ••'h our t"'1U;\IIOn 03) anti th(' tonnula ot MOlllr"th''or Ibf' Idf'.al ClrcuU' around tefftn.

O·z

N-.oort--'l--t-+-t'---1 3- 0"

c/A_IO-t

~..Fig. 10. He '3,urves for tilelo.ded ""ipole."d tlte ,stim.ted'''rves for tile""loaded u"ipole.1" shown by thiisolid ."d d.shedli"es respectively.Tile er"ire/edpoi,,'s representfI.'ues ,ompllted

by Monte.th.

-;;;. zt--t-

!......

1-35

1-36

1.4 Wait, J. R., April 1956, "Effect of the Ground Screen on the Field Radiated from aMonopole," Institute ofRadio Engineers Transactions on Antennas and Propagation, Vol. AP-4.pp. 179-182

(Reprint No. 70)

Reprinted with the permission of the IRE (now IEEE)

1-37

Rt'""'td /,.", IRE TRA.""S."crIONSON ANTE.NI\'AS A,\'D rRorACATION

Volum, AP.... Number Z, April, J956

'.I:<lm 1:< TIn. l'.S.A.

Effect of the Ground Screen on the Field Radiatedfrom a Monopole *

J, R. WAITt

nate system (P. fI, z) the antenna is considered to becoincident with the positive z axis and the surface of thegroulld is defined by: =O. Uenoting If.(P, :) as the m:tl!­nelic field of the antenna in the absence of any groundscn'en and till.(P. z) as the change due to the presenceof the ground screen. it follo\\'s from an earlier paper.'that

where IJ ... 2.,,/),. ), .. ffloe space \\'a\'elength, " ... (jJwll1) II:

(;,/1 (surface impedance of the ground). jJ = 411' X IlI-T•

to) =anJ'(ular frequency. ",-1201r (intrinsic impedance offree space). , '"' tan-I zip. and where J I is the Besselhlllctilln of the first type. In the abm..e, lJ.- refers to thefield of the antenna over a Oat perfectly conductingI:roulld. This expression lor the fractional change of thelIIal!netic field is approximate and neglects terms ofhil:her order in ("/,,.). It also assumes that the attenua­tion of lhe I:fOund wa\'e hall a dependence with distance\\ hich is independent of the 5izt' of the ground screen.till.!J/. can he rt'J'(arded all lhe fractional rh:\nJ'(e of thet'fTr"li"r ht'iJ'(ht of the :\ntt'nn:\ due to lhl' prl'!'\l'nrl' 01 thegrounrl snet'n.

;\s "n example a quarter-wa\'e monopole antenna isconsidered with an assumed sinusoidal currcnt dislrihu­,j,,". In this case

SOME INTEREST has been shown recenth' in theradiation characteristics of an L.F, \'ertical an­tenna with a radial wire ground s}·stem. The in­

vestigations,- a have beeu primarily concerned with thcinput impedance at the terminals of the antenna. Thiswas shown to be mainly a function of the nnmhl'r :111(1length of radials and the J'(ronnd condncth·ity. It W;IS

assumed in most of this work that the ground w,l\-e fit'ldfor a gh'en current on the antenna was not appreriahlyaffected by changes in size of the I1round scrcen. Fnderthis assumption. the radiation el1iciency of the an!l:lIl1;!is determined mainly by the input resistilnce.

In an earlier paper.' an approximate method wa~

given which was suitable for estimating the depelldclll"eof the ground wave field on the size of the I:rOlllll1screen. It is the purpose of this note to show that quanti­tative results can be obtained which support our e.1rlicrcontention that the ground screen has only a small efTccton the ground wave field intensity for a specified currenton the antenna,

The ground screen is assumed to be a perfectly con­ducting disc of radius a lying on a homogeneous flatground of conductivity 11, Choosing a cylindrical coorcli·

• Manuscript recei,-ed b)' the pG!\r, Oecen.hcr IS, 1?55.t l':alio",,1 Rureau nl St"udarrl•. Rnulrler. Cnln., F. R. I\hhnll. "P,.i\:" nl buri,d R.F. I:roulld ')"1'111<.- I'.,'C.

IRE, ,·nl. 40. pp. 8~fl 851; JIII\·, 1952.1 J, R. \\'ail and W. A. Pope, "The characleri<ric. 01 " <err i,..1

anlenna wilh a radial cnnduclor ,round .),.tem," App. .'ici. RtJrtlrrh."01. R~, pp. 177-195; l\larch. 1954,

• I. R. Wail and W. 1\. Pope, "Input rt.i~tance nIl.. F. unif'nl,aerial.,· lVi,t/,JS Ell'" "01. .12, pp 131-138; M..... 1955

f · 1/ "(p' 0). ~. JI(fJp'cos6)p'dp'

,'_, IJ.-(Po :)(1)

1-38

180 IRE TRANSACTIOSS OS ANTENNAS AND PROPAG•• TIO,V

The integral, apparently. cannot be evaluated in c1ost.'dform. It is"not too difficult, however. to evaluate it by agraphical ~ethod. This has been done for the case 1:" 0which corresponds to the grollnd wave field. Letlin!:

-il IIIH .-(p, I) ~-- rill"'.'" - (." .1. ", (3)

211'p

It then follows that the fractional change of the lield isgiven by

2!~.Cllcu"!'".rnc! o! ~~~), ""'0'1"1""9'"

Ole '-_--'-....;;:::;;...••__.......__...-._~__,""__ __'o

110

Ilil

I"

~11M ,.--;

IIII

091

(2)

'I•

cos /I

cosC Sin/l)

. ~. rill'"·'+(~l4JlluIJ.(~p' cos /I)dp', (4)

-if11 -(p' 0) .. -- e-ilIl,·'+(~1tl'1., 2lfP'

bH.(p, I)

11.(p, I)

and since p»>.;

Fit:. I-Th~ r3(io of th" Iround "';1\'" liel" str"ns:ths or the ,,"Iem,,'(5) ..·jlh and "'ithout thl! lround ICl'ftn for" sp~ci/ird antenn" cllrrenl.

Some numerical values of X. nnd X, are Riven in thefollowing Table I.

Since & has already been considered 5mall, these areKiven adequatel)' by

.t ~ I + ax.

with &... 1'1/'1.1. the integrals to consider are

and

f'''/~

XI". sin (p' + 11"/4)1/1 - 1I'/4)J.(pjdp. (i)

where the amplitude ratio A is !:iven by

.A ... ((1 + ax.)! + (oX,pJII!

and the phase factor is 4> given by

ax,4> .. tan-' --- .

I+H.

(9)

( Ill)

TABLI' I and

..~ ax, (radians).

The factor &can be obtained conveniently from the fol­lowing relation

a - 0.00;5 (f..../tt)I/'

whereJ- is the frequency in mc and IT is the ground con­ductivity in mbos/meter. (Since displacement currentsin the ground have been neglected throughout, the for­mulas are valid only when &:«1.) Usin!: (9), the ampli­tude ratio A is ,ho,,-n plotted in Fi!:. I as a function of21raf)., the circumference of the I:round screen in wave­len!:lhs, for typical valucs of &.

It is interestinJ: to note that for small Kreens theJ:rollnd wave field stren!:th is :lctu:llly slil:htly less th:lnit would be in the absence of the screen. As the screenbecomes larger, the amplitude ratio increases somewhat.bllt is still only :l fe'" per e'cnt J:realer than unity for ale~s Ih:lll a wavelength. Thinking in terms of the rccipro-

(Rl

2rrtJ/'A XI X,

0.0 0.000 0.000O.S -O.OU o.o~o

1.0 -0.130 O.IPII.S -0.211 0.-1112.0 -0.209 0.1002.S -0.102 09H3.0 O.OU 1.0933.5 0.155 \.1314.0 0.111 \.1334.5 0.11.1 1.li85.0 0.050 1.3005.5 0.050 \.4086.0 O. \19 1.6126.5 0.205 1.614

The ratio of the field with the !:round screen to I hefield H.(p, s) without the ground screen is then gi\'en by

bll.(p, :~1+----- .. A~~11.(", :)

1-39

1!I1i/i Wait: Effed of 'he Ground S~un on 'h~ Field &dialed from a Jlonopole JHI

cal situation where the monopole is regarded as are·ceiving antenna and the transmitter is located at somedistant point on the surface of the ground. the increaseof A above unity is characteristic of a ·recovery" ellect.Such a phenomenon occurs in ground wave propagationfrom land to sea.·

Another interesting feature of the (:urves ill Fig. J isthat the minimum values of A occur approximatelywhere the input resistance of the antenna is a minimnm.It can be generally concluded. however. that the de­pendence of the I..F. ground wave field strength Oil the

• G. Millington. -Ground "'3V~ propagation ov~r an inhomo­leneou••rnooth nrth.· P,tIC.IE.E. vol. 96. pp. 5~; January. 19-19.

size of the ground. screen is of minor significance com·pared to the dependence of the input resistance 011 thesize of the screen. It should be mentioned that Page andMonteath' have used a similar method to calculate theradiation pattern of a vertical antenna over an irregularground plane.

ACKNOWLEDGMENT

I would like to thank W. A. Pope who carefully car­ried out the graphical integration of Xl and X,.

• II. Pace allli G. O. M....teath. "The vrrtic..lI radial ion p.111f'rll'of ntNium''''a\"r broradc:iulinc aerials.· P,tIC. lEE.. vol. 102. pp. 279­297; May, 1955.

II1IIIII

1-40

II

1.5 Frood, D. G., and J. R. Wait, January 1956. "An Investigation of Slot Radiators inRectangular Metal Plates," Proceedings of the Institution ofElectrical Engineers. Vol. 103. pt. B.pp. 103-109

(Reprint No. 72)

Reprinted with the permission of the lEE

1-41

AN INVESTIGATION OF SLOT RADIATORS IN RECTANGULAR METAL PLATESBy D. G. FROOD, B.A., M.A., and J. R. WAIT. M.Sc., Ph.D.

Wrilten conlribution, on ~Pfrs pubhs"c~ .ilhoUI bcinl IUd .. __linp are"'.lIcd (0' coaaide,ahon _llh I "leW 10 pubhullon.

Mr. f,ood an4 Dr. Wail_ere (ormerly a. Ihe De(ence RcwatdtTdeaDmrnunialicMDEslabbsbmcnl. Ontl,io. Canida

Ms. frood i1 ia the [)cpltllm'nt or Theoreclw 'h,1fC:J. Uni ..enirr o( L.C't'J'OO'.Of. Wall it .. lb. Cen"al Propa,llIon Labontot'). NII"oeaJ Buteau of Slao4ard..

CoIOfldo. U.S.A.PROC£fOINGS I.E.E., VOL. IOJ, PART B, No.7, JANUARY 19~6 110) I

(3)

. (I)

SUMMARYThe radiation from slots cut in conducting surfaces of limited e."ent

is discussed. Equatorial plaoe pattcrns of an axial half-wa\c slot in arectangular melal plate are measured in the X band. Thc experimcntalfaults compare favourably with thc calculatcd patterns on thc a«ump­tion that the plate can be rcpr~n1cd by a thin elliptic cylindcr orribbon of infinite length. II is obser\cd thaI, if the Icngth of the platcis equal to or greater than its width, the pattcrn is within a fcw per ceotof the corresponding theoretical pattern for a plate of infinite length.

The admittance of the 5101 in Ihe plale was also measured and com­pared wilh the compuled conductance. The agreement is seen to bequite good.

LIST OF PRINCIPAL SYMBOLS(p, t/J, z) - Radial. azimuthal and axial co-ordinalcs of a

cylindrical co-ordinale SYSlcm.V = Voltage across the centre of the slol.

'10 Intrinsic impedance of free space (= 120 ohms).Ie 2Tr/A., - Distance from observer to centre of slot.'. - Distance from observer to lower end of slot.'2 Distance from observer to upper end of slot.w Angular frequency.8 - Polar angle"" arc tan zip·

S(8) = H-plane pattern of slot on infinite sheel.Y "" Self-admittance at centre of slot.Z ... Self-impedance of slot (= 11 Y).

cp, cJ>, z) - Cylindrical co-ordinatcs centred at ODe edge of sheetfor principal E-plane.

CP', cJ>', z) - Cylindrical co-ordinates cenlred at second edge ofsheet for principal E-plane.

F(s) ... Fresnel-type integral with upper limit s definedin eqD. (6).

SI' 52 "" Limits for the integral F(s).H o = Axial magnetic ficld of incident plane waye.K .. Faclor of proportionalily dcpending 00 the geo­

melry of the slol.v ... Voltage induced al the centre of the slot by the

incident waye.,(/cd) - Shunt conductance of the slot in the wayeguide

divided by the characteristic admittance of thewaveguide (= fUDction of kd).

Q, b .. Inner dimensions of the broad and narrow faces ofthe waycguidc.

-\ ... Effective wavelength in guide.

(I) INTRODUCTION

Slot anteMae are becoming very extensively employed inmicrowave radiating systems. Their his lOry of development hasbeen rapid. It is difficulr to say exactly when they were invented,but certainly the conlributions from Warson· and Booker2 andtheir collaborators were of major importance.

It is surprising thaI lillIe allention has been paid by previousworkers to the effect of cutting the slot in metal surf3ces of finileelllent. While Ihe de:veloped design procedures assumed thaI thee:'lterior surf3ce was infinile in extcnt. the: need for a pr3cticalanlenna system limit~ the: ph)'sical ~i7e of Ihe: mclal surface onwhich the slols are cut. 51evenwn,J who de:veloped an cle(;anltheory for the radiation of resonant slols in a rectangular wave­guide. assumed for e:onvenience that the exterior region wasequivalent to a half space. He admits that the assumed infinitedimension of the face of the waveguide for the clIlerior problemis a severe limitalion to the validity of the CApressions for theconductance of the slol.

It is the purpose of the paper to investigate in some detail thesignificance of the finile CAtent of the metal surface or sheet onWhich the slot is cut.

f2) THEORETICAL DISCUSSIONThe: radiation pattern of a thin slot cut in an infinite plane

sheet of perfect conductivilY and vani~hing thickness can beobtaincd by an application of an e:lc:ctromagnetic Babinet'sprinciple.2 By using this technique. the well-known results. forthe thin-wire antenna can be transformed immediately to thecomplementary problem. For a half-wave slot oriellled in thez-direction of a (p, t/J. z) co-ordinate system and centre-fed by a\rollage V, the fields are given by

-jV (r:-1l.\ r:-Jl..,).,H =-- --- +-- £''''

• 2"'1)11'1 'z

jV [ £-1". r:-Il.,]H. = --- (z + ),/4)-- + (: - ),14)-- r:1""

2'"101' '. 'z·V

E. = :=..!-(r:-Jl" + r:-ib,)cJ..'2Trp

where r2. '. and, are the distances of any point P to the upper,centre and lower end, rcspeclh'ely, of the slot and where'?O ,. 120Tr ohms and k = 2Tr/(free-space wavelenglh). Eqns. (I)refer to the slot cut in an infinite sheet. When the point P issulrJciently far away from the slot the equations simplify some­what to

E. ~ j(2Tr,)-I5(8) exp (-jk, + jwl)lH. ~ (~JE. sin 8 (2)

H. ~ - (~JE.cos8

where 9 = arc tan (zip) and where terms containing ,-2, ,-J,elc.• have been neglected. The factor 5(8) can be defined asthe radiation panern of the slot and is given by

cos (~cos9)5(8) - V ~ .

sm9

Following the method of Carter' for the thin-wire antenna,

IIIIIIII

, -42

II

....•••• ...L---,,--_-If._-,._.1.'_'

Fie- %.-E·planc pallcm of the slot for "'.. '''':;lltely large sheet.

Fig. 3.-H.plane pallem of t¥ slot for an infinitely larcc shcct.

It would now be ellpectcd. in view of the sh;IJIC of the H·planepallem shown in Fig. 3, that the efTect of the finile value of Lwould not be pronounced. This is later borne out by e~JlCri.

menl. lIo"'1:"er. the finite value of cl can be ellpeeted to lead toa considerable impairment in the pallern, since the field in thebroad~ide directic'O of the SiN is very signilic;lI1t. It is possibleto tre:1l the problem of the slnt in the finite plate by an appro~i·

mate theoretical procedure, if the edges separated by the distanceZtl arc considered to dilTract the primary field of Ihe slot as ifthey were semi-infinite half.planes. The inleraclion between theedges is neglected, and on physical grounds this would seemjustified if 2d is somewhat 5,eater than the free·space wavelength.Again, this supposition is bome out by experiment and co.m.parison with a more rigorous treatment, such as that whichrepresents the plate by a thin elliptic cylinder.' ·' .

To facilitate the discussion it is desirable to conSider Ihe slotas a receiving element. The vollage induced in the thin slotfrom the wave incident on the plate is then proportional to thetangential magnetic field along the ~Iot. This stalement followsfrom Schelltunotr·s equivalence principle.4 From FIg. 4 a plane

, (4)[

/4 c~'

y= -v2H.(O,O,z)eirHczdz

-l/4

The integrations can be carried out to yield the result

Y = 2·06 + jO'97 millimhos

which corresponds to the case when the slot is allowed to radiateon both sides of the sheet. The centre impedance or the slot isthen

104 FROOD AND WAlT: AN INVESTIGAnON OF SLOT RADIATORS IN RECTANGULAR MEfAL I'LATES

the self-admittance Y at the centre of the slot can be obtained '00'

from

z = II Y = 365 - j212 ohms

It should be noted that Begobich6 recently gave the value362'5 + j210'5 ohms, which indicates reasonable agreementwith the above, except for the sign of the reactance. Apparentlyhe made an error in his derivation. The result derived by theauthors can be checked by comparing it with Booker's statementof Sabinet's principle, and connecting t~e impedance, Z, of theslot w '\I the impedance, Z", of the complementary wire antenna,such " ·t

ZZ.. = TJM4 = 360()Tr2

Using Carter's' value 73· 2 + j42' 5 for Z .. leads back to theauthors' expression for Z. Bc:govich erroneously employed thec:omplex co!\iugate of Z".

When the slot radiates only on one side of the sheet the con­ductance G would be one·half the real part of Y given abo\'e,and hence G .. 1·03 millimhos. The corresponding susceptanceis one-half the imaginary pan of Y plus th:: susceptance of thefeed system.

It should be noted that the pallern of the slot on the infinitesheet is essentially omni·directional in the azimuthal plane. Itis of interest to consider the etrect of truncating the sheet. Forexample, the slot is Cllt in the centre of a reclangular metal "lateof width 2d, and orienl~d parallel to the other sides of length L,

Fig. I.-Thin half·wa\e slot cut in the centre of a RCtanl\llar melalsheet and co-ordinate system.

Fig. 4.-End view o~ the sheet showing a plane "'ave incident frombelow.

as illustrated in Fig. I. The slot is fed by a waveguide such thatradiation takes place only on one side of the plale.

The ideal £-plane pallem in the equalorial plane. correspond·ing to a value of d very much greater than the wavelength. wouldbe as shown in Fig. 2, where the plane of the sheet correspondsto cJ> - 0·. In other words, the sheet is assumed to be suffi·ciently wide that difTraction around and by the edges isnegligible. Of course, if the slot were allowed to radiate 00

both sides of the sheet the pallem would be a complete circle.The corresponding H-plane pallem for the half-wave Slol issimply the function S(8) shown in Fig. 3, where 8 - O· is in theplane of the sheet.

wave is shown to be incidenl from below the thin conductiogplale of width 2d. The incident wave is polarized such that themagnetic vector Ho is parallel to both the slot and the edges.Clloosing a polar co-ordinale system centred at the left·handedge of Fig. 4, the field H. at P of co-ordinatcs (p, cJ>, z) due todifTraction around this edge is Jiven by the classical Sommerfeldformula'

H• .. Ho(t:"J··COI(~-.oIF(sl)+ rJlaCOI(---.IF(silJ . (5)

where clio is the angle the incident waves makes with the planeof the sheet. where

1-43

t

'0., C

&Z1IIUTM. ".t

...o

'/-'-7"---+---'" --- - --­~tI

-/0

Fill. 7.-Behaviour of Ihe £.planc pallem ror dircclions" above andbelow lhe plane or the sheel.

V.lue 01 '"' • iftdicaled on 11M' cuncs.".a.-<,_ ..._Ied.

~

.. -41---+---+---~~<:""4----+-..~

Fla. 6.-Principal £·planc paltern of a slot in a shccl of widlh 2dsymmelrical aboul 90'. The plane of the sheet corresponds to 0'.

id- 141, -to'... -,.,.

".a.-<'u... II _1Cd.

4r--~r-----,r-----,r-----,r----r-----.,

0·1---i-

which apparently docs nol accounl properly fur the energydiffracted around the edges. The calculations were also carriedout for kd = 94 and 234, and these are shown ploued in Fig. 7in the inleresling transilion reBion for angles within IS' aboveand below Ihe plane or the sheet. II is noted thaI the field OIl Ihe

FIC. '.-SChemalic or Ihe "·3vecuidc·fed slol mounted on lhe lurntableror lhe pallcm meaSUf'ffl1CnIS.

• (6)

FROOD AND WAIT: AN INVESTICATION OF SLOT RADIATORS IN RECTANCl!LAR METAL PLATES 10'

F(s) - aJft/4w- I /1J~-J"dx- ..

Sl .,. (2kp)l/2 sin (I> ~~O)

s2 = - (2kp)1I2 sin (~.~ cliO)

and

Similarly the field H; at P due to diffraction around the olheror righI-hand edge is idenlical in iarm 10 eqn. (S) if p, <I' and flloare replaced by p', cII'and tlIo.and H ois replaced by Hoe-J2Mcos •••

The vollage 11 induced in the centrally localed slot 00 the sheetis then given by

v -= K(H, + H;) for r' ~ r = d and <I> - <1>' -= 0

where K is a conSlant which depends on the dimensions of theslot. It then follows thaI

Ivl -= KI F[-(2kd)I/2sin(tlIo/2>] + F[ -(2kd) 112 cos (<I>o/2>J1

• (7)

which is applicable in Ihe range clio = 0'-180".When the incidenl wave is incident on the upper side of the

sheel, sleps muSI be taken to combine Ihe fields. as delerminedby the two knife-cdge problems, in Ihe proJ'Cr manner. It isconvenient to choose a slighlly differenl co-ordinale syslem asshown in Fig. S, where Ihe angular co·ordinalcs <I> and <1>0 arenow measured from the bOllom of Ihe sheel.

Fig. 5.-End view of the sheet showing a plane wave incidenl fromabo'·e.

The magnelic field H, OIl P, regarding the shccl as a semi­infinile half·plane wilh the edge OIl (0, 0, z). is given by eqn. (S).and a similar expression is oblained for H;. The voltage "induced in the slol is Ihen given approximalely by

11 -= K[H, + H; - HoeikdCOSO.] • (8)

The lerm Hoejkdcos4>. is Ihe primary field of Ihe incidenl waveal Ihe slol and it musl be sublracled from H. + H;. since theprimary field is included bolh in H, and II;.

The slol vohage for the wave incidenl on the upper side of thesheel is Ihen given by

Ivl ,.. KI F[(2kd)l/2 sin (1110/2)] + F[(2kti)l/2 cos (<1>0/2)]- II

. (9)for ~o in Ihe range 0'-180'.

As a numerical e:'lample. Ihe £.plane pallern for a Ihin 5101cuI in Ihe axial dircclion on a melal sheel of widlh 2d is com·pUled using eCJns. (7) and (9) for kd ~ 141. The pallem plolledin Fig. 6 is normalized in Ihe broadside dircclion 10 OdB. II isinlercsling 10 nOle Ihat the field in Ihe direclion tangenlial tothe shccl is 6dB below the maximum value. This is a charac·terislic which holds consislenlly for all sheet widlhs grcaler Ihana few wavelenglhs. It is wOrlh menlioning Ihal Booker has (livena value of 3dB ralher Ihan 6dB for Ihe redUClion of Ihe fICldalong Ihe shcct. His argumenl is based on physical grounds,

1-44

106 nOOD AND WAlT: AN INVESTIGATION OF SLOT RADIATORS IN RECTANGULAR METAL PLATES

(61 lui - •• l. - Id '41 boJ - 10. l. • Id

Fla. '.-Principal £·planc pallems ora thin halr._ve slot in the ccnlR or. rec:tanlUlar ShCCI ur Icnllh L and width 211.

-- EAPI'ri_laL---- nw-tital.

• - 90".... - to·

......·7n_ .. X-.

rear of the sheet diminishes very slowly with increasing valuesof 2d. In fac:t. it is easy to show from the asymplotic: behaviourof the integral F(s), that the amplitudes of the envelopes of theback lobes are approltimately inversely proportional to ,jU.

The above approximate method for calc:ulating patterns ofslotsin metal sheets of finite width cannot be expec:ted to yield reliableresults ""hen 2d is of the order of a wavelength or less. A morerigorous approach is to represent the sheet by a thin eUipliccylinder of vanishing minor axis. Computations based on thismodel have been c:arried out preViously' for values of led rangingfrom 2 to 8. It was shown thaI the double knife~dgeapproxima­tion is accurate to within a few decibels if ltd is IJUter thanabout 6.

(3) EXPERIMENTAL PATTERNSIn this Seclion some experimental patterns will be compared

. with the patterns computed both from the double knif~ge

technique and from the elliptic<ylinder method.The experimental work was carried out on an antennll nnge

at a wavelen.th of 3·2c;m. A narrow half-wave sial was CUt inthe broad face of an X-band waveguide illustrated in fil. 8,with the centre of tbe 5101 approltimately tbree-quaners of a

JUide wavelength from the short-circuited end. The slot wasparallel to the axis of the .uide and was offset approximately0·1 in from the centre of the broad face of the guide. A numberof thin rectangular lIluminium pilites were prepared with ledvaryinl from 4 to 141 \\ith lenlth/width ratios from ~ to 20.These plates were desilned to be easily mounted on the broadface of the guide. The waveguide assembly was then mountedon a suiUlble turntable and illuminated by a transmitting dishantenna which was located at a distance of 100ft. The radiationwas incident normally to the axis of the guide and was hori­zontally polarized. The outpUt from the waveguide was detecledand amplified and ils qrying qlues were planed on an inkReorder which had a logarithmic scale.

Measured patterns are shown in FilS. 9(a}-9(h) for led '" 4. 6,8. 10, 12, 16. 28 and 141, for the case where the slot is centrallylocated in the plate. Theoretical data are also shown for com­parison in some of the curves. The theoretical curve for thecase led - 141 is shown in Fig. 6. The vertical sc:ale is arbitrary,Ind for the sake of convenience the experimental and calculatedc:urves Ire matched in the direction of the maximum field. Theelliplic<ylinder method of cakulation was employed for ledvalues of up to 10 usin8 eqn. (8) or Refereac:e 8 with 8 - 90",

1-45

FROOD AND WAIT: AN INVESTIGATION OF SLOT RADIATORS IN RECtANGULAR MF.TAL PLATES 107

9r

!•

(.) 1<tl - 12. L - U

AZIMUTH••••

soZ5

150 ZIO Z70 SSO

so15

&Z'"UTII....

Fie- 9--t:oll"IIurd

­..Ii! I'~5 0..•

(&, 1<tl - 141. L - un

and "0- the anlUlar co-ordinate of the slol. equal to 90°. Forlarger values of Jed the double knife-edge technique was used tooblain the theoretical curves. It should be noted that theazimuth scale of these filUres are chosen so that the broadsidedirection from the slot corresponds to 0·.

The agreement between experiment and theory improves forthe larger sheets. The probable reason for the discrepancy forsmall sheets is that the diffraction by the waveguide behind thesheet is becoming significanl. For Jed co 4 the plate is onlyaboul SOy. wider than the broad face of the lUide. and il istherefore not surprising thaI the pallero differs from thaI cal­culaled on the basis of a thin elliptic cylinder. 1\ is also inleresl­ing to observe that the experimental pallem for Jed ~ 4 issomewhat asymmetrical; this is due. no doubl. to the facI Ihalwhile the slol is cenlrally localed in the piaIe. the guide is diS­placed slighlly. The asymmetry is also seen to occ:ur for Jed - 6.bUI to a lesser extenl.

To iIIuslrale the effect of the length of the plale. pallems wererecorded for L - d. 4d and 2Od. keeping Jed constant al 6. Thesimilarity between these curves. shown in Fig. 10. is striking.and substantiates the earlier supposition that the azimuthalpallems are detennined mainly by Ihe lateral dimension (i.e. 2d)of the plate for an axial slol. In facl. it can be seen from thecurves in Fig. 10 that the piaIe can be regarded as infinile inlenllh so long as L is Ireater than about 4d. The main effect

or finile length seems to be an increase in the level or theback lobes.

In all the above-mentioned experiments the slot was situatedin the cenlre of the plate. If 1M slot was displaced towardone edge by an amount d/2. the angular elliptic co-ordinate

aZIMUTH....,

son

fil. IO.-Principal £.plane pallem for lhe shecl of various Icnlths wilhconslant widlh.

Pon", .......... -I.. -...... -... L -",~.zo,r

1-46

o

~I

oG

••,

0'1-,--t--i~--::~..-t,-~e---jr----;:,

thcorrlic:al

,.

.,r-------------------,

Fla. 13.~parison bclWftn the experimental andnormalized conductance.

1N.'.-TIlo.-.iuI.-.rI«_II.I·20.1--E.pr,.........----T-.

,- I• I

" 00 ~; I• I.. ,

I01-

,).

Fie. t 2.-Mcasurrll norm:llimJ conllll\,'1ancc and susecptanc:e or lheslot :I, a func:lion of the widlh of lhe sheet.

Slot _ h - AinSkM 'h -lin

1- J·ZOan,A•• 4·......

By 3dapting Stevemon's theory) for longitudinal slots nearrnon:Jnce, cut in the narrow facc of the wavclUide, it is C:Jsy toshow th:Jt the normalized conductance ,(kd), as a function of kd,in terms of the actual conductance G al the eentre of thc slot,i\ Ii\-en by

480 Q A, • 'Il'A 1·03,(lid) ~ iii h 'i C:~- 2'\" - G' . (10)

Ste"enson's formula c:orlC5Jl(1nd~ to eqn. (10) when G is repl3ccdby 1'03; this corresponds to the casc of 3n infinite pl31e(i.e. I,,: _ co). Employin, the theorelical results for G, men­tioned abovc, the value of ,(lrd) - ,(co) Jivcn by eqn. (10) isploued in Fig. 13 using Q .. 0-90in, b - O'40in, A .. 3'2em,-\ = 4· 48 em, along with the corresponding experimental curve.

There is rcason3ble aarccmenl between the cxperimenlal and the

110 110 ZsoIS

101 ROOD AND WAIT: AN INVfSTlCATION OF SLOT RADIATORS IN RECTANCULAR METAL PLATES

As is c:usl0fIW')' in _wquide measuremenlS. the Ioqitudinalslot is represenled by an equivalent shunt admiltance across lheequivalenl lnnsmission lioe of the _vclUide. EmpIoyin, aslolled line in conjunction wilh an adjustable .hort-eimJiltcnninalion in the paidc beyond the 5101, the conduc:tane:e andsusceplance are dctmnincd usin, a standard technique.lo Thesevalues are normalized by dividin, by the c:haracleriltic admittanceof the ,viele, which is real. 1bc normalized conductancedenoted by I, and the norm:alizcd SUSC%ptanc:c denoted by b.,are plotted in Fi,. 121S a func:tion of kd.

becomes Vo - 60°. The pallcm lhen became quile asymmelricalIS shown in Fig. II. This Iype of as)'mmelry was also presenlin the theoretical comput3tions of Ihe offsct !dot or the Ihinelliptic cylinder.' The olhcr curves in Fig. II corresP<lnd 10 thepallems for Ihe slot covered with 13)'e",", of pl3Stic eleclric:1Itape of Ihickness 7· Smils. The number of 13)'ers is indiC3tedby the value T. It is interesting th3t the 3zimuthOli p311erns arcessenlially unch3nged in Sh3pe, which is in 3ccord wilh theoryfor a thin 5101. II is nOI possiblc 10 dr3w 3ny further conclusionsfrom this sct of curves, since it W3S not feasible at the lime 10mc3Sure the change of the v01t3gc: sl3nding.wave r3tio in thelUide for the difTerent thicknesses of the dielectric co\'Cring.Furthermore, the dielectric properties of the plastic tape arcDOtlaJOwn.

FIa- IJ.-Princ:iflal F.-plane paUern, for Ihe ,101 di,pl:lI:ed loward lheri,hl-hand cd,c by an amounl d/2. (The values of T correspond10 the number of layers of lape coverin,lhe slOI.)

U -,4.' - 90". Y. - w.

(4) MEASUREMEl'I.'T OF SLOT ADMITTA:"CEAnother important Ch3r3cteristic of a slot cut in a met:J1

surface is its admillance. In the e:Jrlier part of the paper it \\'35shown Ihal Ihe admillance at the cenlre (If a thin h:lIf.wavc slorcuI in an infinilely Ihin conducting sheel of infinile e:"enl. w.2·06 + jO·97millimhos. If the slot is fed by a waveguide,which is localed on one side of the sheel so that it radi3tes onlyinlo one of the half·spaces, the conductance at the centre of lheslot is I ·03 millimhos and Ihe susceplance, IS mentioncd earlier,is dependent on the nature of the evanescent structure of the fieldwithin the guide.

When the sheet is of finite size the conductancc is no 10nICr1·03millimhos. The variation of G with Ihe width of the sheetWIS investig:Jled theoretic:Jlly by using a model of a thin :nialhOllf·wave slot at the centre of the broad facc of a Ihin ellipticcylinder.' G was obtained C'tplicitly by computing the powerradiated from the slot for a specified voltaIC at the ccntre of theslot. It was shown that G was an oscillating function of thewidth 2d, and it approached I ·03 millimhos IS 2d approachedinfinity.

It is now worth while to examine the admittance using anexperimenlal procedure. The 5101 of width 1/16in 3nd !eolth5/8 in was cut parallel 10 the narrow face of the X-band wave­lUide. Means were then taken to mount a scries of plates ofvarious widlhs flush with the narrow f3ee of the wavelUide in asimilar manner 10 thaI employed for the paltern me3Suremcntsof the slots. The slot was cut in Ihe narrow facc ralher thOlnIhe broad face. so lhOlt the effect of plates of small width couldbe cumined also.

1-47

(7) REFERENCES(I) WATSON, W. H.: "Wave Guide Transmission and Antenna

Systems" (Oxford University Press.. 1947), Ch:Ipter 6.(2) IklOKEll, H. G.: "Slot Aerials and their Relation to Com·

plementary Wire Aerials," Jounuzl 1.£.£., 1946, 93.Part iliA, p. 620.

(3) Sn\'E!'o'SON. A. F.: "Theory or Slots in RCCI:tntwlaf WaycGuides:' Jour,",1 "1 Applktl PII)'sirs, 1948. 19. p. 24.

(4) ScHELItUNOFf, S. A.: "Elcctromal/lCtic Wayes" (VanNostrand. 1943).

(5) CAllnll, P. S.: "Circuit Relatiom in Radiating Systems andApplicacions /0 An/enn.., Problems," Pf//("(wlinrs 0/ tIlt:Inslilutt' 01 Rml"w &gin"t'fs. 1932. 20. p, 1004,

(6) BEGOVICH, N. A.: "Slot Radiators:' ibiJ., 1950.38, p. 803.(7) WAlr, J. R.: "Field proouced by an Arbitrary SIC'! on an

Elliptic Cylinder," JoufttOl "1 Appli"tI Physirs, ."5, 26,p.451l.

(8) WAil, J. R., and WALPUU. R. E.: "Calculated RadiationCharacteristics of Slots in Metal Sheets," C_dilUlJOluttOlolT,clrnDlOIY, 1955,33. p. 211.

(9) SoMMEllfELD, A. N.: "VorJcsunpm ubcr theoretischePhysik," VoL 4. Optik, Wicsbaden, Dieterich Verla..1950.

(10) SILVEIl, S.: "Microwave Antenna Theory lind Design,"Vol. 12, M.I.T. Radialion Laboratory Series" (McGraw­HiII,I949).

(6) ACKNOWLEDCMENTSWe wish to thank Mr, R. G. Sinclair Who ably assisted with the

pattern measurements and Messrs. R. E. Walpole and W. A.Pope who carried out most of the computations. Valuableadvice was also received from Dr. A. W. Adey in connectionwith the admittane:e measurements.

(5) CONCLUSIONS11 has been demonstrated that the radiation characteristics of

an axial half-wave slot in a rectangular metal plate are mainly afunction of the width, rather than the len,th, of the plate. Themeasured pattern and radiation conductance of the slot acreedquite closely with theory over a wide ranae of plate widths.The experimental phase of this project i5 continuin8, and par·ticular attention will be paid to the clfcet of the fillite widthand depth of the slot. It is also hoped to examine the effectof curvature of the metal plate in which the slot is cut.

FRooD AND WAIT: AN INVESTIGATION OF SLOT RADIATORS IN RECTANGULAR Mt.7AL PLAnS 109

computed curves. The disa8reement for the smaller values of kdcan probably be accounted for by the fact that the theory does

, not account for the dilfraction by the wavc8uide behind the plate.II should also be mentioned that the theoretical value of "which is found usin8 Stevenson's procedure. assumes that thesusceptane:e is much smaller than the conductance (i.e. the slotis near resonance). It can be seen from the experimental resultsthat. althoup b" is small, it is not ne81iaible and would, nodoubt, also be a source of discrepancy.

1·48

1.6 Wait, J. R., and D. G. Frood, OCtober 1956, "The Radiation Patterns and Conductancesof Slots Cut in RectangUlar Metal Plates, Proceedings oftile Institute ofRadio Engineers, Vol. 44,p. 1467

(Reprint No. 85)

Reprinted with the permission of the IRE (now IEEE)

1·49

R".;"".! l'u,," fir' PROl-t:EDI:-;C;S OF' TilE J R E.,"OL. ~~.I'O. 10. OCTOBER. IO~6 PJ' 1467

'.,~ llf '1tll: 1:.5.A.

'J. It \\ ••, ."••, M f-' \\',.11'"'-" -'·al."t..,.., I "-'.1lin" rh.s••"'f''''''''. trl 1 In ."", ,I .. !t ·I •• I'"t1. Col•. J 1n\ ..., " "" In·',~ ~\ .. r '0$' ;''In.',.'1 II won' U "" c.n-7n: '_n''''''" '~'"• n ~ F'''"'''.rtd f It \\'alt -1\" '''\'''_1'•.111'''' nf...,. ,,,,11:,,,,,. ,n , ... ,."cllia' ull"lo1Itt''l .. "'ot '".'" Iltt. Itfl. ,nJ-ltn; ., ". '''~fl

JOil. ,1

II ..."s fOlln,1 lhat the IIIC"""fl'" "11l1•·..lelllale" .101 C.."dIlCI""~, "s ... 1"".li.."..I "lal" "i,lth. :l1I......d "rr~' dn.rl) •." "I"Ih:lt the nK"':'\"lIr'-11 C1"'lhlt·t:ltlt-l.~ '\t'Il' ;aJ·...a~·, ahoul III per ,..."t lo..·('r Ih"" Ih,..,,,·, ;,",,1Olles. This di...r"p.,nC')· nt... ~· ha,'" bc'cII d,,~

to Iht' f"et lhal t1,..or~· "",nir lh" ~h..1'"nllih (II :u..! .1"1 "Nllh Cu,) 1.. 1 'Ill'" ,k.,

CONDUCTANCE

1,,\:,. (;~ ) » I •

Clearl)', thi- ....... Iitinn i. ,"cry h"rd I" I"Ihllin praclice.

Aftt'r Ih,' I'r(',\I<:II;un nf Ihi. 1\I,,,,r,lurlht'r reoc:lrch w,,, d,..'C .." Iht' ",..,h'MI 01"'f'llSU,inllslnl prol'l'rtin. h ...:,. fou"d Ih... 1

lht' lcchniqlll.' uocd abo,'" (lhe qn"r1 ..rw:I'·"nlrthnd) i" nnt. in CI"nt"'r:t1. :u s:nucl ..,,, :I Itt·"

",«'thuet "..hid, i" iudC'l'cndl"nl f'r Ihe ,'''",rand any refleclio". froon the probe ill lht'sloll..d line. Til.. nsenti:ll' of this work willbe publishl!d shortly. The full ntathem:ltir~1

and """rimt'ntal d..tail. of the ft"l "1,,tranlen"" di!'CII.....1 here h.,,·e lJe.o" ~i\'l·n.'·'

}. R. \\'.~n.

Cenlral Radio Propal!"tion Lab..N"';OII3I Bureau of SI:lnd:lrd•.

Boulder. Colo.

D.G. F.",,,'.rl...,ic' Ikpl ..

Ullin",ily or Lin·rpool.Lh'erpool, F.nlll:lnd.

ApI,I)'illl: silllple ph)·~i,...1 lUIl"""''''-.su~h a. were uoro to prroict the radi."i.."patterlls, it an be .hown that the cOlld"~I·

alice Itl of a rcso'l3"t slot ill a llal Illt'l"IpLllC i. " <1"111"'-.1 oscill"linll f""(fi ..,, ••1I.lal" ...i.lth ( ...) "'ith " 1,,'';••1..I "I ••", ..,,,.frt'(' .,\"T ",;","·I"nl:,h, ~.. (Fi~..11.

"00 1ne-....llrM lad~'li'M' .... ll<:r". ,,~,~.... I\"ry clllIel,.. but for lIarrow pI"I... Ih.·..·-as Sl)me al~·lIunctr~· in the pattern!J dm~ tndifT""'Iill" lOfT'octs caullC:d by Ih" .. a,...~"itl.·behind Ihe plale. For widlh. I:..•... 'er lit .•"two or lhr"" lrec sp.1\·" ".1,·"Ie"II,h., Ihi­efreet ..... ""IliCible, hu...e\'er.

".'

--

il~i:.

o'

l::'rcrimrnl"I"· it ..... 10lln,I Ih~l Ihe",.Ii.llion p.,tlrrn .>/ a "ble of ~;\'en "iclth(111 "'3. indere"denl "I ill Ienglh (LI il:

/. '11" i= I.

fur ."IIi,;enl"· I..nt I.'~te- lhe r..te"laled

F'I J-,-'.Im"th or .!."I."P fadlallOft p.lU"·M 'Otrtl.l••• i-tlh, wo. 01 at-tull. t, IS a•• In .,."r.."eth

"Is I

I'·~:~.:~.:.-:,:,::,_.~ ·"~lJ/.·-·-~ _-._-.- ,/.:..1 ~/' ..

j -. f'l~

the slot (whose axis is perpendicnlar to Iheplane of the paper) and let C repr"""nl thewa\'quide feeding po"'er to the slot. \'e~'

far aW3l' frOm the wavquide and pIale isa receivinR antenm P.

When the slot is pointing to"'a,,' P.i .t.•in the O· direction. a large field slrenglhresults. bUI if the slot points away from /'(in the • SO· direction) Ihe field strength issmall. its ..ctual value beinl: delermined bythe fraction of the tOlal power r:U.i.lIrowhich is difTraclro behind Ihc shet't to r.In the inlermcdi:lle dirceli"n. 90· and 2iO·the field stren,lh al P would lie bel..-et'nthe values obtaintd for Ihe O· and 180·direclions. Thus. in Ihe "b"""", nf f"rlht'rcomplications. the ra.Ii.,lion ""lIt'rn 01 Ihemel,,1 shet'l 3nlt'nn" ,.,,,,Id rc""",l"le lloed3.hed line of FiR. 2("1. ACIu.,lIy ..nc: i",·portant fUlure has been overlooked in Ihisdevelopment.

h will be appreci"lro Ih31 Ihe edllesE. lIn.1 E. of the met:ll shet'r foml a dis·continuilY tn the field t"weli,,!! oUlw"rdfrom the slot and alonR Ihe sheel. R:ldi,,'inCline sources will. therefore. be inducednear the edlles E, anti Eo of the sheet andtheooe will come in an.1 o,.t of ph"oe ...ilhe3rh Olht'f a' the anlenna is rol:lll!d. ,.10",.insle"d of Ihe radialion pallern looking likethe dashl!d line of Fill. 2(al. it "'ill resemblethe full line of the same firure. The numberof inlerference fringes in a complele rOI"lion01 the plate will be grealer for "'ide sheel,than for narrow ones si"ce a snl3l1er incre·menl,,1 rnlalion is required 10 form a gi"enpalh diflerence for a "'ide .heel. The ampli·lude of Ihe fringes ...ill be .111.,11.., for "'ide.hftt1 than for nnrrnw ont'S. hn"oeo' ~r••incethe strenl:lh of the induced li"e !IOurC'Csdecrea'eo :IS the ,h~1 "'idth is iner' , ...d

Fron' "'hat h". been ..,id. lhe rndi:llionp:utcrns Inr n:urnw. I11twtrr:Ht'lv wicl.... "'ilir.and i.,Ii..ilrh· wi.lt'" .11('1"" will he- like the'fullli"t's 01 FiR-. 2(:11. 2(h). 2(d. a"tI 2(d).re'f1<'Cli\-.ly.

The Radiation Patterns and Con­ductances of Slots Cut on Rectangu­lar Metal Plates·

RADIATION PATTERNS

It is e:a.~y to predict the form of the prin.cipal £-plane radiation pattern obtlinedfrom a rad~"lting slot cut on a long mel31pbte. Con~ider. for e~3n,plt'. Ihe pl31e ofwidlh W. shown in Fig. I. Lei S repre!enl

1-50

1.7 Wait, J. R., October 1957, "Pattem of a Flush-Mounted Microwave Antenna," JournalofResearch of the (U.S.) National Bureau ofStandards, Vol. 59, pp. 255~259

(Reprint NO.1 07)

Reprinted with the author's permission

1-51

Journal of Research 0/ Ihe NCllional Bureau of Slandards Vol. 59, No.4, October 1957 Research Peper 2796

Pattern of a Flush-Mounted Microwave AntennaJames R. Wait

Thr nlllll,-riclIl r,.,cllll" for th" Car ZOllr rnclilllion fro... 811 asial ~Iot 011 A rirt'Ulnr ..,'!i'III,·rof perfect C:OlldllCth'ity alld illlillilr ll'lIKlh aft' di'"l'lIlOII4od. It ~ ..hown thai lIMo """",,, forInr,;c dinmclt'r c:yHllde.... rail be- l'sprt'l'!C{'(1 in a unin.....1 form that ill ~lIitabll" for It.lIlI,-nlC:lllclllntioll" for nrrll"" or ~lol" 011 n II'-nth' l'1If\·c'flllllrfll..... Thp work j,. romlNln'fl ,,'ilh arellllrd ditrrlll'lioll (,rohlNlI cOII"idc'r,-d h~' j:OI'k.

,l/(r••) = I.\llr.•)1"""'·'.

rorrt'Spollds to the axis of thl' c'.'"lillclt'r. It hlL~ b"I'11shown (91 that tht' el('('tric fi('lcl, whil-h has ollly a IIIrompont'nt, is given b,)" •

TIlC' IIll1plitllcll' l.\/(r••)I 111111 tilt' Jlhll~" l'Orrl-l-tielll.llr,.' of thl' ,.\·Iilllll·r 1l1)III'1' flll·tor nrl' !'lhoWII ploU",1ill lil!lIre'!t 111nl12 fllr .. (nllll n III limo (or,,=ltl, 12,I;i, IR. nllel21. 1'0 l)rl,\Oe·nt. tnlll"II~IIlII' o\O(·rlllp"illl!.lh" ordinall'S arC' shiftl'" for ('PI'1r 1'lIrn' I,," n 1'OII!ltllllt1I1l101lnt. '

(I)

':hl' Jlhn!'ll' fUlII'tioll a(r,.) i~ n rOJlilll.'" \·lIr."ill~ flllll'­hon of. for tbl' Inf]:t'r r \·nI1ll'S. I"or thi~ fl'1I~)fI. itill "e'!tinlhl(' to t'xl)rl'll~ a ill till' ill1I1l1 illll 1,·,1 tl'l!iCIII

(0< '.1< ,,/2) as a ~1'OI",'lri"III-tII)1 i"l1l It'rlll pillS I'rorr('Ctitm fador ~(r,.), O~ follows (91

a(r,.)=a(r,O)-r(J-I'tI!l 1II)+~(r,.). (!ill)

III thl' !'llIltlo\\" rl~ion (..fl<I.I< 11'), il ill tlt'!tirnhlt· tilt'xl!rt'Ssoa all a ph~'!I!"ftl-tlI)liC'IlIl.i·rlll !,Iull 11 l~orrt,t'li~IIl,wlm'h IS alllO dt'SJlPnah'el ~(r,.), III till' followlI'",~ ~

llIallnt'r.

8(6)=1;: 6L" nz}t'tI·"'dz, (2)

AJ(7,1II)=}- i: t,,":·:,~'O!lmill, (:I)"z ••, Il. V,)

r=!'" sin 6, 1o=2,,/(rt't' spa,l' WI1\·,-I"II/!th, to= I,f.. =2(",#(1), ancllJo;,"(,,) i!l th(' cll'ri\"llti\'1' of thl' lInllkl·1fUllt,tioll al)l)rol)rintt· for a till ... (111'101' I'X" (icJ)..:11"11 tioll (I) is nlicl for £or !'lill 6» I,

fht' fUIlt'tioll 8(6) i!l tlll' !'lplle'l' flll'lllr of the' !OInt.or an arnn' of rollilll'or !llots if thl'" W,'I'" ('Ill in 1111

infinilt' 1)111/11' C'olllhlC'tinl: !'lh,·,·I.' The' fllnc~1 inn.\I(r••) ill t'"IIc'CI thl' "t,.,·lilltlc'r j;JIlIf'C' flldor" II!' ilfulh' dl'!!I'ribC'!l thl' I,(fl·,'t. of thl' (illitt· elinnll'l,'!' firtilt: ".\·Iincll'r Cln tht' rlltlilllielll pntle'I'n in holh lilt,~ ~lId ~ clirl·('lion!l. ....lIr 11IIrllt'S"!I IIf e·OIIII'"lnlifln.It l!'l ",nltt'n

1. Introduction

2. Theoretical Basis

, , .." ...... 1ft Ilnockfo..."'II...'.'.... 'It."'.,,"' ................t I .....n.. '" Ih.. I........

TJIt- radiation f!'Om f1ulIh-lIIl1tllltt'd lIIit'rowa\'"antennas has arous('d ('onsidt'rahl(· int('rcst in rt'ct'ntyears (1-6}.' From n tht'Oreti('al standpoint, tht'proble!Jl would lK'('m to ht· simplt' ('nough; tht' surfnt·('on whJ(~h the antl'nna is 1JI01lntl't1 is considert't! to lit'adElquatt'h' reprl'S('n tt'd I,,' a spht'r(' or ('\'Iind('rwhose ractius of cur\"atur(' is IIIlltc'hed to thnt of tht'Jocal (~urvaturt' of the aC'tunl surfaC't'. Unfortunntt'h'the ~ormal solutiou of till' pmhlt'm, in tl'rlllll of tli~('lasslC'al harmoni,' s,-ri,'S ilwoh'ing integral or 1:1I1f­inttgral ordt'r U('S.c;l'! flllll'tions, alwn \"S c'on\'t''l!',!!vt'ry poorly ",ht'n th,' I'ndius of C'urniturt' i!l 111'l!t','olllpar(',{ to tht' wan,lI'lIl:tlr, 'I'ht'r" art' two \\"t'II·kn,?wn a!t('rnath'" r,'pf(~"ntl1tions (6, ii, how('\"'r,wlul'l, CIUI h" us,-d to 1\(1\"I"'tal~'- in c('rlain C'all('!!.In tht' i1huninatt',1 rt'~ion of tht' aull'nun "','oUlt'ldt'at

• • • • "1:"

optICS IS most sat1sfal'to1'\' , wlul,' dl'('p in tht' shndowlhl' rigorous hal'moni,' !Ii'rit'll I'IUI ht' cOIl\'('rtl'Cl illiotill' mpitlh" ('OIl\'I'I'J,tillJ,t rNlidut' !It'rit'S.

The c'abllatioll of till' rnelilltioll fie·lel of ft f1I1!Oh­1~lOlIntt'cl anttlllla ill tht' tnllg"lIt plullt· (lht' dnssil'ullt~ht-!lhaclow houllelnn') is 1I0t re'aclih' lrl'ale'd I,,·c,ithl'r gC'olllctrit'al oJltii·~ or tIlt' rt'!\ithll' ~t'ri,·~. III tilt·(ol'Jllt'r e'allC tilt· fie,ltl ill illell·t,'I·lIIillllllt, alld ill till'latter cast' the' C'OIl "t'r!!I'III'C' ill t'xtrl'lIIt'h' poor 01111woule.1 actllall.\· clive~t· ill till' illumiluitt'cl rl'~illll,Dcspltt' the fart thnt tht' harmemil' lK'ril'!l is ('lImhl'r­IIOmt', it is valid in thi!l tmll!litioll zont' Ilt't\\"('('1I tht'iIIum!"~trc1 ~1IIt1 shadow rl·~iollll of spal't', Tht'rl'­(or~, It IS dt'Sll"ahlt'. to atll'mpt to IlClapl thC' hnrmonil'.st'rlcs r('prelK'lItatloll to sllrfal'l's of lnrgl' rOllill!l of(,"rvaturc. This is tht" purpoSt· of thl' prt'sc'1I1 IJI11)t'r.

A thill axial slot, ('ut on a drcular C'\'limlt'r ofillfinite length and pt'r('I't I'onclu('tivit'· is 1'011­sidered because it gi\"t's rist" to a plan~-polnrizl'Clradiation ficld. The c.ylilltl('r is takt'n to bt' of rntlill!ltJ and coaxial with a ('\·tillllriC'ol coordinatt' S\'sh'm(p,lII,z). The slot that (,':'I:tC'lIcls from £, to £, 1l1'1II=1Ihas a voltage clistrihutioll ,. (z) throlll!hollt il~1t'lIgth. Tht' racliation pnUt'rn is bt'st uprt'ssc'" illtt'rms of spherical coorclillntt's (r,III,6) wht'r(' 6=0

255

1-52

~. It. \'rrUl'UllOIk' bllhlrtnl Int ........ M.n"f'.

..---

IJI(r,+)I:r Ia1111

wlwfr

Dt'rp in till' shatlow, that is 1+1>90° hut Ilot 11"111'900

• .lI(r,+) has thr form of a d81l111NI stnlltli,,~ WII\"C',81111 ("8U be chara("t('rizNI b)O a fUIU·tioll of tilt' form

A(+-./2}t-f"~-.I2I+.t1{:lr/2 -+)t-1'(7-·)

"'or 1+1<900 but tlot Ilc'or 911 0• til\' \'nllll' of til\'

Spll"1' f8("tOf is quilt· W(·l1 appfoximotrcl b," till'gt'OlIll'lrical-opties approximatioll, That is

.t1(tJ}l::tI'Xp (-iT{1rfl/X)li ~I

with T=0.808 ('xp(-;r/:I). TI:is c'xpou"lllilll f01"l1lror .1(+) is tile fir.>t lrnn or a rntlll'r rompli"lltl',1rMlichl1' Sl'ril'S (iJ, It is onb' vlllici for hi» I. TIll'hi~lU'fordl'f tl'rmll (hi~ht'r modl's) urI' ol!'o pOfllllU'lri,'ill (ru/X)"tJ, wl-rrr tJ is !!Oml' Rn~lIll1l' clist!IIIC"·. ] Iis, tlwfc·forl', lo~i("81 to c'xpl'c'l M(r,+) to hI' purunlt'lri.,ill (r/2)"(+- ./2) and (JI2)l'\(:l1ro/2) - +1. In fllc't, ifthc' ril>plrs afr iF:llorc·tI. thc' flllll'tion M{r,t/l) shouldhI' R fuurtioll onh' of (r/2) "(t/l- r/2). TIU' dottl'dclln'C'S ill ligllrrs'l 11llCI 2 orc' h,·1ic"'c'cI til Ill' Ih,'oppropriotr form of .\/(r.t/l) wlll'lt Ih,· 1I111Ilclilt~·WII\·'·

c'ffl'l't. tlUI' to thl' iute'rfrrill!! Iro,"c'lill~ WI1W fmlll Ih.,othc'r !licl,- of thl' (',olillclc'r. i!'l f'·lIIn\·c·,1. Th,'"''"smnolllt'd" t'un't'S Sf;' tllt'U f'·I,lolt,·,1 liS II flllll'linllof .Y, wlll'f(' X=(r:2}'~(~-1r/2). TIll' r..!llllts lire'

51ut\\"II ill til!url' 3n whl'l'C' both tlll' \"l111l"S nf Illllplilllel ..I:'.«XII olul th,· phlL<;" rorre'I'linll Ai.'" fnl" r= Ill,12. Hi. 18 ollcl 21 foJl 011 till' S111llC' s,'l IIf C'lIr\"l'S fllrIh,' rIlU!!c' of .Y iuclic'ote·d. TIll' IlIlIplilllCI .. is nl!'loShtlWII ill fil!lII'c' :11,. bc·illl! plou"cl 011 11 In:.: !l"nll'.

1I0"ill~ thl' 51)01'" flwlor fnr 1111' c'ylilllle'I' plnll ...1ill Ihi~ IlIlin'rsol f"rlll !l1I1-~"sts Ihnl IIIIt' llIn,\" c'x­Ir"pnlu!c' tilt' rl'l'lIlt!l 10 1111~"I' \"Ilh...s nf r. TI:i~

C'Olle'c'pl, C'lIIhoclil'd ill prc~'lIli,,~ till' l'I'!'IlIlls ill I,'nll'"of n l>lunm,-ll'r X, is not lInTc·IIl"·c\ tn tIll' IIntinll orn"~III11r (lislom'C' thllt hUll hc'c'lI 1I11""c'!"sflllly c·lIIplnye'.Iill Ihc' TI'llI'c'!'c'lIlnlioll nf lic·lci !lln'"~lhs of II 11"1111'"llIill"f III IIIIW' rl1l1~C'S 0\"1'1' " !'plll'ril'lIl c'III'lh II lIloI" tIll' Inllc'r i"!lhIllC"', till' oll~lIll1r clislllllt"· i" el.,­lill,-,I us till' Ullltic' suhte'lId,·cI nl Ihl' /Tlltt'I' nr lIlt'c'l1rl ll IlI'lwc'l'lI thl' hllriZCIII!l of Ifllllslllitlill~ 111111r,'c'c·i\·ill~ allle·UlIIlll. ]11 till' IlfC'!"'1I1 !lilolllllinll, 11°,·oll~lIlnr clishllll'" is .-r/2 Filii'" till' !IOIII"'I' 1I111.'lIl1nis (lit till' C'\"limlc'r nlltl till' ol~'r\"t'r is nt inlinil" .

While, IiII' l)r/'S('lIt pmhlc'llI is c'nllc'c·I"II,·,1 clln'''' Iywilh IIII' rncliotioll of 1111 oxilll slot 011 n c'ire'ulnr nlill­c1,.... Ihl'rl' is n clifc'c'l oppli,'otioll IIf till' rc'sllll;; I,; lilt·",·,'ipnll'1I1 c'osC' wlll'fl' a plolll' wn \.,. is illl'ici/'lli 1111 Ib'c'.\·lill,Io'r (lhC' E "C'I'lor i!' to b,· P"I'I"'II,li"lIlnr lei lilt'c'ylill,lt'r IIxi!'). TIll'1I il is Iinssihlc' Iu I"I'l!nnl ,\I (r, 8111!' h"illl! pfoporlioll"1 10 till' llllffn",' C'1IIT"1I1 ,'x,'il.,c11111 till' ",olill,lc'r. III 01111'1' wnrck IIw uxiul !'llIl "1111I", n'!!nr;I"cl as II rt'I'l,i\Oilll!: (mll'nnll 1111.1 tilt' sUlln'" i"Ink"11 10 I,,· 01 illlillil\".

I II \'i,'W of till' rC'e,jimw"1 lin Ilin' or Ih' Jlwltle'lIl. i1i" ,I,'"in,hlo' t" c'nIllJl"fl' till' pn~'111 n',,"II" will. 1111,,,,,

...

'.".."6 ...

..

.....

u

U

II

II

II

U

..1111

v

..u

..u

v

..

.. I r I , I

" .. • • • •A...........

FIr.no: 1. .."nplit",lt of cy/i",'u .poCt fOclor for norro,I'uri,1I ./ot.

}'Ir.loRl: 2o Phair 0/ cy/indtr .p"u I"dor lor norro,I' "ri"/ .101.

:-:. 11. ",",ltm_ ,,",If" bI ~hltt .... l lu, f":H'h nln,".

256

1-53

AIIlplil"tlt tlntl "',,,.r tI/ _,,"et fodn, ill PtJ,u­",tl,ie /"'1110

3. An Application;\11 illll'rc'stin!! npplic'lItil1l1 of tilt' fnn':':Clill!! lla-Ilr,'"

is till' c'lIlc'ullltiuu of the' pntll'tIl IIf 1111 "IIlI-lin' IH1'n,",(If !'lol!' Oil n !!,,"tly c'IIt\'",1 surf"c"', Fn .. ,'XII III pl..I'CllIsiclc·t t'll' IIrtn.,· of 2S+ 1 plll"llll,·1 111111 :lxilll slClt..:011 u c""lill,lti,,,,1 ~lItflll'C' of rllllius II illlli...Il"11 ill iil!lIn'01. Till' 1I11!!1t· 4> ,It·jilll·!' thl' cli,ol·,'linll 11£ I;~I' ClIt""I"'""

of Fcwk II II fur tIll' C'III-rc'lIl~ c'x"itc'cl UII " c'lIn','cl ~II"­fn,'c' hy ItIl illl·ielc,"t IIh1lll' WII\"I'. Fnc'k ii ..;:t "ulI~icll'lo;:

tb' c1ifrralC,ticm of IIhllll' WII'OI' ".'0 II pllnlltului" 1111", illIlnrlic'lIlltr, foc'lIsc'S his IIt1C'1ltiCIII ClII lile' hllll!"lIlilllIn"!!IlC'tic' fiC'leI ill till' pC'1IIllUhrllI r,'!!iun (IIC-1I1" II",1lIIlIlIclnr," of lil!ht nllci ShllClow), '1'1:"11, Ilf!(,.o 11111'111­IlIlillJ: tlml thc' surflll'" c'lIrl'c'llls lin' ullly cI"'ll'II,I"1I1cm tile' Inc',,1 ruclius of C'lIl"\'llllln', h,' ulllllill~ II 1'1'111"1'­sC'lIlaltion for tht' ("tlrrc'lIt clistrihlltioll thllt is 0111\' d('­I>c'"clc,"t 011 thr pnrnmrtC'r I/tl whc'r,' I is t:1t' clistlll""­from the' lil!ht-IVlIulow houllllur~- nllci ,1 i~ HII' wiclthuf tht' IH'lIl1mbrnl r''lrioll, j'oc'" I lle'll Ilc'lic""'~ tllll Ihis rc'Sults arc' rC'nsolluhly "ulicl fUI" nn.'" shllll,'d "odi,·!'scllun!! liS Ih,' ruelills of "\II"'IlIIll'C' is Illwll.'"s \"t'ry 11I ..!!c'C'OIU1m rc'cl to till' wln',·I,'"!!lho A!' 1\ c"'IWilll t.'S1 utFoe'k's 8,>,>mximAt.· (OI'llJlllal, hilll'l'!'lIlts lit,· plott.,c/ illii!!.....•:111 Alon~ with till' c'~'lill,I"I'C'Ulllplll,'.II"ltn, Tit"RI!I'C"'JIlc'nt is ,·C't\·!!IlOCI. 1'l11'rc' nJlIIl'IU'~ III It" II slil!hlIli!'c'n'I)Illlc'~' for lorgc'r positi\"•• .Y \"lIhll'S \\'hi,·11 ig IIllt

c·lItirc".,· ulIl·xp('C·tC'cl hl'c'nusc' smnc' of Fcwk's ""stri,'­t iOlls IlfC' ' ..'c'omill!! ,'iolll 11'11. l\" .'n·I,tllt'lc'ss, c'(llIsid­"1"""1.' jllsli/i,'lItioll is !!i\"t'1I ICI tl:.- ,ollric/il,'o Clf 1111'c'xlrllpoillticm of thc' ,',\"Iillllt'r ,'urn's III 1a1l'!!I'I' "'Ihll'Sof I..

E.. :II..

0....

'.

..

..

..

..

..

..

­•

.'·nll. FClC"k.

.,.,.J

-~~ "'"• .......\

• \I- I- - ~ ) - - ri - - - -~ -

. --- .-'i \i"""\\ I

I

1\• IfI ,j

!:-f--~-

---t\,

-· ~

- ,- -__L~r,.,'I-

~1-1-

, - I-1\1- I-f-0

I-K --f-, It'K ti-f-

• \i""

u

u

u

'"'"i

--.·ro.n rtrtular eylin Irr d:U3;

I'UNYIlWIr--t"--

"H-~

111I~

I I"i\..

"~~r"{

u

u

1I

u"0/

-:= Lt

":IUJ

UI

HI

FIr.t;IlE 31>. "".pli/lld, 0/ _",JU 'ndO' in p"'tllllrl,ic fo,m.

•·,...n carnt., . t'ylln....' ".hl. .',.:1 lit: "'. .'tclt""",i,. tlingrnm n.f ,.",/·firt· f,rr,,!, of ~/,./!t ,,,, (I

cur,.,.,1 rond",."", .",.f"r,..257

1·54

( S1) ( S)nS)=s 1---'----- sill 4>---24 ({'II) % 2 (kll)

with rcspt'ct to tht' IIt11'lIlul at 0, tht' t'I'lIlt'r of lhl'array. The arra\" alld the ohSt'r\"l'r art' lakt'lI to hI'in the principal plulle (i, t'" the plallt' of the pllp!'r),, is the distall('c from 0, mt'asurl'd along tht' <,ylilldt'rsurface, to P, tIll' location of UII\' Ollt' of thc ('1('III(·nts.The field due to the sour('e at Pis thclI proportiollalto

Blltl

" [(11' .) S J[k"J'S.\=- --4' +. .. .2 {'II 2

(SC')

whcre [(~B) is the rt'llIti\"(' strt'lI~th of tht' I·II'lIlt'nt, yis the projc(,tion of, 011 til(' my from 0 to thl' oh!ll'rn'rand .11(,) is til(' ('tIITI'c:tioll fu(,tor to geometri('111 opt iI'S.III tt'rms of the Cum'tiOlI lU, .11(,) ill gil'clI h.r

where

.A(s)=M(X) CorX~O,

:1.'1-1'-

~ M(X), a for Xs:O,

(till)

(lih)

Whl'lI ka t('nds to illfinit\' lIllI'h that the surfllc'(' is('fT(·(,th't'I.\' flat, tht' Plltt(.!·,i h"('OIll('S

N7·(~)=1;.("')= E ,1 (I'll'll-a,,,!, (!In)• __ ~V

wl:('n' reS)=Ssin. anti Rim't! S= "'as,

wht're !(=t:iSlsin ~-mJ.

AC'('orclillg to tht' allnlysis IIf Hnllj;('11 11II(1 '''111111­~'artl (12). tht' l11aximum I!uill ill thl' ('lIcl-fir(, ,Iin,(,tillll(i. t'" ~=900) is ohtnillt'd wh('11 m~(N+'2)/X.

TIlI'll, tllkillg tI,t' "'I·('tri('nl spllt'illJ! 11I'tw"('1I till't'1t'1II"nts as 450 or ;1S=.,,/4, till' nrrn \. haj; li;j 1'1,'­nlt'lIt!l, J\"=:12 DIUI tilt· totlll 1"II:!t" of th,' Ilrl"ll.'· i" l'wnn'lt'"~t"s. Th,' pntll'rll 1~t"') ill this ('Il~" ij; 11111.,'(Jc.fint'cl for tilt' rl'l:ioll _!II1°<"'<~lII°. Tht' IlInill1011(' nllli till' fir.;t sitl,' Inll(' 1lI'(' shoWII plotlt',1 illfil!lIrt· ;) lIormnli7.,·(1 !lllI'h thnt. lilt' IlInXilllUIII fit'lel i"tI (Ih. t"!lilll! tht' ~IIIIt' \'nh'l'" IIf :\" 111111 as. tilt' "0'-­rt'!"poJl(lill~ !J1I1l('n, 1'11') is plntl"ll fur ~.,,:- :WII.TIll' I11l1in lout' is !lI','1I to lit' !lllIII C'\\"hll t ('hlllll!"c1 ill

(ia)

lih)

I·!t·lIIt'nts

Th(' Cactor y is ginll 11\' y=c sin 1~-(812a)J wht'll r:is thc chord length OP iuul bt'('ausc

C=[2a2(l-co!ld))'S~R(1-2~::}

it follows that

( .) ( ).~". ,.y;;:.,~ 1-24;;2 Sill <I>-2a

suujt'ct to s<<a,Heml:'mht'rill~ thut th""t, art' 2X+ I

spacl'd at ill t('n'uls t:i.~, it is S('t'lI thut

Ill' ",......••..•..L-_':'-_..l-_..J-_...L..._-L_-L_--L__

•

...

... .

!t ..;,

l...

..

_1.£..·IGl·ll& 5. P"'r"" of ntd-firr arrtll/ of ,'or. on G cllrwd 'IIIrfner .

...... I.-.;'---ft;--- ._-21'.. .1(:\1-1; ......' Jrnl'h-~: nllm·......,r_b-15.

wht'J"e L is the' 1"lIgth or tilt· arm '", Tht' SOIII't't' isnow writtclI .

/{(s)=c'"

where r is till.' propngatioll ('onst/lIIt or til(' ('xI'itill!!wave alon~ the arra)'. For the pl'C'Sl'nt purpost'

r=+imk,

wlH'rc m is a t'Ollstall t rl'al lIulllh(·r.The total field oC the o.rrIlY is now proportionul to

NE e"'t-1hllA(s)• --N

where '=1I,t:iS with 11=-1\", -1\"+ I, . , . -I. Il,I, 2 , , , N-I, N. Illtroducillg dimellsionlt'Ss pu­ramders, defined by

t:iS=kt:is alld S=lIt:iS

it follows that the Jlllttt·rn T(c1» of thl' army is I!i\"('nby

....T(~)= E e1l1·(-,,-a,"I.A(X), (8a),,--.,.

258

1-55

IIIIIIII

III

II

I

II

Th(' nllthor thllnks "-ill ill III Bri:!l!s fill' Jli!lI1!l!li~tlllll'"

ill th,' l'II1t'ullltioll of th,' JI"tl"rns ill fi~un'!l :, nllclli,

5. References

4. Concluding Remarks

It is l\l'1'1I that till' rlltiinti'lII plltt"rJI of n llu!'h­1II0ullh'd nllt..1I1I8 is iuflul'lIt'l'tl. ttl Il '·'lIIsitl'·l'lIh!t·I'Xtl'lIt. In' thr ra,lins of 1'III"\'I,tlll·'· of th,· 1II1rfll"" Oilwhi,'h it is mmmt..,\. 111 IIIU!lt I'IIS,'S, th,' mllill III'IUIIis till,',lupWltrtl 8\\'11'- from tlll' !lllrflll"', IIltllCllIJ..:h tilt'tutlll wi,JtII of th.. b"'1I11l "111I h,· 11rl1l1l1l'1I"tI illto tilt'5hllluw. Similar ph,'nom"1111 hllli h"I'1I ohs"I"\'",1 illth,· c'I\I"III"t4.ocl paU('rlls of !lluts UII n l'lIIl1h"'lill~ hlllf­pllIlII' 11:11. It. \\"uultl nl'JI"lIr tJlIlt. nllY nUI'IIlJlI. tiln',ltIt·" 11lC' !lid!' lulu'!! IIf 1111 "IIII-lil'l' 111'1"11" Is,· lI!~illJ: IL

,Iilfnll'lill;! ,',I~,' or surfll"c will 1"1111 to I~ III:lIl1t1"lIill~of thl' mllill h,·nm. Jt ill nlso (1IIil,' IIpplln'lIt thntIIII' r"!lllltnut paU,'rll of II f1u!lh mlllllll",1 IIl1t"1I111l is11111 simply uhtoilll'tI hy lI1ulliplyilll! till' fn',' S)III1'"Jlultc·rJI hy th,· patlt'rn of n sill:!It, sllli (1/1 Ih,' ,·ul"\·",1!lUrfll"I', SIIl·h a prtlt"'!l~ WCIIIItI 11I',"hll'" 11 plltll'rllwilh IlIlll'S ill thl' dilfrnl'tioll III' !llllltlflw r"J.!ioll whil'h,Ill 1I0t l'xist ill rl'l1lih' for Il 1I0Ud(lSl'li slll'fll,·...I'nrlhl'rmorr. this "Ulllltiplil'ntioll" 1t·('hlli'llIl· wUllltl1101 pr('tlic·t thl' brondl'lIill~of th,· h"1I11l ill thl' tnll:!,'lItplIlIlC.

II) C;. :-i'II''''ir. Th,' 1",1l"rll' ..r ,lnll,·,1 "~'Iill""r :II,It·"":1'.I'roN'. I. It. E .. 36. I.JS; 11\l1~1.

(2) C. II. 1':111:". 1t:llli"li,." fru", :I Inlll""'''' .Inl ill :III i",fillilt, r'\·lilld'·r. J. '1"lh. 11I111 I'h~'" ~H. 22; (1!I;;IIl.

13\ ('. l'il\"l'r':U1I1 W. K. ~:"",,I,·r•. 1t:llli:llinll frnm a Inlll-­\"'r",' ~Iol ill II t"irr'III:\f 1'~'lilld"r, .J. '\1'1'1. I'h.,· •. 21.;·I!I (I!150).

HI l'. l"'II<ipl'r, W. C;. ~It'rll" ,u,,1 T. T. T,,~·It,r. ,\ fmllll'r"llld~' of Ih,' pl\lI"rt\' IIf .illl:h· ")",, "" ,.jr,·ul:tr C'nll'dlll'liull 1'~'lillfl"r", 11:11"'1' 1''''''''111,·,1 :It ~:I'rill!: I""'r,1I:1'inll,,1 ~l'i'·lItifi .. 1t",1i" I'IIi"" (\'. H. :-. I.l ,,,·..1;111:1.\I'ril 21. 1!I,~Il.

1:,1 J. H. Wnil ,,11I1 ~. 10\ "It:III11, (""k"Ia"·,1 ,,,,It"r,,' IIr "ir,rnluf"f,oll,i:d !"lut" I'll " dU"III:!r "ulultll"'illJ,: (o.\"Jjlu1t"r,("III. J. T"l'hlllli. 33. ;; f I!I;".;".

(IiI 1•. I.. n"iIIill, Till' r:"li"tillll lit·hl "rlllh..·,·,1 I,y a ~llIt illI"r!:,' l'irl'lIl"r l'\·lillcl,·r. Trail". 1'. C;. ,\. 1'. (I 11,1 illl'"IIr Ihuli" I-:lIl1:i,,;,'r<) 3. 1211 (HI;;!;).

171 C:. ~. W"I~ulI. Th,' dilrr:l ... i,," flf r:"li" wa....' hy Ih.·,'"rl h. I'rur. IlU\·. l',,,,. 95. !Hli ,l !ll !I).

181 H. Bn'lIlll1c·r. T,·rr,·.lri,,1 r:"li" """"'''. III1'my flf l,r"I"'­Il,,'ioll. I'h"I''''r 3 (I-:/",'\-i,'r )'ullli"li"l: Co.• A""l"r­,I"",. 1!1.J!11.

I!'I J. It W'lil. 1l".li,,'i,," r1l11r:II· .. ·ri"lk. "r ",i:ol "fll" fill II"U."llIl'lilllll'~·lillcll·r. Win·I,·., J-:1I~r. :S~. 311; (1!1!,,!j).

110) K. ,\. :'\orlflll. I', I.. Hil'o'. nllcl I.. E. \·ll~h·r. TIll' """ of""Il"har ,li.IIU'"'' ill "'li"mtilllC Irllll"",i,..iull I""", I'rUl·.I. II. 1-:.•3. I.JAA (I!I;..",).

I11I \'. Fu..k, Thc' fi.·..1 uf II 1'11111" \\,:1\'0' IlI'ar tl", "urfa,'o' of II1',,"t1I1r'lilll: hu,h', J. I'h~'" (I'. ~. ~. IU 10, 3!1!1 (1!I.Jlil.

(12\ W. W. tlnll~l'n luicl J. H. Wflud~·lIr,l. ,\ III'W "rill..i"", illdin'l'lionn/ 81111'1111" ,h.,.illll. f'rol'. I. 1(. J-:. %41. 333( HI381.

(131 J. It. \\""il ..11I1 H. E. WIIII",I,'. ("1I1l'III:1I"cI radi"tiollrluu:u"""ri .. tir!4 uf !'lfJt~ rtll in ..wlnl !"hf·"l!"l, (":tn. J.T.,..hll"l. 33. 211 IIH.',:,>.

., .\\,

\,,... I,III: I.,I

c ,r~.. .,.;;..

·n

...

.g ... .. ...ANGL[ •

[orlll, pllrticuhLl'ly iII tilt' el iI"'" t illllll ""1\ I' ...= !l1l0which is the tllllltCllt plllll" of thc ""IIlt'I' of till' nrrn.".The d('ribcl1cv('1 of th(' fit'1I1 ill th(' diffrndioll lOll"(eI»900) is se('11 to d""I'rnsc lilll'al'l,'" with III IgII' ,Thrre nrc 110 10hrs fOl"llll'd ill this fI'gioll, 1t is ofilltl'rcst to know how thc ('lrr"atUl'" of til(' arrn,"itsrlf wOllld moelih' th(' Pllttl'l"II. Th"rl'fon', fursakc of cOlllrarisllll. th,' plltl"rJI 1'{eI» of till' 1'11,1­firc anny 0 isolntl'd 1'!t'III"IIt!! Oil J) ('in'ulllr arc(ka=200) is woo plott,·tl ill figufl' 5. Thl' workill~forlll IIln IlI're ill iell'lI ti"111 to t'(1I1ll1 ill II (RII) wit h.ti(.\') III·ill~ r,·phLl',·(1 hy 1 ",','nu~(· ,Iilfmdioll ,·If"t·tll111'1' 110 1ul...l·r p...·S'·lIt. Th,' pllth'l'lI i!l nllllo!lt i,I"lIti­('nl to thnt for nil Clld-fir" nfl'ln- wlll'n' ,·I'·III(·lIts arl'Oil a stl"8i~ht lim'. Loh,·s. ill ihis "II!lt', an' forll ...dOil hoth si,II'lI IIf till' IIIl1ill 1II'IIIIl nlthllll~h tlH','" 1\1"1'1I0t (Iuite SYJIlIIII'tri('nl II11I1Ut it.

Thc \mttl'rJI 1'(4)) is 111'::0 ('III11J1l1t,'cl fOl' ktr= lilli,alltl is S 10WII plott"11 ill fi~lIn' (j nlllll~ with th,' ,'urn'for ka= ... alld thr 1'111"\'1'11 nrm,' of isolllh'd "I"III'·lItS.

It would srrlll that all ('Iul-11 ...• amn' of slots Oil a"lIrvI'I1 slIrfa('r has SOliit' rntlH'r illh'rcsiill~ proJl('rtil's.Th,' mnill clfc(,ts of th,' 11I11l-lIl1tll"SlI uf thl' slIrfllc',' isto tilt thr maximlllll of tlH' IIIl1ill "1'1111I up sli~htly.111111 to ('xh'llIl t.ll(' totlll width Ilf th,· IIInill h('nm illtothr shndow rl'~ioll. Th('I'" is 110 ,',"iel"II('C of IlIh,'s011 this lIid,' of th" h"11 III , IIltholl~h nt Inrg,· nll"!!"slei> nppront:hillg HllIO). th.'n· i!l thc pO!l!lihilit.r thnt thchnck lobe frolll thr clld-fi ...• nrrny coulll ('n','p nrnlllultht' surfa('c in the oth"r din·"tioll nlld forlll loh('s.This rlTI'I't trolll n prtwti,'nl StlllldJloillt, how('n'r, is('olllplctcly iIlSigllili"llIIt fOl' kll> lUll.

I

!o'!CtiRE G. Pallern oj tnd-firt array oj ,101, on a c"rrtd ,.,,/otr.

1••;:O;~ie~:.:;.~t •• IOO; --- •• -100, A(.l1-J; 1UT3)" Ifl1lth.~; nhm· li.,," •.n.:n. CUI.O., ~Inn'h 2~. I !t:"ii"

259 I1-56 I

I

1.8 Wait, J. R., August 1958, MA Study of Earth Currents Near a VLF Monopole Antennawith a Radial Wire Ground System," Proceedings of the Institute ofRadio Engineers, Vol. 46,pp. 1539-1541

(Reprint No. 121)

Reprinted with the permission of the IRE (now IEEE)

1·57

'00C<lI.CUL.lJ(D Q,IIN£S--- C-IO·--ce-o·'---- C-II)'"

- Ea.,.,~........

p'" F••, '.'"'5 5kc.... noo "'" IlOt

II

It

F~. 2-<:omparitoo 01 calculated and o....rvod cu,·no,. 10 ndl...rir•• I.... N .U. 96. an4 191 rntli..sfroID lOP 10 IIDtlom,

In Fig. 1 the computed current..... normal­bed to unity al ~-O. is shown plotted as afunclion of , in feet for It -100 feet. ).-63.400 feet (IS.S Icc). N- 192. '-0.03(,,-1 milli.mho/meter), CS¥IO-' (no. a

It, .. Cal ...

,'k,,,,F"'1Qf155kc:

".~5ID""I1o:.O~... 1lIt~-~

~ _~-===

:. ..~• •• III til ..,Ik

, '" Ftt! ,.. 155kc:

"'I~'._·'".. IIlIO IlOt... ~----.-----.. ----- .-.. . .. _- ...----_.._-;=:=

.,. . .. ". ...,,,

'il. I-Decaya/ current 10 ndl.1 wi,... a''1lKttoa 01 dlllafta: " from bue of aO\ItDD&

'>,~__•.1...._....l ..J,..L-_..,..~--..-:!::---:!_

X-I~I-I-I II, 1 + ,liZ.v2,

where

, -12Oft/V2

24D.'P P,-~)nNC'

C-c/).. P-p().. The amplitude of the cur­rent in one radial wire..... is then given by

. _!.!::l_ 2.!-. " X.I. N 2.N vp'+'"

,"'n'

and

the total IfOUnd colftnt. 1,. il essenliallylIumerically equal to H,· and is composed ortwo parts. 1. the total earth current and.1.. the total wire current. We can now wrile.for pu,..-es of computation.

where .. is the len!:lh of the radial wires, II isthe permea!;ilil~'of the ground (~rX 10-'),w is Ihe a"glliar fr"'luellc)', _ is Ihe ,roundcomh'l·ti\,il)' in tnhn.. /mercr•• is the dielec­tric CUlnal .• nt (If till' ~rullwl. ,i. the r..diu. ofthe radial wir~•• :Ind 4 is the spacing b....t\\'een the wires. (If Ihere are N radi..1wireseqllallr spaced about the base of the an·tenna, d~2rp/.\'.)

The validily of (2) for the composite sur·face :mped.1nce ",'as not established in theabo"e mentioned work. The expression usedfor ZOo the equivalent shunt impedance for awire grid. ",'as t.1ken to be the same as thatfor an infinile p.1rallelwire grid in free space.A recent an..I)·.is indicates thaI the equiva.lenl shunl impedance for an infinite wiregrid in the plane interface of t"'·o homogene­ous media is indeed allllost identical to Z. forthe isolated grid if the inter...ire spacing, 4.is alwol)'s IIlllch less than a wa"eleneth in theelectric..lly denser medium.' In the presentsiluation. this restriction is equivalent tostaling that 4 should be some,,·hatle.. th3nan ek'Ctrical skin depth in the soil. that is4«(2/__>'''. In n'OSI pracliul cases thiscondition is OIet.•-\ question also rises as tothe applicability of the formula for surf. :impedance of a parallel ""ire grid to a radial""ire grid srSlem ""here the wire SpaCinl isnot conSlanl. Funhermor.., the wires are notof infinite length being terminated by roclsor some other means at a rinite discance fromthe base of the antenna. The assumptionthat the radial wire behaves locally as aparallel infinite "'ire grid would seem to bevery difficult to justify on purely theoreticalgrounds. It could be expected. however.that, if the length or the radial wires is largecomp.~ted to a skin depth in tbe soil. the"'a,'e rellected from the end or Ihe radial""ires ...ould be highly attenuated,

As a check of the ...ire grid theory foranlenna ground loss calculations. an operi­mencal tcst "'as carried out in Cutler. Me.•""hich is Ihe proposed .ite 01 a high.power(I-megw) vIC transmitter for the U. S. Navy.A small test antenna was erected on the siteand radial ",ires ""ere inscalled in the samemanner as in a permanent installation. Theactual currents carried by the ...irCll weremeasured using a small loop pickup placedin proximity to the wires. The average cur­renlS carried by the Iround were also meas­ured. f'ow the radio of the tocal currentcarried by the wires. I •. 10 the current inthe soil. I .. is equal to the ratio of the ItUr·face impedance. ". of the soil to the lurfaceimpedance. Z.. of the grid.' Therefore. lucha test of the splitting 01 the current betweenthe radial wires and the ground is a goodcheck on the validity of the theory.

Before discussing the experimental re­sulu' calculations of I. are presented forpertinent values of the parameten. Denot·ing the effecth'e Intenna height by A. thel:ln,elltiill ma.:nelic f,e1d. 11,-, on the groundplane at dillance p from the base is ,iven by

1. Itn--- ---, = 2. vp' +It'

for ,,«w.1velen,th. AClually. H,·. is theradial current density in amperes per meteremanating from the antenna for an idealizedperfeclly conducting ground plane. In thecase of the i'l\l'etfccI conductinl groundwilh Ihe radi.1 ""ire ground syltem present,

i,.".,d 4Z. - --In-

2r 2.t

and

1 f·AR ~ real part 01- (1I,-)'Z(p)2.pdp (I)I.' •where I. is the base: current of the antenna.H,· is Ihe tangenli,,1 l1lagnetic field 01 Iheanlenna assuming a perfectly conductingground plane. p is the distance musuredalong Ihe ground plane from the base of Iheanlenna. and Z(p) is the effective surfaceimpedance of the actual ground plane. Z(p)was taken to be equal to the inlrinsic im­pedance 01 the soil, ", in parallel wilh thesurlace impedance of the radial ""ire grid.Z•. That is.

• RKeivN by the rRE. M:uch II .•9sa'F. R. Abbf)tt and C. J. Fisher. ·0..11" of

~,ounl1 S"'tm n( R.lJ.,,1 Cf'IIulurlort f,n a VI.FT,.'ll'Inlllltr,· tl. ~ ll'.1W'y Fln-unn". I•.tb. Mtl" No.WS. I"fbfU.aty 10. IIJ"".

• J. N, \V~IC and \V, A. "OOl'. -Thf chuacltrialbof a vfrtical anlfn":! with a radl~ cond"Clor pound.ySCfm. e Appl. .s'l R.J .. B. vol. 4. pp. 177-19S~19H

• J. R. Wail and \V. A. Po~. -Input rftlJl.ft('~ofI. f. unioolc 31~nal•. - U'i,,,,U CII,., vol. J2. DD. IJI­IJI. Ma,. 19U.

t J. R. \"au -On Ih~ theory of rfllc<tioa from a.... irf .rid p.r.II~1 to an inCf1'la('t tHe_fleD homoll"­M'Ou. media. e A'''. SCI. Ittl .. B. Yo1. O. I'D. 2S9-27S:19S6

• \V. G. Hutton and C. E. Smtltt:-Na.", VLF SileI..oralton p,ft;«I. - Final Rf'D. Iimlth Elecuoftk. Inc .•Ck........d. Ohio~ 19J7.

where

"Z.Z(P) - --Z- for p < 41,,+ •-" for p > 41 (2)

A Study of Earth Currents Neara VLF Monopole Antenna with aRadial Wire Ground System-

The eRicicu.:y of ;luteuuas for very lowradio frequeucics is determined to a large ex­tent by the ohmic los.<es ill the soil ne.\f theb~sc o( the allfenll.l. Ir is t.'lIstolU.lrv to in..SI.,11 a r,..li,,1 wire grollnd s)'slem bu~ied jllstbelow the sllrface 01 the earlh. The purpose01 this wire grid is to pro"ide a 10""-105s re­turll (\.\lh lor the i1ntenllil oose cllrrent in anerrort to improve the eflil'iellcy 01 transmis·!tion.

The rules for groulld s)'Stem design illthe past hilve bee.l usuallr empirical andbased on the results 01 experiments on e.ist·illg illstall,1Iions. The first atlemptto designan optimum s)'stem was carried out byAl>l>olt.' £xtensive cakul.llions of the inputresistance of a monol>ole with a radial wireground srstelll have been cilrried out byWait and Pope.·· It should be emphasizedIhilt in this Iiltler ,""ork no i111empt wasnl."le to e,'aluate the losses associilted withhigh"'oltage insulators, tunin.: coils, andcoppcr losses. The attenlion was de"oted tothe ohmic tosses in soil and their dependence011 numher and le"l:llI of radial wires.Fllrlhermore. the radial wires were assumedto be in intimate contaCI wilh the soil beinglocated just below the air-eanh interface.The working formula for the component ofIhe input resislance. ,).R, due 10 ohmic lossesin Ihe soil i. g;"en b)'

1-58

wire, awg at IS.S kc). In Fig. 2, compUledvalues of X are showu planed as a funClionof "f). for N-48, 96, and 192, respeclively.Various \'ah,es 01 Care indicaled on thecun'eI.

In Ihe experilllenlal selUp. Ihe anlennawas a monopol" wilh a h"ighl of 100 feelwith a circular capacilY hat Wilh a radius ofaboul 200 feel. In view 01 the difficuhy inclearing land in the hea\'ily wooded androcky terrain of I\I"in", il was nOl feasible toen.ploy nlany r,ulials emanaling in all di­rections from th" anlenna. To elleet a com·promise three radials at azimoth angles of0·, 180·, and 270· were installed. The seclor~tween about 60· and 100· "'as then se·lecled as the region for further teslS. In caseI, this sector was f.lled wilh radials at anangular separalion of 7.S·, in case 2 il was3.7S·, and in case 3 it was 1.88·. For thesethree separalions the equivalenl value of Nwonl..l ~ 48, 96, and 192 respeclively.

The indicaled poinls in Fill. 1 are meas·ured current values in t,,·o of the cenlral"'ires 01 the Ian as a funClion 01 p. The onli·n,lIes are norm,"i,ecl 10 unily at .. -0. InIhis case the lIase currenI 01 Ihe anlenuawas kepI constanl at about 1.0 amper".There is a general agr~ment with Ihe calcu.laled curve. Discrepancies could ~ .scri~d

to the varying nature of the lOil conductivityalong the radi:ll wires. Measuremenls olooilconduClh'ity by Ihe four-pro~melhod indio

cated a randonl vanallon of SO per centaboul 1.0 milli-mho/meler. The abnormally1:I'Ile measnred ,.IIues 01 .. for the smaller\'alues of .. in Fill I mighl ~ ascri~d to Iheasyml\lelri~al variation 01 Ihe tOlal earthcurrent re.uhinll Irnnl nsing a fan of rad~~"

ralher Ih.1n a cumplele array equally spacedfor 360·.

1\ mnre appealing ""periment whichovercom"" 10 lollle exlent the .I)·mmetry ofIhe experimelll,,1 ..,IUP is carried out al fol·low•. The r""ei"inK cuil il mO\'ed in a direc·tion tranS\·e..e to Ihe radial wires at a con·Slant heigh I and constanl dislance ". Themeasured n"ll:nelic field i. then essentiallyconslanl for Ihe region ~tween the wiresand rises 10 falher pronounced n.uima overIhe wires. The ralio of the maximum field10 Ihat ~tween the wires is denoled by Bwhich is ai\'en by Ihe relalion

B 0:><1 r.I/2wl- 11.I/cI

where J is Ihe height of the recc\vlllg orpi~kup coil above Ihe wire and d is IheirlC~r~,tion. :\ow. linc~ ,I:~ tur",,' i .. 1M wi,.MlllllIe ,,,"e,,1 i" Ih, I'ou"d by 45·, it follo.osthat'

or

'X I r. I [ 2.4" JUt. - r. + r. - 1 + 2A + 2.• '

with

... _ '\12., Bd •

Values of .'C calculaled from the experimen­tal dala are Ihown plotted in Fig. 2, lorN-.8, 96, and 192 respectively for 1-4 feetand tl-2~"IN.The agreement ~Iween theexperimenlal points and the calculaledcurve for C-Io-' (no. 8 wire at IS.S kc) isquire aoad fur Ihe two cases of N-48 and96. The departure fur the calC N-I92 can~ allribuled to contribulion from Ihe cur·rent in the wires to the measured field in Iheminimal position.

In aenera1, one may say that ulisfactoryagreement ~tween theory and experimenthas been reached. Therefore, further juslifi.cation is given to the validity of the pub­fished calculatiotll of the (round loss com·ponent of the base resistance of a monopolewilh a radial conductor ground system. ItIhould ~ mentioned. however, thaI Ihepresent atudy has not con.idered losseswilhin the aperture or near the base of IheantenRa. Recent Itudies by Guslafson,Devaney, and Smith' indicate Ih;1t theseIouee may ~ quite large in lpecial ..ntennal

. for high.power vII installation•.The aUlhor wisba to thank A. D. Watt

and T. E. Devaney for helpful advice. C. E.Smilh and W. G. Hutton for permission toquote their experimental results, and Ana·~th Murphy for auistance with the calcu·IatiOIlL

JAKES R. WAITNat. Bur. of Slandard.

Boulder, Colo.

Reprirlted /'0"' rhe PROCEEDlXGS OF THE IREVOL. 46,1\0 8, Al'GUST, 1958, pp, 1539 - 1541.

pan.-nD 01 TBE V.SA.

1-59

, -60

1.9 Wait, J. R., and A. M. Conda, "The Patterns of a Slot-Array Antenna on a Finite andImperfect Ground Plane," L'Onde Electrique, Supplement #376, pp. 21-30

(Reprint No. 128)

Reprinted with the author's permission

1-61

extracted from: L'Onde Electrigue , Supplement '376, 19587

LES DIIIC;RA'rIMES DE R,I YONNE,\fEf,TV'UN£ ANT£NN£ A FENT£~ PLAC£Es Sl'R

l"N PLAN HORIZONTAL Ll.\lITt ET IMPARFAIT

TilE PATrERNS or A SLOT.ARR,\Y ANTENNA

ON A FINITE AND IMPERn:cr CROUND PLANE

J. R. W41T • A. M. Co:liDA

N.tio.... Hure.u ../ St.ndard,. Boulder, CoIor.do (U.S.A.)

Thc radialion flo. a .101 apcrauc OD a perfced, coDclucWl' half-plaoe i. di.cussed in _c dellil. 'ThcealCD.io" 10 eod-rare un,. of .Iou i. alao ueslCd......eriCliI co..putatioo. arc carried our for radialion pu·tcma. II is .ho.n shal in I"nclal. shc .ain be.... !. tiltcd up ..d ••• y fro. lite ed.e of she 1t.ll-plao... 'The.isuation .hcr.. !he h.lI-pl.oe i. I,i". 00 she .urface 01 a hDmo...neou. llat nrlh i••1.. con.idercd. The rc­oulu loa we applic.,ion in sh.. de.ill' 01 f1u.h Mounled .nl..nn.. of lit .. COl6-lir.. "pe .hcr... beCllu.e of " ....lical Iwillti.... she """nd plane i. of fiaise C'"ct1l aod ••, Itc Ioc.led • she ....fac.. of a .10••'. pl."ecar!h.

The co"ccpt !hal hilhly dit..cli... anleona. ca" be ..ounl..dnu.h to • MCllllic 1Ur/.CC .uch a. lite fUlel.~ 01 • hilh .peedaiter.lt 10.. beeD .uccnsfully uliliud io _n, insllocn itt re­ccot ,ears. AI"''' shon w••clc"lths. she .ud.ce ••hiclt !he."'CMa i••ountcd Or _beddcd cao ohen be coasidered llat.nd infi"ile in cale"l for parposes 01 des ill" Itt some esses.h_e.el. lite effeci of lite finile Uleal 01 she .r""Dd pl.n...uSIItc co".idcred. Aa es...ple i••hcn lite pallcm i. of !he .ea6­fite. r,pc IUch as a Iiaea. arra, 01 .Iols uciled I., a u ••elliolw..e. The ..a in lobe in shi. esIC is in lite COl6-lire direclion ilshe lrawellinl •••e loa•••eloci" C'Jual 10 or .Iilhll, leu th.ndie .peed of Ii.hl. 'The finile eSICOlI of lite ",...nd pl."e lends10 .educ. lite diteca.i" .nd produces a lill 01 lite _ia be••..., boG !he aurface.

. Ano!her i..porllnl ua..ple i••hcn a lhIoellin••••C .nlCOln.•••ounled on • mCllllic pl.ne DI finil" eSlenl which ilself i.I,io. on the .ud.ce 01 lite e.r!h. Anleon.s 01 litis "pe are Dsed10 obtain a 10-an.l.. Ioea.. 101 aitc.all laodin. de.ices.

It is the purpose DI Ihi. paper ID .Nd, lIti. proble. Ir_ alItcorelical .tandpoinr. '110.....odel cho.en fDr lit........d pl.ncis an inlinilc.im.lly Ihin re"ecdy cDnrlllclin. h.lI.pl.ne. Thearr.y i. considered ID be lDade up conlinuDu. and di.cr.. , .. dis.uiloulion. 01 puaHel ..aJllelic .ourc... or sloll mounled on one( ..... uppcr) side of !he hall-pl.ne. '110.. 101l0.iDI e/f..cII .iII be

'lSi.red : lit.. dill.nce 01 the .rr., lD lite edl" 01 lit.. h.lf.,,1.0... Ihe lenllh 01 the .".y. lite weloci" 01 the eaeilinl In'.ellinl nd the elecuic.1 constanlS 01 lite half.space 00wbich lite h.ll-plane i. Iyio••

E.rll.r In....tl..tl.n.

There h••e been. nUDlber of prior in.esliplions .hi ch .r..c1o.ely relaled 10 !he .ubjrel of this paper. It is d..sitable if!hue .re ..ention..d .nd pl.ced ia lIt..ir proper per.pecli.e. Ra.dialion pallera. 01 • 1D.lnelic lin.. lOurce .dj.cenl 10 • pcrleed,i.ola,..d h.ll-pl..... were co..puled 10, W.il' for ....rie" 01 ca.

.•es. '110.. rnersliution lo.rbirr." .ourc... such .. an .peroareor finile .1011 10.. Iteen c.rried OUI by T.i' .nd o!hero'••• '110..raeli.,ion froID. p.r.llel ••rsy of lin.. m."..elic .ourcCl .000nledon • h.U-plane 10.. been con.id..red by flurd•. lie .CIII.U, p_lIenled co"'puled p.llem••hich iIIusu.'ed !h.. lor.", lilt ofcn6-fite .".,. due '0 lite linil" dislSnc.. of the ....y Irom Ihe

. cd.e of she h.U·plane.

, , " •••",.4 ~, " t (.s.~. t " ",,-,. eN

~I.I_"".

The inllueace 01 a h_oleneou. 10••, fla, e.rlh on !he r••di.lion p.llem 01 • we.licalule.... i. included in Ill.. eo..pr..he••i.e .... I,.i. It, NorlDn•• '110.. eflecI of • circular arouad.cree. on lite ... IICIII of • ar...nd ....ed aonopole .nl..n.......been .PPlosched 10, • nu",loer of in.e.tip_.'- •• ThCle DIe'thod••re u.u.lI, ....ed on !he .ssu.ption lIt.1 the influeDce oflit.. ilDpcdecI ..ou"d beyond lite edp 01 the .c.een CID be u •ted'" • pcmub.lio...elltod ia conjunclion .ith .n lI!'Prui le(Leonlo.ich) bounds" C'OtIditi.... A" ca""Plion i. the wOlk ofBekefi '0 who pro"ide......i.Ii••1 f_.lation lor • ...rlicalcfecltic dipole .1 lite ccnter of • ci.cul•• IDe III di... laid on •plsne e.nh. It appe.r•• ho.e.er. lIt.1 cIoae 10 .n o.ersilDpfifiedchoice of uial "'"clion 10. die ...,....li.1 clecuic lield o"er Ihe.round. lit.. deri ...d re.u Ir. lor !he pror.JllIlion Con....1 of 110...urbce •••e is in .eri.... c....mcl .i!h re.li". ACN.lly lite.eloci" 01 she IUrl.ce w••e .hould .pprosch the ""loci". c.01 free .pace .. die relt.cli"e indea II of lite around .pploachesinlini"••here•• Bc.... fi·•••".ce ........Ioci" is .pplo.chinl• ..fue less !h.o e/l0. 'Chile lIti. MiPI 1101 arcady IIDp.it !heimpcdance calcul.Ii.... il w..ld It..e • profound cHecl OD Ihed..ri.ed radi.tion panelll. It wODld he worthwhile 10 i.pro.eBeke/i·•.ae!hod b, choollilll • aore recondile ltial "'nctioll.

Itt II "e" rece"l p.per C.rwell ••d Fl._c. U h••e c.ICII.laled lite panCl'll. 01 • "enical clecuic dipole Ioc.,ed on • h.U·plan...hieh i. loca.cd on she surface of • h_o,..n..OII. Ilale.r!h or hall·.pac... V.i"l .n .rll""'''''' ....ed on Greco'. Ih..o­lelD !h.., deri.e •• apression lor !he radi.li.o lield itt ler",. 01!he (••keO.II) clectric tanletlli•• field 0ger the canh·••urf.ceoullide lite hall-p1I1De. It i. !hc••ssuMed lItal litis electric fieldi. a".odified Ity die presctlce 01 the half·plsne. 51ach .n • s­....prion "iolsl... Ihe cd,e .i..... lsri" .1 lite ed,e of !he It.U·plane. It can be .ho.n. howncr. (.ee .ppendi.) th.1 .uch .n.rrrosch is p.....bl, ... Iid SIlo. an,le••nd for. Itilhl, condue­Ii... ear!h.

A ••lhe...lic.U, rill_••rrrollch 10 Ihe rrobleDl of • pl.­.e ...e inciden. 0" • eonduclinll hall-pl.n...hich i. Iyin~ on •h_o.ene f1.1 earth ha. be... canied in .n ..Ie,.nl I••hionloy Cle ow I:r. 'lite .c.llered field i. apreued ...n .nsul.r.pcCcrulS of pl.lle ...... lor the I""eral cue. 'Chen lite iocidenlw••e i. doc 10 • line •••oelic ._rce 00 lite imperf..cd, conduc·lin, fill. c.r!h .nd lit.. tecei.er is 01 .he h.ll-pl.ne....riou.r....oo.ble .pproai••lion. can be _de .nd lhe lin.1 .Dlulio..Cor 110.../troun6-_lroa..d. field cao be espre••ed in terlll. 01 •.odified Fresnel inle,rsl.Cle_ow·. an.ly.i. i. bolh eSlCa.i.eand c_plic.,ed. e.eo lItou.h !he solulion. lhem.el.e. reduceto '1.ile .i"'ple 10.... in _n, of lite es.... of pr.clical inler..st.In .i... 01 !hi. I.CI, Senior '" h•• "'IIClI..d • 1II",hod .I,ich i.• union .1 ray !h. ,ry .nd riIlO..... dillr.cli"" theor,. The p...senc .. 01 lit.. pl.ne .....10 i••ccouo.ed lor b, lite inuoduclion01 ••uil.b1e i_Ie lield .nd lhe proble", lIt.....b' reduced 10 onein .hieh fWo lield••re incident obliquely apan • pe.fectly con-duclin. hall-plane in free .pace.Since Senior'. _Ihod i. clo lyr.l.,ed 10 Ihe one used in the pte. COlI pap... aore .ill 10 id"boul il is die folio. in, te., .DIII .1.. in th.. appendia.

21

1·62

51.... Stet _ • HaI",I•• (61))

-,t2a oil

where '. - 1. 'ro • 2 em 'I 0) ••d J.., 2 ekp) ia • Be...1 ....c­Iio.. of order ..,2 .nd .,,.._ne kp. l' e .econd i.

(8)

(7)

(61)

'.A.. =J. ., (~) 0I~

!I ...

,a .. - (41r pl_)'; .in ./2

.17 .'(~).S• !...-.-.! f.... --

../2.. a

..C~).·1r 2: .'fJ.P. eO'.P...... A.. F(u.)

..1..he,e

.hele" (~) I. che Ira"...e,se (cOlllple.) ..oleap .Iont: die "ch01.1 ..i~ che ph..e referellce .1 r • 0 ."d P • P.' From cheprinciple of ••perpo.ilion il chb 'oil_a ch.1 die panem ofche ua, in die ....Iorial or principal plane is •

n ......ioa 0' die lac....., to ....m,. ., par.U.1 N••i.1 .1010 .1 PI' P.. ",p .. , •••p N "- che cd,e 0' dI. hal~planeis no. e.rried .n. Th••i li.. In iII••cra.ed ill fi,. 3a .he,ech. ....erftl i••••i" _.d to be in Ihe .aloli.1 plalle(z - 0) d. is che 180" e_pl_e.lof che alli...chal .",Ie ~..ed p ioul,. Eacb 0101 ia .1I_ed • ba... a" .rbiuar,. di..lribali_ 0' ..ola.e I' .(~) .nd lurche, Ihe,. ••,. be 0' .rbillaryle••hI2'... nc ncil.don coemcie"l lor ncb .101 i. defi"cd!ly

(4)•• p •••• F?... .~)

I.here p - (2lrp)2 co, ~ 12). Fo, p,ue.r purpo". che I.ft"10_ i. deair.ble and it ~. be .rileft i. r_a of Frnul i..te,rala .a '0110••

.here Jf (,') ia Ihe di.aib.nion of _ eae nlca.e .IOft. ehe• Ioe aad .bere '} i•• "acti.. of Icp d che ..wuchal .".Ie.T.o .Ie...ti... ropr••••cau.....re ailable fo, Q; d.e lirae ia-

Q=i 2: '.. I·" coa (11I .../2) J..'s ~p) (2)••g

Wich "'erean •• co....e..liOft.1 c,.li..drical coo,di... ,e a,.._(Po <t. a)•••Iae 01 le.,ch 21 is loc.led ... pe,fecd,. condue­ria. hal~plane aheee defined b," • 0.2 •• The .101 ia ,....lIel•••d • di•••ce p fr_ che ed.e of che half.pla..e aheer(·).The .10, Ia .......ed • be ••no••nd radiaeu 0..1, Oft OIIe aideof che .hue. Tbe eleeeric field ill che far 0' ,adinlio. field i..che .....rial plane (i.e. a - 0) haa ..I,. .. ee-pone"l .nd ia....... by

F... ,.~lrp [J+I "('')oIa'] eJlP(-alrp. +i.... ')'} (I)• .1

CCs)" r..,·H,') oil,

•

s,.) = rajll(~ IS) oil,

•

_.I WMft fJ. i. ph..e IaClor .laiell .cmea die lelauft ph..e0' each 0' die .lolS. For e...ple. che ph••e diffelftcc berwee"p. and ' ..+1 in p. fJ. - P..+lfJ•• , ••di.... la ..h.1 '0"0•••A i••1...,. re.1 aad fJ. i•• CO"aca"l .hich ia replaced .i....pl1 by fJ. 1a oilier .ord•• die .1010 are ucieed by • e..... lIin•....e .ich • phase ..eloci" u/fJ. F.rdlec.ole. che alol••,e• be -.••11,. apaced .ilb ••eparalioa " beawn. ce.cera. Itche. IoU_. Ibnl

..d che leallb of che .r.,. ia

and ..he,e che .pper +) ai,.. ia 10 be e..plo,ed .beD • in ,.0..Ii....nd die Io.e, o"n (-) ..he. II ia .e..li....

,.he ""clion F(· , cha..cee,i,.. ch. radialio.. paltem 0' •• ill.le ..inl alol r.d•• linll Oft Ofte .ide of • hal~pla"e c_"c­ein. aheee i.. lermo of p che diocance of ch. alor 10 che ed,. andche azi..ueh.1 Ie 't. Fell) io rcadil, coRlpuled ,,_ .bl•• ofF,.....I i..le la. To ilI.aaace iu propeni•• IF(a>l i. ploftedin fi.a. 2A '0' II > ....d in Ii•. 2B fo, a < 0. The ph....... defi­ned b,. ... - - .,. F(II) ia ploned in fiS•• 2C '0' • >° a.nd Ii••m lor a < •. It can be .een lhae fa, a ••fIiciend,. po.ici.eIF(II)I i••pp,oachin, .nily indicacin. char Ihe hall-pl••e i. b~ha.in. euenei.U,. a••n infini,e plane C..dvClac. AI a - ° die•alue of Fell) i. idenlicall,. O. S. When a bec..... Ia,.e in cheIlep,i.e direclion. F(a) ..y"'p.o.icall, approache. UICI. TheliRlieill. beha..i., of die IIInceion ia au_.,ized i. che 'oll...i".

(9)

L =M

The panem ca" "0. be .,inc. co....eniendy " len.. of d."diRleoo.le.. paraRleleu ,. X .nd a :

if ... (/J 0 Ir) LI•• X = (2 Ir L~ .in ~/2) ...da. 0/(0 +L)(s)•

(6-)'01. 4 +_

,..1- ..

• -I -.'F(a) I: I +_e__ e 2

"'2.. a

(.) S- "._ I. fS) , ... .._ .... X 10 , .."" ••• __ .1. I. "1. 11.

22 1-63

HtI/f-P/tIM

.Ilae

'1'1Ie paltera ., a COGlialloua dia"ib,"ion 01 uial alOl. ia"caiaed by le,cia, N llee_ ilIdeliaitel, lar,e Ian Hepia, Lr,nd ..ell ....1 •

L .. N Jp

[(a •• l)2]•

• aX -+---• l-CI N II

-here

T1le _npoodill, palle. I. die inlioile "....01 plaee (ie.oIiffraeti_ .Ikeu i....d) i. ailaply

Wilen the plane Ia ilIlilIite ill aleat, die pattera ia ideaucalto die abne il '(a) ia replaced by ..oil)'.lo thia caae, diffraeuoaelfeeu due to die edp 01 die ball-piaee are i....r.cL Farth__., if A• ••re ..il)' corrnpoaclio, • a lIIIilo_ ..pU.de ofualatioD, the aaiea cu lie __ed • lift

C(lP) • 2 [. 1_ +XI} (...!. ~ .!.)] .1. [(as +Xl)!] (10)

lL ap "'" . l-CI 2 N...[(,. +XI} :N]

(2 1 )1_l-X I._ l-a

( 2 a~lWI·J[ --'.. • 1-

.l"(2,_I_)~I-a

r •• (2, ....!...)t1- a

o.\ I I I _

F"9"e2B1111 FlIlClion ~luII .

\ r .\ ---I 1+-\ \.luM ...-.....1COIe1

.\ 1-..

'"I

'1-0..lVM .." ..... KalIl

~I ...~

I I I I ",

~~.F"1IJI"2D . --Phase LCI9/ Iof FW _I

V I I~.

- -J. I - _.

_/ II I_. - -. 1/ I/1 Ii

- .- - .. -,oo ·1 ·Z ·3 ·4 ·5 .. ·1 -. -•

U

QI

QZ

Q5

•

Q'

100

zoo

toll

500

Q4'3!!:..u

1I 1-++-hr--f---1I--+---1I--+---l--j

.,II

D$

,4

}og-40)0 .,

IJZ..Z 3 •

U

1·64 23

pla.e .Dd • ".If-p".e lor I - 0 and q - 0. The coeffici~l~ ~'e,pecified by the c....it Itt., me .ioor lobe. 10, die IDfulllep"ne case should be of 1 ••pliNde. For ~i. reoa... rhe.rr., i. c.lled • Tcllcb,.chcff allOy'. lollo••n, che worlr ofDolph ...d Ribler. UaiD, dacit procedu,e, it is found ch.,

(I) 'h 1 ,.~ h ,." , 2S •••,•.,,_ct••••,.

Ie i. io,e,earia, I. DOle cha, rhc .iDor lobe, for me half.p"oe Cue arc 00 -au of C'lu.~ ••pliNde.•Dd die -i"! I~eis ",.."ed •• i, w•• 111 Itte .Dlf•• endoftrc arr.,. A a.ID.larelfec, ho. Nee receatl, obaened .., B.rnerrDDd T.i" .ho c.l·cu",d me pallem of • Tcheb,.chdl .".y of .lecDic currcalele_a.. lo die .,ieilli" of. holf-pIoDC.

In rite pre<'edin, calcu..'i.....tceDliOD ha. been confinedelIlirel, ro rlt _ illl.'ioo .here Itt. half-p"ne .heel i, in free....ce. AD lol~.~.liD' ellICD,i... ia .hen Itt" h.lf.pl.ne i. re••rin, on me ..;I.ce of • h_o.eDe......ound. In Itti, case Ihepaaem of ,.,,, .rray of .101. for aft ob,ene, io Itte .ir i. obc.inedb, iocroduclD, aD ••,. _,t. on whe .nder ed,e of .me h.U·p"ne. The Iuco"h of whe ....'e .rray i••odified by the Frcanelrefleelioo coelfacieor .",.".,e for (.enicall)'-pol.rizcd) pl.ne..... locidenl on rite h...o,.neOll' .....nd .r ••r.zin.... ,Ieof ". 1'1Ie lormol. 10' rite paacro f•••iD.le el...eor ., di 'ra....ce p fr_ dac ed,. ia &i.eD by

(13)

A. 0:: A. = 0.632 •A. = A, = 0.67' •A. = A. c 0.881 • .nd

A. .. A. = 1.000.

,•• (8pIA)T ai" ~/2)

.,. p .c, "IF F I,. = e (a) + R~) (••)

.here

.ad

where Ie i, che c.pln dieleccric COIIaranlof che "ouDd rel.li.e'0 free .pace. The .bo.. C'lU.,iOD for I' i•• ,oed .pprOlli/llllliDnff" ftlues of" Iro.. 0 ro 111)0 and i, .pin rc'Dicred 10 Iheprincipal 0' ......,ori.1 pl.ne. 'The UNre of !he .pprollilDaliOD.arc di,Aud io che .ppendill.

'The parre,. IPI for ••io.le alor on • h.lf-p"ne i••how"in fi,. 10 for • IDules, 0' dieleccric llJound lor Ie ronllin. fro'"I CO 100 .Dd dlrce ..foel of piA. The cue 1( .. I. of course,eorrftpooda ro 'rft apace belo••• well •••bon che h.If'plane.A, 1( inc,..... me diffr.crion dfcci. "'e rD whe edl:e of rhehalJ-pl.ne .ppareDd, .re de-e.ph••izcd. When 1( i••1I0wed 10.pproach infinity .idl " > O. wher. II'I .FJ'roaches uni.y. Phy•,icall,. dli. correspoad, CO c...plele refleclion frOIll Ihe ,roundplane (If ~) - I) .nd die pallem has die u.u.l oIDDi.direcliDn_1ch.rocrerisric 0' • ,i••le .Ior on .n infinire ideall, conduclin,.,...nd p"ne. 11 i. inrcresrin, 10 note rh." for) < " < _, if 'IIi. red.ced CO Zero che parrem II'I i, .1••)', zero. In olher wDrds,., ID_ .n,Ie, die field of rhe "'.,e _rce ,cncI, 10 c.ncel Ihefield of Ihe direci '_ce. In ,he Case of Ie .. I Ihe p.nern IPIi. idenlicaU, 0.5 ., " • 0 ,ince Ihere i. nD •• ICC .ource. Onche ocher h.nd for" I.r,e. che panern II'I i...cilia lin, .bDUIuniry. The.e ripple, ••,lDpIOIically .pproach zero as pIA .~

proaches iofini" .nd mey .re due 10 die inlerference of Ihedireci rod.,iDn Ir... Itte .101 and che •••e ,nrrered &DIII cheed,e of rhe h.lf-pl.ne.

The picN.e i. nor ch...,cd ......eci.I.ly if !he ,round i. dis•• ira ,i • e• (ie. 10ss,I,.heDCe" h.s. nepli.e ...."in.ry ... rt. Theparrem II'I for. cOlaplel dielecuic con.ranl of I~i••ilh • _4. 16. 64, 25(,(1) .,e .h_n in fi,. II f. "0 ••Iun of piA.Apin u 1I bec..... Iar.e Itte _,nirude 01 n (") .ppro.ches.ni" .nd 11'1 .".rDaches uni".

The pre.cripri.. for calcul.rin, Ittc p.nern 0' .n .rr.y on

l------1

which of cou,... ia well IuIown'. huill, a .a''',ude e._I '02 aiD III whe,e 1- (,,, + X2)/2.

E.ployin, C'lUa,ioD (101. che paClem Ie ('f)/I LI is pl,,!led.a • "Dcli.. of che p.,_eICl X iD fi,_ 4 fOI , • 1 and ~"ou,

ftluea of N. Thia ia che .iNolion whele che .11., eOft""a ofN eIi.creIC .1....." fed wich • c••ranl ••plillldc. '1he condi­cion , • 1 i. the cche,ioa for opli•• piD of ncb .n .11., ••band b, H.na...nd "ooc1)'ardll• Ir .ean, char the elec~inl

lenada 1 T. of che .rroy is "rodiana leu rhon fJ T• • here fJ IS rlaeph.ae co...nl of che •••e eacilin. the .rro,. The pallerna infi,.' 4 .re ahown for bach po,ilin .nd nepli.e ..Iue. 01 .\'. Inche coae of .101. elDbedded in .n inlinire "ound pl.ne ...1, po­.i,i.e ftluea .re phy,iaUy re.liuble. 00 che ocher bond. 'heft,ion of neptin X is acceuible if che elelDenu .re loc.,edin I,ee .pace. Ie ia .eeD chal •• IV becolDe...earer chao .bo,n8 che partern doca DOl depcad ... N .1 leUI lor Ixl less chao ,.10 olher wor-ta. che ..ray of di,crele ,ourcea can be ccp"ced byconriauoua .,roy (N =_) if IV ia DOl ,_ ._11.

[ (II 1)]=.[('" + X')~]

EJ!) - 2.. i (,e + X') - + - L ') i (12)iL 1· II 2 \6" + X '2

Fi,3a

Uain. C'luolion (2) Ihe influence of ..'yin. , i, iIlualroledin fa•• , lor die cOllrinuous unifor....rroy. For.\' > 0 chi. corre,.pond, 10 .n .,roy of len,lh L elDbedded in ... infinile ,roundplane, or if nepri.e ftlue. of )( .re 10 be included. die .rrayia loared in free 'pace. Here il an be seen dI.r Ihe eflecl ofirIcreoain, , bey...d uDi" i. 10 reduce che -ior (eDd-fire) lobecon,iderably.

'The info....li.. in fi". 4 and 5 is, of cour.....elllr....14

bul il ia desirable dI., il be pr..eDled here fo, c_paria.. 01 .•"'U '011_•.Fr... rhc boaic C'l.ali... (9) for N di.ccc.e elelD.,lI. Oil •

boll-p"ne. 1hz partem IG ('f)/IT.1 ia plorted io 6,. 6 lor checondilioDa. A w • 1. IV - 8. a- O. , • O. In 'he ca.e of auch ancod-fire .nrenu ... a h.lf-p"n". borh po.ili.e and De.ali.eftlue, 01 X .rc .cceuible. The corre,poDdiD, pUlerD lor dieCOftlinuou, .rray eN =a)••i,h o4 w - I. a - O. , - 0 is .1'0.hown. For purpo.es Df cOlapariaon die ... rrem of die ".e co...liD..... ...., on • infallile ,heel i, .1'0 illu.cr.,ed in fi,. 6-

Th••"eci of .."in, " ,he .1Icirali... par.IlIeter, for che.nilona ~nriDUou, .lIa, (A w - I) on • h.lf.p"ne i. ,ho•• in6,.7 lor a- O. Con"." 10 !he n,n of .n infinil. pl... ,hftlor for .n iaol.,ed .rr." che _in lobe .ppcan 10 be ••r",.UIlor, • 0 ,.dler dI.n I • I.

When me .ltay i. ",o.ed .w., &0.. che cd•• of the h.lf.p"ne (a> 9) ch. parrem i, .,.in "'odified. 'Thi. i. iIIu'D.redin Ii,. • tor 'he a ... of the unifor. coolinuous .rr., on mehalJ-pl.... wilh II • 0 .nd a ••"in. fro'" 0 10 0 .9. In e.ch CueItte p.rre,n of Itte .rray on .n infini.e ,r...nd pl.ne (a- 1.0) i.aho-n for c_p.ri.oD.

The lore,oin, ha, been r..uicrcd co .niforlD Dci,.,i....he.reiD A w • I, 'he efleci of raperin. Itte iIlulDinalion i. shown inIi.. 9 for die ca.e of. di.crere .,ray (N • 8) .. 1Iolh ... infiDile

241-65

II

.~_ .. - L--

1.·01"'01

U PATTERN OF END-RRE. iANTONAS OF LENGTH u NaG

Ie f'lO. SA i29';

'.'~,_....l~__.~I__.';......-L.,_-+_~_~I--,,......'-+......-!

x• .,I2I'C .... ...-z

FIG. 5Eu .

u

••

,,-flIlTTERNS f7 INFORM

Ill· END- FIRE SlOT IJIffUYf7 N ELEMENTSrlaURE 6

u

u

II

..

12

F"IG.SCII

Ie -

-'-''"::::"+

F"IG. 50~-II

It

..J

$ It+-.!2

0.4

u -

F"IG. "A

..IJ

u·

It

•X

•

THE PARAMETER XIZ VERSUS

ANGLE '"• . FIGURE 3b

1.2'.It

0.4 .

-'......:;2 R'

1-66

"Ih~ half'rlan~ lyinC on a h....orn~OII. ,round i. identical 10rIta, 01.0.&. For ~..mpl~ if C ('1') is rh~ pale~m 01 the _, onche isolaced hall-plaa. rhen rhc deairc.J pallem 7; ('1') .. ,i._b,

• • ,.!. "F (0) - (l + I) J e 2 Jl.-

r: C") - r; ('1') + .'1 ('II) • r; (-'!')

To illu.lIlIl. chi., rhe palt~m It; t'I')/, LI 'or Ih c..~ 01 _i­for. coalinuGUa ~n""ire arra, 0' I~ a,rh t. wich II - 0 and I - 0ia ah_n in IiC. n 'or .ariou. re.1 ftlue. it, r. and ia 'i,. Ufor che _plu .aluc l()oia.

ia eh e fa..CI;oa prui...I, defined. No. if It - 0 Ih~ at.-se"ft i rhe h.lI-p"'e peee ., • di.coace pI... rhe eelac.Ie Un 101'-. rh.,

(16)

Cottcl...lo. a c (8P/A~ co•••12

II ia lhu••un dlAe the field". (P, 0) h•• c~ ••me O2illlulh.1de~ndencc ... "'••a the ••di.lion field 0' ••101 elcmeftC .,(P ,0). nis i.,.' cou.ae•• con.equenC~ 01 ehe .eciprocity rheo­rem.

Tlae ..dial c.-poMllc 0' rhe elecrric field i. oLe.ined frolll

Th~ polrern 0' • flu." mOllne~d ~."'ire aneuna is .un CoI.e .arkedl, influenced I.y rfoe auncalion of rh~ t;lound ploa~.

The pneral rendeney .. 'or che lIIain be.1II co M Ii Ired upward.nd .100 Lro.dened. A. rh~ diarance 0' ch~ .rra, 'r.... Ih~ ed,eof rhe h.lf-plone is incre..~d rhe po',~m approaches rh., 'oron infinir~ pound plane. Such a L~hnior i. in '1u.li.ri" .,r~e­

mene wirh rhe eaperilllenllli re.ulr. 0' F.lIiollsa for CorN,.'eof.urf.ce aae~IIII.S and Simon .nd noLieua s. 'or dielecrriC"ahe~1

onCe..... Ie is DOC possible co ••ke • decoiled c_potiaon wiehcheir work aince the eareri_n.1 condiriona do noe correspoodco che ..odel cho.en in rhia po~r. Ie ia ho~d in rhe fanate Coel.llOr.ee che model co include olher 'or..a 0' n_unifor.. eac;'.,ion", IS auch .s welocity ..odulalion 0' the .eacitin, wneand ....iou. 'Ollll. 0' ••plieude .odularion.

I: (p .)__. _I_ 3 If, (P, ..)

p' "OIp 3'1>

.nd in rhe piau. - w it foll_. chal

I:p (P.w) - 1:. [e,. aln ••]

(17)

(18)

TI.i. '111.ariey ftriea •• (Ipr} •• I, aPrr.chea aero .here..for Iorr.c lop ••, ..plOcicyllf Lcc_.

AcllllO.ladt-OIIt

We would like co ch.nk Nr•• P. Nurdock and Nrs. A. NUlph,(::or their help in rhe prep.rarion 0' ch~ paper and ~. Joha C.II..... 'or dra'eio, rhe m.a, ilIusrr.,iOlls.

A".....a

.here

••• , •••• [ 1. ,1e•• 2si••• e • F .(8p/A)2 ai""'.12r

•• w/. • •• p ( 2)1 ••.c e -- co,-• Ie p 2

(19)

" can be im,.edi.,el, .~en rhor t·: 1/1, has rhe .. Iuc orp,o".ial~for ••~rrically pololi.~d pl••e wne .e oblique incidence on •

(20)

(22)F.p (P••) - E. + e.11 • n~)]

fl. (P, .). H, .. 'I. 1:. (I + R ('1')]

• &lP1·iflp. w/41 [ 3 ].. . ._----e'-.'2 • up I •

V-(2' ,)'12 .i,,' of ., p ,ill' 2·

.nd

Ie can reodil, be nrilied ch., '-p (P,O) • 0 which i. rh~ .cquir~d!.-ndary condirion on rh~ I.c~ 01 rhe half-pl.ne .I,eee lhe.~,enri.1 fielda on chc face 0' e~ pound ••e cOllnni~nrl,.IIU~II

wh~r~

lhe qu.nlil' e, i. rhe .eneclecl elecrric 'ield &om ch~ edr indie plane of rhe half-plane as a_.liaed ro rh~ ...plieude F-.f rhe incideae .Iecaic 'ielcl. •

To iIlu....ee eh~ "ri.riOll 0' rhe _,nilUde 01 die rell~cled

.r scaaered field frooo che ed,~, I~J is plorced in fi,. I' as ..funceion .1 frp ••• _CliOll 01. (- " •••). When lop > 2.(or p > A), IeJ i. I... rh.a lIT' for ...,Ies ~ leu eh.n .bOII,10'. F...rhellDore, the fieLi is d~c.Jinl: .. p 2 indicaein, IhaClilde .cliaci•••e. place ia thi, direceion.

When rhe I_er half-.pace (+ w < '" < 2.) i. li1I~d wilhhomo,eneou•••ceri.1 who.e (c_ples) dielecrric cons.nl i•... rel.cin co fru .pocettrhcn the .01udOll •• propo.~d by S~lliorfor the _ pecic 'ield (P••) in chc upper h.1f .pace is

(m

I •• p ••• ~.It} [ 1. '" •••J "I". (p,.) - ".! e • F (8p/Ai' co, -2- +

+ e'" ... ~ .'t.) F[ (8p/A) teo. It ~~.] I

Tit. Holl-p/o... on ,It. Surfoco 0' 0 Holl-spoco.

The dillr.ceion of a pl.ne •••e b, • hall-plone shul of~rfece conducri"e, lonred in f,u .p.ce ... solwed b, ArnoldSommerfeld in 1899. The correspondin, soluciOll for Ihe silUalioa.here rh~ half-pl.ne is loneed in rh~ plane inl~rface bel.eenfWO hOllloacneous media is • much more complicaeed ploblem.Cle_o_ U p.e a formal solucion co rhi. problelll in 19H bul,eacepe in apecial caoea, it ••• aol .uicoble 'or nulll~rical C_

pucorion. A••encioned, Senior h.. propooed • lIIechod baoecl Ona c_bined uoe of ray rheory and ri,orou. dilfracrion rheory,ieldin, ••imple .olulion. 11 i. rhe purpou of chi••p~ndia 10cli...... rhe li.. icolion. in Senior'••olucion .nd i,••pplicabilityco rhe reciproc.1 r.di.,ion prol.l~m _hen ch~ .ource i. on ehehalf-pl.n~.A compari.on _irh Flamm~r'aJ.l merhod is .1.0 _de.

A h.lf-plone locaeed in rh~ inl~rl.c~ MI.Un c.o ho.o,..a~ous .~dia i. indicae~d in Ii;. 1.(. A cylindrical co..dinale.yscelll (P, ., ,) i. chooen .uc" ,h., rh~ half-pl.u i. delinedb, • - 0 (01. - 2.). A pl.ne .i,h rh~ _,neric .ecl.. paraUel10 rhe ed,e 01 rhe half-plane is inciclenl ., .n .n,le ell. .idll~upper .url.ce 01 rh~ h.lf-pl.ne. Th~ incidene 'ield 'or. ci_'eclor eap (i 01 ,) i. rhu. lina by

w~re H. is rhe .mplilUd~ 01 rh~ incidenl field.If rhe .hol~ re,ion (0 < 't < 2.) is lree sp.ce, rh~ SOIll18e,.

'eld .ol..rion lor rh~ resulrin, lield i.

261·67

I

II

I

II

I

I

I

I

I

I

I

I

II

II

(27)

1 '!! F (M) + If ('1') F (...) (2S)

which ,. ill .~ftlllenc wid, tqIt.cion (24) d.",..d •• '''ll 5""'0" •• elhod.

The " , coincide"ce of the cwo .pr.olli_ce for.... I.. for._11 .. Iu of the pu;", ."Llle '" lend. confide"ce co cite .~proach d .. lhe \aocIy of che ,.pct fo, Cllculatint "ane,ns ofco~fire a_.00. half-pl.ne which.rc l,inC on. h.1I .p.ce.

F.. (s', 0) - 1.. h • R (1'») .in .. e" ••' ••• ". (26)

..hich ne,leclS cDlllpleeel, the p...."ce of the hall-pl."e fo'

.11 ".,.ei Iue. of lI. F..,th.,.ore, F.. (s', 0) d.... DOC h•••the conce, e. of .in,..lari" It •• s'. O. Aft., .o..e .I!:.b•• ic,..ni""la,i_ th. re...hinl ClIp,esaion fo, the ."scnei.1 .acn"tic field 00 lhe uppe' .",fa... of th. h.lf-pl••• ia

T - 1· co• .!. + (F (II) + R ('1') F (-..)) co.!2 2

The .....nei" '". (P, 0) .... the ••me function dependence on "'.•• the ,.diatioe field of ••ioll. oaial .Iot at a di.cance p " .....the edle of the hal...,lane•

II i. of " ....est '0 c....,.'e "'e ,..e.en••echnique wi.h one.uUes.ed .ecend, b, Fl._e. for ••imil•• I"oble",. Wi.h .efe­,.nu co fi•• 14 .nd the c...... ian coo,dinae•• (So ,., .) eheincidene field ••h ••n ....Ie of incidence • wieh ehe "el.ei...s aai. foll••in..... c1eti..eion of FIa_." die inee,••1 eque;...for the ."••••ial ••I"e.;c field on the ,,_nd pia". i.

wh.,e

27

H] -211 e"·' •••• +• •,-,(25)

Thi. Cln be wrine. io .e.... of the (IInl"._,,) .",e"ei.1 elee­aic field 1:. (s', 0) b, lIIinl thc ,e...1e

f8 H.,t-8--] = i , (oJ F.. (s', 0)

1 ,-a

. • [811.]+!.l "."I(k(s's'») - u'2_ 81 ,-a

(I) J. II. ".ie, .R.eIi.eion frOftl • Lin. SOIf.ce A.li.cene 10 •

Con....ceina lIall Pia..... ' Jou•. of App. Phy•. , Vol. 24."0.12, pp 1'2.1'29. Dec. 1953 (Ll.

(2) C. T. l.i, .R.di.eion fro", Cu"....e r,1_enu .".1 Are'N.eSin ,he P....ncr ot • Pr.fecll, Conclucei"l: II.1f PI.ne"lechftical Report No. 4', Staft'ord !'e....teh Insli...t.,Stanford, C.lifornia, J.. ly. 1954.

(3) J. R. W.ic and n. E. W.lpolr, .Calc"l.eed n.dialion Ch••racee.i..ic. of 510.. C..l in ~ellli Sh~e.. " C.n. Jou•.'echnolo!lY. Pc. 1. Vol. H. rr 211'227. M.y. I"'" .ndPe. 11. Vol. '4, pp ~70. J.n., 19'6 .nd n. G. 1'.00.1 .ndJ. R. l:.ie, ,A" InyeSli,aeion of 5101 R.di.eors in -"elalPI••r .., P.oc. I••e. E1ece. En!:.s., Vol.103, P •. n, PI' 10l­110, J••.• 11)5(>.

1-68

(24)

g·(B·k)~

·z .1 0 I Z

Xo~ Sin ./2

- L ----""0)

.. = (Sp/A)l .in C'f/2)

- 2" .' ". P ••• '" (,.. (..) + .,: ('1') ,.. (-.. »)•

riG. 7D

riG. 7E

H.(p, 0) - ". (p, 0) + ,'r ('.) ". (p, 2.)

Pl.TTE~ OF \lNlFORMI.Z ENO • FIRE SLOT ARRAY

ON A HALF PL.1NE SHEETLO No'"

IlZ

14

15 riG. 7c

0....--------'---14,----- .

14

15

:1 ~I~ 7A :J9\f\;\/J:~rIG.7BgoiO'------------~----0Ir---------;---r--;-----,

lZ+~".....

1.0

Q.

Q5

Q4

QZ

0.J.~ ·4

:i! o•....2-06~

04

0......-------­11,----------

where

h_aneous " ....nd. Con.eque"tly, die field e, (J - R ('P») donnot utist, the'bo,..,da" conditio" ....e it h.. b..........n that• t i..... I1 ..cepe close ... the edce .nd "ether_,e fo, • hi.hl,c._Ctia. ."",nd, R ('1') i. ncar uniry. It ••, ehe.efore becoacNded that the boundary condition i. only .iolaeed for •._11 ...ioa of the IIOlI.d ...rI.ce ne.' the cd,. of the "OlI.dpu•.

A....mia. th.e this proced.., .. i...lid, the ran,..oei.1 •••pecic fi.ld on the IIOlInd plan. becomes

I

u

'....

:;" IS~ ...!l

•.. 0 1 L

• ·0lIIL'O

II

,.IGURE •

'4 • J

.·CZ'LJ"" ....1t'21

I'llTTDlN f7 lH'IlRM DCl·RIEON A HALfolUNE SHEET IrCI

!t>.-

. 11

•

u'r-..........,,.....,"""T......,.-.-................, r·.....................,......,....~...-.,-,......

II

II

•_II·;I......~~:=;:~~:::::~ r-:::=:::::::~==:::::~• U.... r-

.i ..

(..) L. B. Felsen. oRadiarion frOID Slou on a Perfectlr Condu.,.UD, Wed,e ". popel prennred ar WescoD. Lo. An~les.

Au....' 1956; also ne ~elDo DO. H fr_ ElecrrophrsicsGroup. Polrtechnic lnsuN,e of BrooklrD, ....r. 19'6.

(,) R. A. Hurd. oRadiauoo Parrems of • Dielecuic CooredAaiaUy-SloCled Crliadero. Con. Jour. Pbrs.. Vol. 3-4.P' 638. JIIlr. 19'6.

(6) It. A. NORon••The Propaprion of Radio \r....e. Ooer rheSorface of che £Srllu. Proc. IRE. Vol. 2~. pp 120~1236.

Sepr.. 1937.

(7) J. R. Waie, oThe Chsracreris'ics of a Vertical Anrenna.idl a Radial Cooncror Ground Srsre.l. lnsr. aadioEo,rs .. 195«. Aloo Appl. Sci. lies•• D•• Vol.... pp 177­l!n.l~.

FIG IOC PIITTERN OF A SLOTTECHALF-PLANE SHEET ON AD'ELECTRIC HALF - SPIIICE

• • • •

--~.~.

-- ,'0--- .-,._.- •• 16

-- .·6<:-- .·256

FIG. II B: PATTERN OF A SLOTTEDHALF- PLANE SHEET ON A

DSSlPATlVE HALF-SPACE (K"'O-il)

(8) J. R. Wair••Effecr of die Gr....nd Screen on die Field no­diared fr_ a ~nopole •• IRE Traasacrions 00 AnrennasDod Propoprion. Vol. AP.... No.2. pp 1790181. April. 19~6.

'0

Ilr---.--.--.--.--.....--.--.....--.....--.....---.

u

IPI II

IPI

_.-

-- ..,---- 1It-'-- tc-.,-- "·20-- "-100

, .t.-._~

_/).-1

FIG lOA

",."FIG lOB

• II • • 1I •

II

u

·IPI

+ lOf9ronl

II 1-69

(9) H. P.sc and G. D. Monceach••Tb" V"ccical R.diationP.uems 01 M..dill....W.... Rro-dcascia,. Aeri.I••, Proc.I.E.E•• Vol. 102. pp 279-297, N.y. 1955.

(10) G. Bek"fi, .The "pedaace 01.0 Anrenna A.....e .Circul.rGround Plare Laid Upon. PI.ne Eardu. Cu. Jour. Phys.Vo'. 32, pp 20)0222. Warch. 195".

(11) I. Cu.well .nd C. Fla....er. .Ground I.nrenna PhaseBeha.ior in • Differenrial Phase Neasurin, Sysr"....Con.enrion Record 01 ch~ 1.R.E., Vol. 5. P.n .. pp 49-~.

Warch, 1~7.(12) P. C. CleMenow••R.dio Propaprion O...r • Flu E.nh

Across .. Boundary Sep.racin, Two Pilfercnr Medi••• Phil.Trans. Roy. Soc. (London) Vol. 246. pp 1'55. JlIne. 1~3.

(13) T. B. A. Senior••R.dio Prop.ll.rion O.er • Discoacinlliryin rhe E.rrh·. Elecuical Properrie... Inst. 01 F.leccricalEa,rs. Mano,rsph No. 192R, pp 1·11. All'" 1956.

(17) J. Nciucr, •Thc Deha .ior 01 E lecu_.,ncric Field••cEd,...., R"PDcc 91·72 lasr. of N.ch. Scieacea, New YorkU.i.erairy • Dec•• I"'.

(18) R. S. Ellioc••00 che Theory of Corl\lllSled P lane Surf.·c..... Trans. I.R.c.., Vol. AP·2. pp 71·81, April, 19,...

(19) J. C. Simon .nd J. Rol>icas, .R.di.rion Perlorlll.acl' ofEnd-Fir" Surf.ce W.... Ancenna... Fi.1 Repocc onCona.cr AF 61(514~9H .._ Compepie ~ner.le deT .s.F., P.ri•• No••, 1956.

(20) A. S. Thom•••nd F. J. 7ud"c, .R.di.cion fro.. Modul.redSurface w.... Scruel\lle••, Pr. I. Connnri. Recocd ofIRE. Vol. 5. pp 15H60. 1~7.

(21) R. L. Pease, .R.di.rion frOlll Nodal.ced Surf.ce W....SCN~Nr"••, Pr. U. C_.eDrion Record of IRE. Vol. "pp 161·165, 1~7.

10

.. PATTERN a: UNIFORMEM>-FIRE SLOT ARRAY

r. "~ CN HALF-PLANE WITH16 - /7 \t\ DIELECTRIC

/1 \~ HALF-SPACE, I ' .\ FIGURE 1314 I \

I .,, I

,gaO,,

; I ,0 0 0,

: I,

L/X-15,: I

,,1-10

~ ! / ,,. Naill

~f""100,

IC.:> \if

,01 -

,,!/,-""10

,.._U \~

06 :, \~:/ 1\ ~

04 .1/ \ ~ I'II,,

i/ ,\

oz:/ , ., ,1/ ,-,'

00 5' Ill' 35'

'"

g=Oa=O

L/X=15K=IONeCD

PATTERN OF UNIFORMEND-FIRE SLOT ARRAY(XII HALF - PLANE WITH

LOSSY DIELECTRICHALF-SPACE

FIGURE 12

••8,192 •• tl\. >--·U /' -',

i .'0 '.,• x·,12 .......... .<2.. ..~

1.1. /' i/,)".. '\

1 / 1// \":\I I :1' , \~

1.4 • I ,f't.....•4 \\

1 I 11:'--..-16 \.I I t' .\

1.2 ,1 it' ~i / if \\

1.0 ., J \\

j I I \\.1/ \OJ , "j I • .~

06 if I ~.II \il I .

04', \\ '\\- "...

01

00~--~5"""--Ill'.1-----l--..,:...l--..:...--.L...--JJ:\.

(14) R. W. P. king, .Theory 01 Line.c Anrenn.... Han.rdUni.ersiry Press. 1956.

(15) ':1. W. H.nsen .nd J. R. Woodyard••A New Pcincipl" inDircccion.1 Ancenaa Dcsi,n t. Proc. IRE, Vol. 26. p. 3.M.rch. 1938.

(16) R. I. B.raccc .nd C. T. T.i•• Effccr 01. ".II-pl.nc ShucUpon chc R..disrion P.ceccn 01 • Dolrh-Tchcbysch,,1f Ar­ray', Rcporr 678-4 Irom chc Anrenns L.bor.cory 01 chcOhio Uni.ccsiry Rcsearch Foundarion on Conu.cc AF 19(604H7n. No•., 1956.

1 "'GURE 14

ul""'::-,·-.:;::--- ...,

.'1.,1 ..' .......;E.

~ •..- ·-1

.'REFLECTED FIELD AT EDGE

OF HALF-PLANE".GUllt IS

.~';,----...-7'---....:..:----...:,.----.:.....-~u.....-..l..

1-70"

1.10 Wait. J. R., 1963. "The Theory of an Antenna Over an Inhomogeneous GroundPlane," extracted from Electromagnetic Wave Theory, edited by E. C. Jordan,pp. 1079-1097, Pergamon Press. Oxford.

(Reprint No. 208)

Reprinted with the permission of Pergamon Press PLC

1·71

extracted from : Electromaanetic Wave Theory edited byE.C. Jordan, Pergamon Press, Oxford , 1963

THE THEORY OF AN ANTENNA OVER ANINHOMOGENEOUS GROUND PLANE

JAMES R. WAITtNational Bureau of Standards, Boulder, Colorado, U.S.A.

ABSTRACT

We consider an antenna over a lIat ground plane which is characterized by a variablesurfacc imped'lDCC as in a radial-wire screen. The problem is formulated in terms of themutual impedance ~twccn two venical dipoles, one which is raised. and the other islocated on the Il"ound plane. The IPlund screen is taken to be in the combined form ofa circular disc and a concentric sector. An approximate solution of the problem isobtained and the results are compared with previous invcstiptions of closely relatedw'lrk.

I. INTRODUCTION

The influence of the ground plane in antenna radiation is a subject whichhas not received the attention it deserves. In many cases it is assumed thatthe ground behaves as a flat perfectly conducting surface. Unfortunately,the principal characteristics of the antenna are influenced-in general. in anadverse way-by the finite ground conductivity. At broadcast frequencies.it has been common practice for many years to improve the situation by theuse of a radial-wire ground system. Typically, the approach has been empirical.Apparently the first systematic study was carried out by Brown and hisassociates (Gihring and Brown. 1935; Brown et al.• 1937). More recentinvestigations have been focused mainly on the influence of the ground systemon the impedance of the antenna (Abbott. 1952; Monteath. 1951; Wait andSu"ees. 1954; Wait and Pope. 1954. 1955). It has been assumed usually thatthe radiated field for a given current on the antenna was not appreciablyaffected by the presence of the ground screen.

In an earlier paper (Wait and Pope. 1954) an approximate method wasgiven which is suitable for estimating the dependence of the ground waveon the size of the screen. Calculations (Wait. 1956) based on this workSUPPO" the contention that the ground screen has only a small effect on theradiated field for screens with a radius of the order of a wavelength or less.Very similar conclusions have been amved at by Monteath (Page and Mon­teath. 1955; Monteath. 1958).

t This work was carried out while the author .. as on a visit to the Technical Universityof Denmark. Copenhagen. in the faU of 1960.

1079

1·72

1080 JAMES R. WAIT

In this paper the theory is extended to ground screens which may belarge in tenns of wavelength. The problem now bears some similarity toprevious investigations of propagation across a land-sea boundary (Bremmer.1958; Wait, 1956, 1957). While the literature on this subject is quite extensive,the results are not sufficiently general to be applied without modification.Thus it seems worthwhile to refonnulate the problem in a fairly general way.

p

h

IIt-------- d

IIIII

pLl=:::::::::~====rc~:bJ

FlO. 1. Side and plan views of the dipOles and the lround plane.(Note that tan ." - hI' and tan l(IO - hid.)

2. FORMULATION

The problem to be considered is illustrated in Fig. 1. Two vertical electricdipole antennas are located at P and Q which are located at height handheight zero, respectively. over a fiat earth. The effective lengths of thesedinoles are II and 12. respectively. The surface of the ground is characterizedb, a surface impedance Z everywhere except at a ground screen about Q.This ground screen is made up of a circular disc of radius a which is lyingon the surface of the ground. Furthennore. a sector of radial wires emanatesfrom this circular disc in the manner shown in Fig. I. The tangential fieldsover the ground screen are also assumed to be related by an effective surfaceimpedance Z'. Over the disc (i.e. p < a) this is denoted Z~ and over the sector(i.e. b > P > a• .::12 > t/> > -.::1 1) it is denoted Z~.

The mutual impedance: between dipoles P and Q is written

== =0 ~ A: (I)

1-73

11IE 1lfEORY OF AN ANTENNA, 1081

where zo is the mutual impedance between dipole P and Q when no groundplane was present. Thus ~z is the change of the impedance resulting from thepresence of the ground plane. Having the problem formulated in this wayallows the compensation theorem from network theory to be used to express~z in terms of a surface integral over the area of the ground screen S. Thus(Monteath, 1951)

Az = kI I (Z' - Z) Hpc . H~c dS (2)s

where Hpc is the tangential magnetic field on the ground plane, with noground screen, resulting from a current 10 impressed on terminals of dipole Pand where ~c is the tangential magnetic field on the ground plane, withthe ground screen present.

Equation (2) is an exact formulation of the problem provided that thetangential electric and magnetic fields may be related by a surface impedance.Essentially it is a two-dimensional integral equation since the integrandcontains the unknown field H~" It has been shown in a previous paper(Wait, 1956) th:u such an equation can be reduced to a one-dimensionalintegral equation by employing a stationary-phase principle. Such a methodis particularly appropriate for studying propagation over land-sea boundaries.Here: a somewhat different approach is used since the two-dimensional natureof the problem must be preserved.

3. THE APPROXIMATE SOLUTION

The mutual impedance zoo in the absence of the screen is easily obtainedfrom the known solution (Norton. 1957; Wait, 1957) of a dipole over a flatconducting plane of surface impedance Z. Thus

illow!dz .Zo = -- e-1t8• cosz.po W(Ro, Z) (3)

2fTRowhere

W(Ro. Z) = I - i(fTpO)1I2 e-tD• erfc(iwo1/2), (4)

wo = (I +Zf po,

Po = - ik:o (~r, k = 2fT/wavelength.

'10 = 120fT and p.o = 4fT X 10-7•

This result is valid subject to the approximation that kd~ I and !z/'1olz~ l.It should be noted that W(Ro. Z) can be written in the form (Nonon. 1957;

Wait, 1957)W(Ro. Z) = W, ;- W,

1-74

(5)

1082

where

JAMES Il. WAIl

W, == 1 + R(I/Io)2

is the radiation or space field and W, is the Norton surface wave, and R(I/Io)is a Fresnel reflection coefficient. Provided 1/10 is not near zero, and ifIpo I~ I, W, is the principal part of W. By definition, the antenna patternof dipole Q in the absence of the ground screen is the function W,. Now atnear-grazing conditions where 1/1 approaches zero W, becomes the principalpart of W since W, tends to zero. This is an important point to keep in mindwhen discussing the effects of the ground on the radiation from the antennaat low angles,

Another quantity required is the tangential magnetic field Hp , resultingfrom a current /0 in dipole P. It is a vector with amplitude given by

where

'klo!Hp, == -'_1 e-,kR cos 1/1 W(r, Z)2"R

(6)

W(r,Z):; I - i("p)t/'!. c-IO Crfc(ilVt/'!.), (7)

W - P(I +i.;f,p == _ ikR (~):, and R == (rl + "1)1I~.

2 '10

This result is valid subject to kr:> 1 and IZI'1O 12 «1.Now the tangential magnetic field ~t on the ground plane in the presence

of the screen may be written in the form

H' == iJclo!z (1 + .!.) e-ctp W'( Z' Z)I' 2 iJc p, , ,"p p(8)

where W' is an unknown function of the radial distance p and the surfaceimpedances Z and Z' of the screen and of the ground Equation (8) is nor­malized so that W' would approach unity if Z' == Z = O.

In a similar manner the mutual impedance:: in the presence of the screenis written

z = ilAG",111z e-,tRo cosz ./. W'(R Z Z')2"Ro '1'0 0"

(9)

where W' is an unknown function which is also normalized such tbat itbecomes unity for Z = Z' == O.

Using (3), (6), (8), and (9), it readily fOllows that (2) may be expressedby the equivaJent form

1-75

1HE THEORY OF AN ANTENNA 1083

(10)

W'(Ro, Z, Z') = W(Ro, Z)

+ 2" ::5 +0 JJ(Z';;: Z) e-fkp

e-fk(R-RoI (i . it) (Ro) COS '"P R cos "'0

X (1 + ~p) W(R,Z) W'(p,Z',Z)d'dp,

where the variables of integration are p and., the polar coordinates about Q.Also, ip and ill are unit vectors in the direction of the fields Hpl and H~/

In the case when h approaches zero, (10) reduces to

W'(d, Z, Z') - Wed, Z) I+ ~~ ff (Z';;: Z) (II' .i,,) (~ (I + ;~p) (II)

W(r, Z) W'(p, Z', Z) d' dp.

This is a two-dimensional integral equation for the function W'. In principleone could solve (II) for W' and then using this, (10) becomes an explicitintegral formula for W'(Ro. Z. Z') at any height h. Because of the complexityof such an approach, it is customary to reduce (11) to a one-dimensionalintegral equntion by employing a stationary-phase principle (Bremmer, 1958;Feinberg, 1946). Essentially this makes use of the idea that the importantpart of the surface integral is a narrow ellipse with foci at P and Q. In thegeometry of Fig. I we would then find that (11) could be reduced to

W'(d, Z, Z')~ Wed, Z)b

_ (ikd)1/2 f (Z' - Z) Wed - p, Z) W'(~. Z', Z) dp (12)271' '10 [P(d - p})1/2 .

o

provided kb is somewhat greater than unity. An interesting special case of(12) is when Z' is allowed to be constant with respect to p. Then. to withinthe st:uionary-phase approximation. W'(p, Z', Z) in the integrand can bereplaced by W(p, Z'). This leads to the following integral formula for themixed-path function W',

W'(d, Z, Z') ~ Wed, Z)b

_ (ikd\1/2 (Z' - Z) f Wed - p, Z) W(P. Z') dp2;1 '10 _ [p(d - p)]1/2 •

o

All quantities on the right-hand side are known, since

W(p, Z') = I - i(1:'p)I:2 e-P erfc(ipl/2)

1-76

(13)

(14)

l~

J

III

1084 JAMES R. WAIT

ik (Z')2where p = - -! - .2 '10

the preceding results, borrowed from the previous work (Wait, 1956)on ground-wave propagation over mixed paths, suggest that a plausibleapproximation to (10) is obtained if W'(p, Z', Z) is replaced by the functionW(p, Z') appropriate for propagation over a plane of surface impedance Z'.In the limiting case where Z' was zero W(p, Z') would be unity. Furthermore,even if Z' were finite and slowly varying, but satisfied the inequality

kb IZ' 12

- - < I2 '10 '

it would be an excellent approximation to replace W(p, Z') by unity. In mostground screens, this condition would be met.

In order to simplify the required integrations, it is now assumed that d issomewhat greater than b. Thus. in the integrand of (10) we make the foUowingadditional approximations

I" . I,:::::: - cos ••

~cosl/l~1R cos 1/10

for the whole range of integration. Thus

W'(R, Z. Z') - W(R. Z) [I + D)where

D ..... - --..!!:..- f f e-fkp e-fkCR-Rol (Z' - Z)211' cos 1/10 '10

8

(15)

where

and

x (I + ~p) cos. d. dp, (16)

R = (p2 + d2 + h2 - 2pd cos .]1/2

Ro = [d2 + hZ)1I2.

The quantity D is the fractional change of the mutual impedance between Pand Q resulting from the presence of the screen S. It is now convenient tosplit D into two parts in the manner

(17)

where D. is the contribution from the semi-circular screen of radius a. andD. is the contribution from the sector.

1-77

DIE DlEORY Of AN ANTENNA 108S

4. THE INTEGRAL D. FOR THE CIRCULARGROUND SCREEN

Over the range 0< p < a. it is pennissible to retain only first-order phasetenns. thus R - Ro a! - Pcos ~ cos .po. Therefore•

(18)

•f e-«kp (1 + ;~p) efb-_cos'l'.

---.•

-iJc JD.= =----;-- 2" cos .po,-0

(Z' Z)cos ~ II~ d,p dp.

The ~ integration can be carried out in closed fonn to give

•D. ~ co:.po f e-fkp (I + ~p) Jl(kp cos .po) (Z~ '1~ Z) dp (19)

,-0where it has been assumed that Z~ does not depend on ~. Jt(x) is the Besselfunction of the first type of order. An alternate derivation of (19) isgiven in the appendix.

In the absence of the sector ponion. D. represents the total effect of theground screen. If ka> I the integrand in (19) can be approximated byemploying only the first tenn of the asymptotic expansion for the Besselfunction. Thus. for the major range of the integrand.

Jl(kp cos 1/10) e-fkp (I + ~p)

2! ( 1 )"2 e-a-/~ (I _ ;~-fUp COl Po) efkp,COI Po-ll. (20)2"kp cos 1/10. .

Thus

D. =- J( ; ) f·k (Z~'1-0 Z) (I _ ;e-2Iz coo vol- 2" cos3 1/10

° e-fzU-eOI Pol---:1'::'/2- dx. (21)

x

When Z~ is essentially a constant over the range of integration. D. can beexpressed in tenns of Fresnel in,egrals. Thus

D.~J(2 CO~3 1/10) (Z ~oZ~)

[

1 f,/(~) .... ". . f V(~ kc) cos·,__ "t e-Il./21 ,I dt __' _ :t 2

.r/Io 1/10SID 2 ° cos 2 0

1·78

1086 JAMES R. WAIT

As .po tends to zero, the above equation reduces to

('2}( )1/2 Z Z') [ ( )1/2 f (4 :/1)111 ]

QII::::: '.,,0

(~o II 1 -; ::Co 0 e"'Cw/21 " dl •

Since ko ~ I

(23)

(24)DII ::::: e~or2 (Z ~oZ~) [I - (;;aY'lWhen the second term in the square bracket is neglected the result agreeswith a previous analysis (Wait, 1956) for the transmission across a land-seaboundary. It is concluded that the second term in the square brackets of(24) results from the circular shape of the screen. Usually in the land-seaboundary problems both media are semi-infinite. Further suppon to thiscontention is given below.

It is of interest to compare (22) with the solution obtained on the assump­tion that the ground screen may be replae:c:d by a perfectly conducting half­plane. Assuming an incident plane wave, the voltage induced in antenna Qis calculated by a modification of Sommerfeld's results (Sommerfeld. 1899)for diffraction by a knife edge. The method suggested by Senior (1956) is aunion of ray theory and rigorous diffraction theory. A direct applicationleads to

I + (J _ 2 F(u) + R(t/lo) F( -u)II 1 + R(r/Jo)

where R(,po) is the Fresnel reflection coefficient given by

R (r/Jo) _ sin r/Jo - (Z/T/o)sin ';0 + (Z/T/o)

and

(25)

(26)

(27)•

F(u) = 1 + v(li) f e-cCw121 " dl

°with /I = (4kol.,,)1/2 sin r/Jo/2. The function F(u) can be identified as theresponse in the dipole at Q resulting from an incident plane wave (at grazingangle .po) on a semi-infinite conducting plane. The function R (.po) F(-u)is then the response of an (image) plane wave incident from below the half­plane. Such a solution is very plausible since the tangential magnetic field iscontinuous across the air-ground interface and the fields have the propersingularity at the edge of the half-plane and satisfy the boundary conditionson the half-plane itself. However, the tangential electric field is not continuousacross the air-ground interface. For near-grazing angles this violation of theboundary conditions takes place within a very small distance from the edge

1-79

THE THEORY OF AN ANTENNA 1087

of the half-plane. The result given by (2S) can be rewritten in the form

Q. = J(~) (!\ _1__1_ f \1'(4~.) liD l' c-'(w/2) " dt (28)2 ;/ cos "'0 sin ~ 0

2 2

which is almost identical to (22) when Z~ = 0 and the second integral is dis­regarded. It is also required that "'0 is small. (It might also be mentionedthat a similar fonnula for this situation has been proposed by Carswell andFlammer (1957). Their result is derived by an approximate evaluation of theone-dimensional integral equation of the same type that occurs in mixed-paththeory for ground-wave propagation (W~it. 1956).)

An exact correspondence between (22) and (28) is not expected because ofthe differing approximations inherent in the two approaches. However, thisdemonstration strongly suggests that the second integral in the squarebracket is the contribution from the back edge (i.e. ; - tr) of the circulardisc. As mentioned, for ka» I. this term can be neglected.

Another aspect of the circular ground screen also deserves mention. In thefonnulation of the problem it was assumed that antennas P and Q werevenical electric dipoles of infinitesimal length. The modification of thetheory for antennas of finite length is not difficult although the complexity isgreatly increased. For near-grazing angles. it is not difficult to show that theprincipal results are not changed essentially. For example, if Q is a quarter­wave monopole with an assumed sinusoidal current distribution. the changesin (16), (11). and (18) arc:

1 cos t/Io-- is replaced bycos tPo (tr )

cos :2 sin "'0and

( 1 -+- ~) e-ftp is replac~d by e-.h'(pl+ (A/41 1 1.ikp

For large ground screens (ka ~ I), the difference between the latter twofactors is negligible. For screens comparable ;n size to the wavelength. itmay be important to use the correct fonn of this factor. In the case of thequaner-wave monopole, D. can be written in the fonn

t.

D. = cos tPo f (Z~ - Z) e-l v (%1 +(./2111J1{x cos tPo) dx. (29)

cos (~Sin "'0) 0 '10When Z~ is a constant over the range of :c and "'0 tends to zero, it is con­venient to write

(30)

1-80

t

J

I1IIIIIIIIIIIII

1088

where

and

JAMES R. WAIT

fi. [( "Z)l/Z,,]Xl = - cos xZ+ '4 - 4 hex) dx

o

i.

Xz = f sin [(XZ+;rz- i] hex) dx.

o

(31)

(32)

If the screen is pcrfce:tly conducting

Z ( 'OCII )l/Z ( i.oCII )l/Z8 = - e-C• /4 = 1 - ---'10 a + ;.CII a + ;.CII

in terms of the electrical constants, a and " of the ground. At low frequencies,where CII «a/.,

('OW)l/Z8::: -;- -= 0'0075 (fMe/a)l/Z

which is essentiaJly a real quantity. (In the above, fMe is the frequency inmegacyc:les and (f is the ground conductivity in mhos per meter.)

Numerical values of Xl and X2 for ka in the range from 0 to 6·5 are givenin Table 1.

TAlLE J

lea XI X.

0'0 0'00 0'000O'S -0'042 0'0401'0 -0'130 0'181I· S -0'211 0·4172'0 -0'209 0·7002·S -0'102 0·9473'0 0'042 1·0933'S O'ISS 1'1314'0 0'171 1'1334·S 0'113 1'178S'O O·osa 1·300S'S O·osa 1-4686-0 0'119 1·6126-S 0'20S 1·674

These values were obtained by graphical integration so the accuracy of thelast significant figures are doubtful. For large values of ka, Xl and Xz can

1-81

THE THEORY OF AN ANTENNA 1089

be represented by Fresnel integrals if (xZ+ .,,2/4)1/Z is replaced by x andhex) is replaced by the first term of its asymptotic expansion. Thus

and

where

•Xz e; (2ka/,,)l/Z - ~2 f sin (i IZ) dl

o

(33)

(34)

(35)

u _ (4ka/,,)l/Z _ (Sa/AlI/Z.

It c:an be seen from Table I and from (33) and (34) that Xz is generallyquite large compared with Xl. Consequently, the imaginary part of D. issomewhat larger than its real part which indic:ates that the presence of theground screen influenc:es the phase to a greater extent than the amplitude.

5. THE INTEORAL FOR THE SECTOR

OROUND SCREEN

Attention is now turned to the sector portion of the screen. The antenna atQ is again assumed to be a dipole. If b «d it is again permissible to retainonly first-order phase terms. Therefore, it follows from (10) and (14), that. ....Db - - ik f f e-ctp (1 + ~) e·tp - ~ COl r10 X

2" c:os ';0 ikp_-a .--~I

(Z' Z)x c:os '" II "1~ d'" dp.

This result is analogous to (18) for the c:ircular screen. To effect the'" integra­tion we usc the basic relation

•eCz COl ~ _ 1: ~"e',,·12 J,,(x) cos n '"

.. -0

whic:h is a generating function for Bessel func:tions. J,,(:c). Using this result,it readily follows that

b

Db - _k_ f e-1tp (I ... ~) A(kp c:os ';0) tL; - Z) dpc:os "'0 ,kp \ 110.-.

1-82

(36)

1090 JAMES R. WAIT

where

A(.~) = 2..l.. f fll e'lIlf/l lll(X) J4. cos n; cos; d ;", 11-0

1 ..=- 1:

4fT; ,,-0"'./2 1 ( ) [sin (n - 1).d2 + sin (n - 1) .dl +

.." e II x 1n-

+ sin (n + I) .d2 + sin (n + I) .d1]. (37)n+l

It is Doted that if .dl = .da = ",

A(x) == Il(X)

which corresponds to a circular screen. Another special case is

which leads to

(38)

(39)A(.~) == h(.~) +!.. i .Ja,,(x).2 '" ""0 (4n2 - I)

This particular formula was quoted by Monteath (1958).Equation (36), which generally requires a nwnerical integration and a swn·

mation, becomes very complex if the scretn is large in terms of wavelength.This complication results from the poor convergence of the Bessel functionseries when kp cos .po» 1. An alternate representation for Q" can be obtainedif a modified stationary-phase evaluation of the ; integration is used. Thisapproach is particularly suitable when the kb» 1. The phase factork (R - Ro) is now approximated in the following way

(R - Ro) = [~ + pI - 2pd( I - ~ + ... )yl2 - Ro

=::: R1 - Ro + 7:; (40)

where R1 == [Ri + p2 - 2pd]1/2. Thus we have retained second·order phasevariations. Consequently,

"ik J (Z' Z)D" 0< - " "'-0 e-up e-UIRI-Ro) x- 2fT cos .po Of

.01,

X f e-'t~1/2RI d.p dp. (41)-.01.

1·83

THE 11IEOilY OF AN ANTENNA 1091

This can be written in the formb

(ik )1/2 1 f (Z~ - Z) e-1kp e-lk(RI-RaID,,~ - - --- -- F(kp) dp (42)

- 211' cos3/2 "'0 '0 pilI•

where.I, • I(bd)

1 f v'a;F(kp) = --. e-l(w/2I'· dl

1 - I .1 iN-41, V (ii1f,)

and

1-84

j

'I

~

J

1II

IIIIII

(47)

(43)

(44)

= {C[~2J(;~] + C[~lJ(;~]}

-i{S[~2J(~] + S[~lJ(~]}1 - i

•C(u) - is(u) = f e-(w/21 ,I d/.

o

and

It can be seen that if

are greater than about 5, F(kp) may be replaced by unity and thus the sectoris behaving as a circular screen of radius b. Equation (42) is in a suitablefonn for numerical integration in the general case when F(kp) cannot bereplaced by unity.

In the far zone, where only first-order phase variations arc employed,(42) has the fonn

b___ (ik)"2 I f Z~ - Z e-·tP(1- coo volD" = 2 3/21/10 1/2 F(kp) dp (45)

11' cos '0 P•

I .,'V(bc: Y,)

F(kp) = 1 _ i f e-,(w/21 i l d/. (46)_ J, ,/(bc:, ",)

This is also in suitable form for numeri~1 integration with respect to p(or kp).

An interesting special case of (45) is when "'0 .... 0 and Z~ can be regardedas a constant. Then

1092 JAMES Il. WAIT

After an integration by parts, it readily follows that

Db = - (2~br12 (Z~~ Z) {[F(kb) _ iJ(fr~b) (e~~:'12+ e-;~:1/2)]

-J(i) [F(ka) - iJ(fr~a) e-;~:'/2+ e-c~:I12)]} (48)

where F(kb) and F(ka) are defined by (46). If (kb)1I2 ~2 and (kb)I/2 ~1 ~ I,the first term in square brackets can be replaced by unity. Furthermore, if(bla)I/2 ~ I, the second square brar":t term is negligible compared withunity. To within this approximation, the sector screen is behaving as acircular screen of radius b (e.g. compare (24) and (48».

When "'0 is finite but small, it is possible to extend the preceding result byreplacing the factor exp[- ikp(l - cos "'0)] in (45) by its power seriesexpansion. Then, again assuming Z~ is a constant, it is found that

Do = _ (2ikb)1I:! (Z~ - Z) i [- ikb(l - cos "'0»'"fr '10 ",.0 m!(2m+ I)

x {[F(kb) - _.1 ~ (~:!Gm(- .i_~J. kb) + ~lG",(- i~i. kb))](2fr,kb)l/- 2 2

- (6)"'11/2 [F(ka) - (2fri~a)1I2

(~:!Gm ( _. i~~, ka) + ~lGm ( - i~i. ka) ) ]} (49)

where

(SO)

When "'0 = 0 only the m = 0 term of the series is finite and the result isidentical to (48). When "'0 is finite but small enough that (kb)1I2.po < 1,the series converges very rapidly and only a few terms are needed.

6. CONCLUDING REMARKS

The collected results presented here should be useful in making estimatesof the influence of an inhomogeneous ground plane on antenna radiation.As we have seen, the subject is closely related to the question of ground-wavepropagation over mixed paths such as occur at land-sea boundaries.

In the present study. the electrical characteristics of the ground are assumedto be characterized by a surface impedance which is a (complex) constant Zoutside a surface S. Within S the impedance Z' is allowed to be variable. Inthe case of a radial-wire system emanating from Q. it is appropriate to useformulas which have been developed for the surface of a wire grid in the

1·85

THE THEOR.Y OF AN ANrENNA 1093

interface of a conductint, half-space (Wait, 1958). In general these are com­plicated, but recently some numerical results have been obtained which shouldbe useful in this problem. At low radio frequencies for moderately or well­conducting soils it is a satisfactory approximation to regard the surface Z·as the parallel combination of the surface impedance Z. of the equivalentgrid and the ground beneath. Thus

where

Z. Q,< Z.z-Z+Z.

. od dZ.~ '.!- log. -,

.\0 211'c

Z ~ (iIJ4Jw/a)1/2,

(51)

(52)

and d is the spacing between the radial conductors and c is the radius of thewires. Such a formula is strictly valid only if (al'Ow)1I2 d < I everywherewithin the ground system. If there are N radial conductors, it can be seenthat d can be replaced by 211'p/N where N is usually of the order of 100.

It is admitted that the theory in this paper is rather involved. In order toobtain numerical results it is necessary to evaluate the integrals D. and Db bynumerical or analytical means. The quantity D( == D. + D,,) is then regardedas the fractional increase of the field as a result of the presence of the sectorground screen (as indicated by equation (15». The final results should bevalid when the surface impedances Z and Z' are reasonably small comparedwith '70 or 12011' ohms. However, this is a condition which is also requiredin Sommerfeld's theory for a dipole over a conducting half-space, and is notoverly restrictive.

Extensive numerical results based on the theory given in this paper havenow been obtained. They will be included in a forthcoming paper co-authoredwith Mrs. L. C. Walters [N. B. S. Monograph No. 60, 1963].

7. APPENDIX

The various approximate formulas used in this present work start from themutual impedance formula given by (2). As Monteath (1951) shows. this isbased upon Ballentine's (1929) formulation of the electromagnetic reciprocitytheorem. The author (1954) has obtained similar results directly from solu­tions of the wave equation. The latter method has the advantage that aperturbation procedure can readily be applied to obtain higher order cor­rections. It has the disadvantage that only relatively simple geometries areeasily treated. A brief presentation of the alternate methc:1 is given heresince it sheds some light on the nature of the approximations used in thebody of the text.

1-86

1094 JAMES R. WAIT

We consider a vertical antenna erected over a fiat ground plane. Choosinga cylindrical coordinate system (p,., z) the antenna extends from Zl to %2

on the z axis and the ground plane is z = O. On the assumption that thefields do not vary in the • direction it is evident that the resultant magneticfield has only a • component, H9• Furthermore, in the homogeneous spacez > 0, H9 may be derived from a scalar function which satisfies

from the relation

Thus, in general,

(1 a a a2 )- - p - + k2 + - .p = 0pap ap az2

a.pH~ = ap'

(53)

(54)

(55)•

H;(p, z) - f Jl(Ap) e-·' -/(A) AdAo

where H; is the field of the antenna over a perfectly conducting ground planefor: &: 0, where Uo - (.\2 - k2)1/2. Then from Maxwell's equations

•[ Ep(p,:) ] = Ep(p,O) == ~ f h(Ap)/(.\) Uo A d.\. (56)

_-0 0

On an application of the Fourier-Bessel theorem, it follows that

•ik f/(.\) == - heAp) E,,(p, O) pdp.

'101&0o

Equation (55) may then be written in the form

(57)

• •HJ(p, :) =; f f Jl(Ap) heAp') e-.e- 1101 Ep(p', o) p' dp' AdA. (58)

.'-01_0

We consider that the ground is modified in some way so that tbe new tan­gential field becomes E;(p, 0), This in turn leads to a new secondary fieldHJ. Therefore. the cbange of the field IJH9 resulting from tbe modificationof the ground plane is

Uijl [E;(p', 0) - E.(p', 0)] p' dp' .\ d.\. (59)

1-87

1096 JAMES R. WAIT

(63)

appropriate for a perfectly conducting ground plane. When the antenna is aquarter-wave monopole with a sinsuoidal current distribution

H;(p',O) = - 2;/ . e-1kv'[C_')'+(1/4)')

"'pand, for kp > I,

H:(p,:)~ - 2~p e-1kR cos (i sin t/J). (64)

Thusk_,

n~ cos t/J f (Z'~ Z) e-fvCZ2+1r

2/4) hex cos t/J) dx (65)

- cos (i sin t/J) 0

This is in agreement with (29).Ackno",lecJgemems-1 would like to thank Mrs. Alyce Conda for reading

the manuscript and Mrs. Eileen Brackett for typing. Their assistance in thepreparation of the report during my absence is also appreciated.

REFERENCES

AnUOTT. F. R. (l9S2l lA'$ign of oplimum buried R.F. ground $yslem. Proc. I.R.E. 40.114(,·IlS2 (July I.

8ALLANTINE. S. (l929l Rc:ciprocilY in e:lc:ctrom:lgne:tic. mechani~l. acOU5tiC;&!. and inter.connc:cu:d systems. Proc.I.R.E. 17.929-951 (June:).

8EKEfI. G. (19541 The impedance of an antenna above a circular ground plate: laid upon aplane eanh.C~ J. Ph.l·s. 31. 205-222 (March). .

DRLMMt.lI.. Ii. (l9~S) Propagation of electromagnetic waves. Handb/lch drr Ph.l'sik. 16,423-639. [Conlalns an e:ltce:llent summ:lry on milled-path propagation.)

DROWN. G. Ii.. LEWIS. R. F. and EPSTEIN. J. (1937) Ground systems as a factor in antennae:fflcie:ncy, Proc.I.R.r:. 25. 753-787 (June).

CARSWELL. I. and FLAMMER. C. (1957) Ground antenna phase behavior in a differentialphase-me:asurlng syslem. Conl'rntion Rrcord of tM I.R.E. 5. Pan I, 49-56 (March).

CLEMMOW. P. C. C1953l Radio propagation over a flat eanh across a boundary separatingIWO different media. Phil. Trans. Roy. Soc. (London). %46. I-55 (June).

FEINBERG. E. C1946) On the propagation of radio waves along an imperfect surface:. J. Phys.(U.S.S.R.l. 10. 41Cl-4 I8.

GIHlIIf\;G. H. E. and BROWN. G. H. (I935l General considerations of tower antennas forbroad~st use. Proc. I.R.E. 23. 311-356 (April).

LARSEN. T. A nume:n~1 investigation of the problem of the wire grid parallel to lhe interface:belwc:en homogeneous media. submitted to J. Rrsrarch N8S, Radio Propagation,Section D.

MALEY. S. W. and KISG. R. J. (1961} The: impedance ora monopole antenna with a circularconducting-disk ground $ystem on the surface: of a lossy half-space:. J. Rrsrarch NBS.Radio P'.?PDllatlOn. 650, No.2. 183-188 (March-April).

MO:'IITIATll. G. D. C1951} Appli~lion of the compensation the:orem to ce:nain radiation andpropagation I'rC'blems. Proc./.E.E. (London I98. Pan IV. 23-30 (October).

MOST£AnJ. G. D. 11958l The effect of Ihe ground constants. and of Ihe eanh system. onthe performance: of a ve:"I~1 medium wave: aerial. Monograph No. 279R. pp. 1-15.Insi. Elect. Engs. (London) (January). [This paper conlains a valuable and cnti~1

sUI"'e:y of pre:vlOUs work.)

1-88

THE THEORY OF AN ANTENNA 1097

NORTON, K. A. (1957) The propaplion of radio waves over the surface of lbe Elnh,Proc. J.R.E. 25, 1203-1236 (5eplember).

PAGE, H. and MONTEA11I, G. D. (1955) The vertical radiation pauems of medium Wivebroadcasting aerials, Proc. I.E.E. 101, 279-297 (May).

SENIOR. T. B. A. (1956) Radio propagalion over a discontinuity in the earth's electricalproperties, Monograph 192 R. pp. 1-II.lnst. Elccl. Engs. (Londonl(Augusl).

SMl11I, A. N. and DEVANEY. T. E. (1959) Fields in electrically shon ground syslems: anexperimenlal sludy, J. R~s~arch NBS, Radio Propagation 6JD, No. 2, 175-180 (5ep­tember-OCtober).

SO......ER.FELD. A. (1899) Elcclromagnetic waves near wires, Ann. d~r P/:)'sik und Ch~mi~ 67,No.2. 233-290.

WAIT. J. R. (1956) Effect of the ground screen on the field radiated from a monopole.I.R.E. Trans. .o\P-4. 179-181 (April).

WAIT. J. R. (1956) Mixed.path ground WIve propagation. I: Shorl dist3nces. J. Res~arch

N.B.S.57, 1-15(July). (In equation (10), ('1 - l'/1)should be ('11 - ,/).)WAIT, J. R. (1957) Amplitude and phase of the low·frequency ground wave nC3r a coast

line, J. R~search N.B.S. 58. 237-242 (May). (In equations (14), (15). and (18). q shouldbe replaced by q/2; at top of page 240. the second inequality should read (J = {lRu ...: 0and in the tint equation of the apflCndill ('1 - 'II) shoald be replaced by ('11 - '1).)

WAIT. J. R. (1957) Exc:itation of surface waves on conducting, stratified. diclcctric-elad. andcorrugated surfaces. J. Research N.B.S. 59, 365-377 (O<.'CCmben. (In cQuallon (71)(plw) should be (plw)'; on P3gc 376. r, is defined by r, = r, - 2h C.J

WAIT. J. R. 1It)59) On the theory of reftcction from a wire gml parallel to an intcrfa\:cbetween homo~ncous media. Part II. ..1",,/. Sci. Researcll. /1. 7. 355-360.

WAIT. J. R. (1959) C/ert",mtIJrnt'llc R,u/jullon j"",m C.I'/imlrjca/ Structurrs. Pcrgamon PfC'I~.

London and New Ymk. (Inclull..-s a lli!ICusslon lIf slot ralliators on wedges anll hair·planes. I

WAIT. J. R. and SURrrr5. W. J. (1954) Impedance: of .. lort-I<la<kd .~nlcnna 1>, OJrbi"..ryIcnllth over:l cm:ular rounded ~·rcen. J . .••,.,,/. I'III·s. 25, 553-555 (\laYI. (In equations(14) and l(5). SID (1111 - ,.)shoulll ~esIDI" - I//,))

WAIT. J. R. and PorE. W. A. (19S4) The c:haracteristlcs of a \'ertical antenn3 with a rallialconductor llround system. "'"p/. Sci. R~srQrril. D. 4..177-195.

WAIT. J. R. and Porr.. W. A. (1955) Input resistance: of L.F. Unil'Ole aerials Wjrr/rs,Engillrrr 31. 131-138 (May). (In equation (9). IIp should be 1111~J.

Numerical results based on these formulas are available in J. R. Walt and L. C. W.lllers."Influence of a seclor llJ"ound screen on the field of a vertical 3ntenna," NBS MnoolUal'lhNo. 60. 1963. 3vailable from Supt. of Documents. U. S. Gov·t. Printing Office. WashingtonD.C.• price $1.00.

1-89

1-90

1.11 Wait, J. R., and L. C. Walters, April 1963, -Influence of a Sector Ground SCreen onthe Field of a Vertical Antenna,- u.s. National Bureau ojStandards, Monograph 60,pp. 1-11

(Reprint No. 215)

Reprinted with the autho"s permission

1-91

U.S. National Bureau of Standards, Monograph 60 , April 1963

Influence of a Sector Ground Screen on the Field of a Verti<"&1Antenna I

JSJmes R. Wait and Lillie C. Walters

Tht, li.·ld of a ~hort \'prtil'al a/ltt'II/18 on a hOlDol!r/lt'OlI~ I!roUlld i. phowlI 10 Ix- nllldifil·db~' tI..• prt'>'('III'" of" IIwl"lIir "('r....... Th(' ~rr""11 i~ t"k"11 ill Ih.. form of "rir('''''" di~k "Iltl nt'tIll('t·"lrie ",·rlor. Th,· nlOdilienliulI of th.. fi.,ltt i...~pr'·.~'·11 ill thp forln of "urf"pI' illl l 'llral.O\"'r Ih.· di.k R'I<I th., ...·etur. E~tt'II"i\'" IIl1llwrip,,1 ,p.IIII" fur 'h,....· hn."ie in"'Rr"l" ",.' Slinll.IId • IlIlRlbt'r of "lllllieatioll" an' illtl..',a'..",

1. Introduction

IIIII

The illfJuen('r of thp ground 011 the fields oftll1lpnnas has bNon dis<'ussed sporndicnUy in t hI'literAture for 1I11111~' ~'enrs, III mo!!t propn~lltiol1e.-8lrulatiolls it ill nll.~umt'd thnt thr trallslllittill~

antenna hns n lixrd dipolt' 1II01llrllt and t hI' I!roulldis takt>n to bP II prrfe('t ('ondurtor or possibly ahomogt>lIeous illlpcrfef'tl," rondurting hlllf sp",'r.In prartie.-t>, howf'n·r. 801111' kind of ground s.n1trlJlis used, UsuIII"· this tllkrs thr (orlll of II lI\rllllSt"Tet>n or rndini wirr S\"strlJl whi,·h is on thrsurfare or thp ground or mllY hI' burird sli~htl.,·~nellth the surflll·t'. Thr :,rsiJ!n or !;\II·h ~.\'~I('Ins

has I)('f'n t.'"pil·II\1~' l'lIIpiril'lll. A\lpllrrntl.,·, Ihl'first ttnlll.\'ti(·lIll1pprolll'h WIIS ('lIrdl'( lIul hy BrowlIpt· a1. (I}.' J.,ntf'r works (2. al 1III\'p drill I IJIII illl.\·with the in1luell('e of thr ground s\'slf'1II Oil lht'impedane.-e. III mosl rllsPs il hus !,t'pll ussulllrdthat the rlldiated field for a f!i\'en ('urrrllt on thr

antt'IUla was not nPJ>rp("illbl." "fJr("lpd h.,· thp prt's­('JI('P of thl' lUollnd SI'rt'1'11. III rll,'I. an approxi­IIlntl' nlluh'tirul llll'tlJod ,,"lIS ~"I'II Ilr(','iousl.," h~'\Yuil and )'0)11' ',II whi,'" is !""illl"ll' for rslimutill!!t hI' t11'J>t'ndt'lIre of IItt' I!fOlllll1 WII \"1' on IhI' !lilt' of /I

l'ifl'lIlnf rOmIII srr('rll, ( '/11"111111 iOlls 15J IIIISl'd Oilthis WOf' supportpd th.. ('0111 1'111 ion thlll II S('rrrllhilS 0111." II slJIull rtfI'd 011 IhI' rlldillll',1 fil'ld I'ro­\'illt'11 thI' f1\1li\1!.\ of t hI' !lC'fl'l'lI is of tlll' IIrd('r of /I

WII \'rll'lI!!1 h or Irss. \" rr," !"illl illl r "OIl1'ltllliolls hll \"I'

hrrll nrri\'t'd lit h~' Brit ish WOrk('fS Iii, iI.111 this JIIIJ>rr "ollsi,lrflll i'"1 j" I!j\"'11 III j!fOtllltl

!,,"fl'l'lI!'l whi,'h 11111.'" hI' IIII"!:" ill ' ....III!'l Ilr II \\'11\""1t'1Ij!1 h. Silll'" t hI' Ihrof.'" hll~ 1""'11 If('11 I1't1 lfuilt·I!l'nrrllHy illn prp\'iolls l'"pl'r 18). IIllrlllioll will hrfOI'u,,('d hrrp Oil t hI' 1I111111'ric'II1 ("IIk"l" IiOlls lind thI'"rt'dil'll'd pt'rfoflllll 111'1'.

2. Formulation and Description of Problem

(J)

The situation isdt"Sl'riht'd I\S followlI. A "I'fti,'"lrlrrtril' dipolr iR 10('11II'd 011 II fI'lt hOmll!!"II"""!!~olllld of rond\l"1 i\"il~' 11 1111(1 t1il'!t'c'l ric' "Ollll!" III f.

The vertical t'let:tric field E I1t It distan('t' Ill) undelevation angle "'0 is gi\'ell as follo\\"s

E,_i;:;;:; e-.tlt, ('os~"'oJr(llo,Z)

where

jrrOulltl ,'o"d"rt i"jt ." Ill)

lnU,,7.)~I-i(7f'J~')"'· """I"fl' (ill',',:)

"'''rrrU'o= II +('1,IZ) sin tJofJIo.

Po=:- iHio (?;,')~2 'In.

IIl1d

(:il»

"o=4Il'X )0-7

w=angular (re9uen('~'l=effet'th'e ht'lghl of trltllslllitting dipolrI =current at terminuls or trllnslllittinlt dipolek=2../wBvelellgth,

In the aoo"e. 11"(00 , Z) is /I romplt'x qUllntit ,.whie.-h is a rUIl('t iOIl or I hp surf"rt' illlpednn('(' 7. ofIhe ,round. O\'f'r II pf'rfp('tl~' ('ondul't in, f!roulld,Jr "'ould Ilpproll('h ullit,\". III tht' ('nse of lillitt'

• This ."ark .at .1JOfIfO' It)· 11M' £1<-<,...,,1,.. lI_h IJir"·lor.1f 0111",AI' f_. Can.I"ltl••· II "" La&hOf~tOl.". lIlf..·••, AfIOoSI'- 1Il'tO'''',hCUIU.·J, , ....."".,. III..,., U,,"'" ,0"""'1 1·IIU-4.-5I"o4

.......,... In "''''''10 ....In.. II.. 11I,.,nlu(l' ..I_n.... on _ n

'10= 1207r.

This result is "alid for Ie Ro>> 1 Alit! IZ/'1ol~< < I,If

(4)

t hI' I'xprrs!!'ion for II' roilll'idl's ,'xllrll~' wil h I h,'rl'!"ull gi"rll h~' !\ortoll(1lI1 for Ihl' Sllllll' lIilualion.JI 11111\' br IIOIl'd Ihllt Ihill \"IIluI' of 7. i!: I'xIII'lh'1''1,,"1'10 thl' ratio of IIII' tllllgl'lIlilll 1·1(·,'lril' IIl1ilIIInglll'lir firl,11I for II "l'rli"I1I1," polari1.I'd pllllll'wa\'p illl'idrllt at lin angll' 9()O-",o Oil thl' hOlllo­~I'III'OU!l flat ~roun,1.

1-92

whc'rc', h.,' dc'fillitioll,

,r,= (1 +1l,(1J'0>J/2(l'()l1"(R. Z. Z')=Jr(R. Z) II +ll)

!!~_ i~' r.f,-f.,r- ItIR - H.' (?'_-Z)2,.. ('OS '1o. • '1~

s

111 a prr\'ious f"prr (M) all jlltr~r,,1 ('(Iuation forJr' "'as obtaillt'c bv an applic·"tion of t Ie Lorrlltzrrriprodt," th('orf'ni. Although it would b(' POll­siblr to soh'(' this t'quation dirr('tly using Il digitnl('omputrr it was indif'lltr,1 thnl 1\ firllt ordrr itrrll­tioll WftS sntisfllc,tof\', 111 thill C'lIl1r it was foullclthnt '

/l =IpZ+"1+/,z_2prl ('fle; !p1":

1l~=(d1+hf)ll2. h=ll~ sill l/-n

whrrr II is thr frn('tionnl C'hftll~(' of Ihr fir'" Ilur tothl' prl'lIrlU'1' or thr I'lI'rrrll. Wilhin thr np"roxi­Infttlollll 1It11t('d. thr fll,·tor 11 c'l\n Ill' rt'~lIrdc'II II"

thr modificfttion or th(' rfff'(,ti\"(' Iwighl of thetransmitting ant('nllft. sill(,(, it illflurllC'l'S Jr, alit!W. to thr snlll(' (':((('nl. ., Bf'ror(' pro('rrdin!! furl hrr it ill ,'on"rnirnl IIIIIltrtllhlC'(, II polllr ('oordilllllc lwslrlll (p, 41) Cl'II­t('rt't! al thr sOIJr('e dipol(' 1111 incli('lIt('d in fi~url' I.ThUll 1111 ('Ielllent of IIrell of t h(' !!roulld plllll(, ispd41 "p.

From 'hr "",,Iysill ill '11f' I"'f'\'iflll" 1"'1)('" hy"'"il lSI-it WIIS shown 'hIll

wll('1'1'

nllli

(51

(6)Il lIill ",o-Z;'Io• sill "'o+Z/'Io

is t.hr rltdi"tion or sp,,('r \\"Il \'r firM, alI<I

111 appli('lItiolls to prn"tii'lIl rollllllUlli('otiollprohlrllls it is wry ('OIlHlli"lIt 10 split ofT thl'surfo('l' wavr portioll II'. h," writing

ill th(' Frrsnrl r,'f1r<'lioll ('Orffi('irlll. This dr"olll­positioll of thr lotlll fil'lcl illto "pa('r olld lIurflll'rwave was first madl' b~' !\ortoll (10/ alld il is a ('011­vl'nip.nt pro('t'dur(' ill radio t'llginrrrilllt sill(,(, hydt'finition, Jr, ill thr radiation p"ttrrll of thr 011­tt'nna in thC' prl'!'('IWI' or t Ill' ~round plnl\l'. Itma:,' br thl' dOlllillllllt trrlll in 1111111." ('aSI'll of prtlC'­ti('al intt'rt'St although "s "'~ approa('h('!l z('ro 11',a('tuaU:,' vanishes. J\I('thods for ('Stilllatin~ thc'rt'latin' impcrtan('C' or Ir. ar(' gi"C'n in thr pllpf'N'b:,' Norton (10/.

ThC' ('C'ntral tllsk ill tl\l' Im'!!I'lIt 11I\Iwr i!' to ill­di(,lItt' how a wirr lIIl'!lh or II lIilllilnr IIIrtnl !,"r(,l'1I1~'j1lF on thC' ground will 1II0,lif~' thr firltl al thrrt'('t'I\'inJ1; "ntC'nnn. Thr lIUrfll"" illlprdnnf'r i!' n"­sUllied to b(' lIIotlifi"11 to 7.' O"f'r IhI' Ufl'n or t h.,scr('t'n, but rl'maills thr SIIIIIC' oUllli,I,· th(' 1II'\'('l'lI.

Tht' field E; in the presenc(' of th(' s('rrrll is t!lrllwriuNI

wh('r(' 11" is all unkllown 1'01111'11':'\ Ilunllt il~' whi.'his now a fUl\{'tion of 7.' ill ndclil iOIl to /l~ nlltl 7..Thr (IUllntit~· Jr' redu('f'S to Jr if 7.' = 7.:

E·, i~1 -ftR ",f 11"(/l Z Z'.•= 2"/l; , 0 "oS' ,.~ ~. " • I. Ii)Thr intC'~r,,1 1II11~' h(' r"IIII1"IC',1 whl'lI Ih(' !':hlll'f'

of thl' !!rolllid lIc'r(,(,1l is lIpl'f'ifi(',1. III til(' followin~,"ttl'lIliflll will h(' C'ollfill('11 '0 l;\'TI'I'IIe; ",hi,'h ""1' illthr fflrlll of "lIl'c'lflr. A !'tl"·(·jlll I'"sr jc; " I,jl'('lIl"r(:('rr('n nncl t his is l'of1silll'r('f1 fir!':t.

O"t'r the range O<p<a it il; nssullled thnt th('surface impeden('e is Z'=7.~. Bt'.'"ond tht' s('ret'n(i.e., p>a). Z'=Z. Furtht'rlllore, it is nssulllt'dthnt thl'! rect'i\'injZ antt'1II111 is in the fnr fit'lcI su('hthllt

3. The Circular Screen (' -< i f

~, f.0 i I )no~--.-, "t, (I +:1 J.(~·p "OS "'0)('OS,.o, ,-0 l"'p

e~;:;::Z)dp. (J I)

wh('rt' J, is thl' Bt'SSpl function of 'ht' first ty pe ofordpr onp. Wh('n dt'nlinl! wit h III'1!e screens thpn~ulII('nt kp cos "'0 cnn h(' r('Jrnrd('cI liS n IlItJrl'"unl1tit~· O\'('r tht' IIInjor portiolls of th(' illl('~rnnd.

Thus. J. mn~' bt' r('plnct'cI h~' t he first terlll of itsnS)'mptotic t'xpnnsion. Thert'fore,

Thus

i~' J:0 J:'II -- ,- ••,0- 2" cos "'0 ,.0 •__ ,

X(I +_.1_), .•,....... ~U('os 41 (Z;-Z)"41 dp. (III)lkp '10

If 7.~ dOt'S nol c1t'pl'ncl on 41 th(' int~rntion withrt'Spt'ct to 4111111." hI' r(,llllil.,· ('lIrrit'd Ollt to Jri,·('

i . I"i to Z'-zp -- -'-'0= (2,.. ('OS3 ",) 0 ( 'I" )

t - f,I1-('0I '10'

X(I-it-'''·... 40) ----,",-dr.r "

(12)

1-93

where

G=i (2ka)"' [I-i (_!:_ )"1 ("A/,"I!,.. ·U'a .0

I'Xp l-i(1I'/2)t'W] (I:lh)

When Z~ is essenliaJl.\· ('onstant on'r thp rllll~p ofint~mtion, n. ('all hI' I'xpr('1\s('d ill tl'rlll!! ofJt~resnt'l intl'f,!rllls, Aft('r II dllUl!!p of \'lIrillhle itreadil,r follow thllt

u.~Z-.Z~ ,-,.HG'I.

G= ., i, _" [ I J'UA.toI1l11Ift '4.'"(21'093 oJ-o)'" sin (oJ-II/2) •

, (e.","' roo C4,1"exp l-i(,../2)t'~It-_1_,, .

, I'OS (oj-., 2) • 0

I'Xp l-i(1I'/21f'Ht] (1:111)

As oJ-o I\pprol\ches zero thp IIbon' equlltion rpduresto

Fllr!lUrfRI'I' illlpl'dnlll'l'!l of thI' ~ro"lId \1111111',this Imrpo!ll' it ill l'Oll\'l'lIil'lIt 10 wrill'

Z-Z;_2.. r"'I. -,\'

",hl'rl' S Rnd fJ arl' rl'lIl, If till' ~roUlIII Sl'rl'l'lI i!l II

"'1'1111 !lhl'I,t 7.~ ...., <Z 81111. l'O"!lI"III,'"lly, 1\',.- 0'('ollid hI' r"Ir"rdl',1 n!l Ih" ,'ollll'l"x rc,fr'll'li\'1' i'llll':'\of thl' ~rotJI\(1 ilsl,Jr, lIowl'\"I'r, ill g"III'rul, I\' 11111113 hll\',. 8 1II0rl' ~l'lll'rnlllll'"11ill~ 11'\ 1II'Ii'Il'II hy (I!i).Tnkillg ,\'=:1 anll ~=II°, th,. 1I111\1lilllll,. Illld phll!'I'of I ~!!. arl' shown plott I'll in fi~ItrI'S 2n lind 211,r('!'pl'l'ti\'('I~', as R funl'tion of 1/.0 for "Ilrioll!' \"lIluI'!<of ~'l1. It is I'lIIphn'\i1.(,tI thlll l<IlI'h I'IIr\'l''\ sholll,11I0t hI' n'~lIrtll'll n!' rudilll ion pllttl"'I1'\ hili rill h,'rIl!' lIIu,lilil'lltion!' of tllf' l'If.",tin' hl'ilrht of till'trnll!llllittillg Rntl'nnll dUI' 10 till' J.!round !<I'rI'l'lI.It i'\ Ilppllrl'nt thnl for thl' 1o\\, 1I11!,dl''' ill \"01 \'('11 inII F "ullllllllni,'lItion thl' Irrolllul !l,'n'.'n will inl'rt'lISI'Ih,' ,'If.'din' IlI'i J.:h I of Ih.' 11'1I11"lIIittilll! 111111'111111h.\' II !'ilr"ilil'lIl1t 1l1ll01l1l1. TIll' nihil' IIf .\' (rin'lIill Ihi!' I'xlllllpll' ,'orn'!<polld" 10 II Ilil'l,.dri" ('011­!lIIIIII of:F or 9 whil'lI i!' t~'pi"111 of HI'Y Il'y ground,Thl' I'IfN'I of ('hoOf:illg II IlIrlrl' nllul' of .\" is showlIin figllrl'!il all IIl1d ah whl'rt,:\"= 10 IIl1d 8=11°, Th",'urns nrl' \'I'ry !'illlilllr in !lhn/,I' hili IIIl' o\"l'rullI'lf1'('1 i\'I'III'!'!' of till' Irroll 1111 !','rl'l'n i'\ r,.tI IIl'l'tI!'OlllI'\\'hll I ,

TIll' \'11111" of d, "" ,lc-lillt',1 11\' (1.'i I, ,1.'I"I'I"i'If''''Ihl' /,1111"" of IIII' "III"plo-:o, n'fl'l;,'li,1' illd.'x of IIIf'~rolJlJ'1. For /I ""I'y dry or 1I0IWOIle!1Il'l inl! IrroUluldis 1.l'ro Il!il i'lllic'IIIt'd in li~lIre'!< 21l 10 :l". 1101\'''''1'1',\\'1"," thl' l'Olullll'li\"il~' 111"'01111'" il"/,01'111111 B 11111,"

IIf' J.!I'I'III,'r Ihllll 1."1'0. In flll'l, fill' II highly "1111­IIII ... i'l!! r.rrollllli wlll'l'I' di"plllf"'III1'111 1'11""1'111" 111'1'1I1'l!liJ.!ihll', d 11111.\' 1I\11,,'olll'h 4;;°, 'I'll ilh."'1 I'll II' Ih,·inlhll'lwl' of fillill' d, till' 1III1I'Ii'"d., Illld phll'\I' of1.L !!. lin' !'howlI ill Ii(rlll'l'" 411 lind 4h for hI =20,.\"=-10. IIncl \"lIrioll" \'11111"" of fl 1",lw""1I 0° IIwl4,;", II i" ('\'idl'lll fmlll ,hI''''' ('111'\"'''' ,hili IIIl'pl"l'''','nl''' of thI' 1'011111...1jlln ,'urn'lIt" •l'lul" 10dilllini!lh thl' Illllplilutll' bill i, IIOI'!! in"n'lIsl' thl'Jlhll!l",

It i!l hl'('olllin(r IIp/,lIn'II' thlll Ilt IIII' low"r fn'·'IUl'lwi,'!' /lnd highl~' l'Olllllll't inlr grollntl thI' prl'!il­I'n,'1' of thl' ~roUlIII !,"rt'I'II 11lI!il II '\1111111 ('lfl"'1 011 thl'lolnl firM (for II gi\'l'II !illr,'n~lh II of thl' SOUlT('dipol.,1. To mll!ll rtlt I' t hi!il Jloillt. IhI' 11111 pliltldl'Illlli \lhll"" of 1+11. lire' !lhown ill fil!url''\ 511 Illll.l .'ihfor ko=211 lind 8=45° for X=1tI lind :ill. '1'111'1II0difi('ntion of th(' dfl',{Oti\"1' h('ighl of thl' nnll'nnlli!' I"!l" thnll 2 ,Ib nne! hl'rl' thl' rntliu" of Ihl' '\l'rl'l'lIis nllllost :1 wlI\"I'I('II~tll'\,

(\4),(2ka)'1i2[ ('ill')"j ,(2~'a)I'2G:::1- 1-' , ''''.:1' ',.. S~'" 11'

lind, if ko>>I, this IIIl1y hI' 1I\1\1rOXilllllt('d h,'"

It is interesting to 1I0h' tl\llt. if th(' intl'grlll in(12) is pvaluntPd h~' n stntionllr.\" phnsl' lII('thod.thl' 81'1'0nd Fr<'8nl'l intl'lrrnl in thl' !l'IUllrl' hrn,",,"1tl'rlll of (lau) is 1I0t pr"!lI'IIt. This would ,'om'­sponll to th(' IIpproxinllllioli u!lulIlI,\- ('lIIplo~'I'cl illthe prlll'til'al th('ori<'8 of lIIixI'd-pllth groulld WII\'I'propllgntion. Thl' \"Illul' of G l'orr('spo,"linlr tothill SitUlltioll is 1II'II0tl'd G(I),

I'tJllIl'ril'lIl vlllul's of IIII' illl.'!!rlll!l G 111111 r., I)ur(' giV('1I ill t111J11' I, Thl' \"1I1U1'S of ~'a (d"IIOIt',1KA) tnk(' th(' \'/llu<'8 5, 10, 20, :W, IIl1d 1110, whill'oJ-o (d('noted P.-;I) rUlls frolll 0° to 45°, It i!l illl­III('d ill t('1\' ('\'id('lIt thllt, for slll,,11 ,'"Iu('s of oJ-o(i,I'" n('/ir grllzillgl. thl' illtl'grnls G IIl1d G( I) 8rl'1I0t sigllifi('llntl~' different. As will b(' c-I('nr frolllthp followillg s('I,tioll thl' illt('grlll G(I) wouldI'orr('!polld p!J,'sil'lIl1y to thl' situation wh('rl' thl'

. srr('en is sl'lIIi"irc'ular ill Shllpl' (i.r" • ('xtl'lIds frolll,../2 to -,../2 onl~'),

To iIIustrnt(' thl' IIpplil'ntion of thl' r('suIts illtubl(' I, \'nlul's of thl' cOlllpll'x qURlltil,\' 1+0,hu\'1' bl'l'n (,olllput('d for SI'\'('rul nlul's of thl'

4. The Sector Screen11 is dl'nr frolll thl' prl'\'ious rrsult!l thnt n IlIr~1'

"irl'ulllr groll III I !lrrl'l'n will. inlll',.,1. illlpro\",. IhI'low angll' rudi/lt iOIl frolll II IrroulI,l-bllSl'd \'1'1'1 i"111nntpnnll, How('\'I'r, on(' lIIighl IIsk if IIII~' portiollsof thl' cirrulllr ~roulIJ 5,'rl'I'II ,'ould hI' r('lIIo\'l'tIwithout lIIotl'rilllly nlfl'!'IillJ.! Ihl' pl'rfOnlllllll'1' ofthl' 1l~'lltl'lII, This is I'l'rtoinl~' Il \"alid (IUI'Slion.

III Ihl' first plnl'l' it i'\ kllo\\,nj.'jl thllt Illl' illlJll'tllllI"1'of till' Illltl'nllll is nol nlff'c,It'.1 b,'- 1111,\'1 hill(r hl'yolle!Ilboul olll'-hllJr \\'1l\'I'II'II~th frOlIl IIII' 1l1l1,.I1I1I1,Thl'rl'forl', to tbro\\' SOIllI' liJ~ht 011 Ihl' qUl'sliollI'O"l'tI Il ho\"1', thp I!:rountl !ll'rl'l'lI i'\ IlIkl'lI 10 brillth,. forlll of 8 !<I'I'tor I'Xtl'lIllill~ frolll p=a 10 p=bfrolll thl' bllsl' of thl' trtlll!llllittlllg 111111'111111. I'rolll

1-94

(21 )

(211)(1 )IIIJ" f"(T)G.=i .'- --. ,'dr.

21r a. Zlt2

('XP (-i~'a~~/2)')J}+---- .2~1

The int~rlll G. hll!! hp..11 CVl1luutl'd for 11 rull~Cof nIhIl'S of A'b. To silllplif~' th(' sitlllltion, tll('low('r limit kn is 'ixrt! I1t !i 111111 AI ~= A~"" A. 'I'll ..II II III ('ri(,111 r~slllt!! for G. Itlrnot('(1 (i(Bllllr(' l!iHII

ill tllbl~ 2 to 8 for ~ IDELTAI fllnj!illg frolll 5° tofiOo. Within 1'111'1t tl1l1l(' A,b IKBI \'Ilril'S frolll 10 to11111 IIncllln (pSI) \'Ilri(~ from lIO to 4;'°.

Ae; 11 rhl'l'k on thp 1l1ll1l..ril'1I1 work. a. for lI-n=lI\\"I1S colcuhlt('d lIsin!! hoth (19) I1l1d (21). Also, itml1~' be noted tl1l1t

Ah~r nn integrlltion b~" pllrts it rl'lIdil~" followsthnt

Th(' Frl'Snl'1 inl ..~ral F(z) is normoli7...d 1'10 thlltlim F(z)= I, pro\'id('d AI snd ~7 arl' bOlh posi-.....tin.. III this lin.liting ('nsl' till' lll'~'tor is bl'hll\'illgI'!l!'l'ntlftll~' ss s rlrl'ulsr srr....II. For (·xllmpl.., Olll!mo~' not .. thnt

\\"h..rl' GI is tht' intl'grnl dl'llrrihl'd h.," omil till!! till'st'rond t..rill of (I ao).

An intl'rl'Sting llpl'l"iul ('II~1' of (19) i~ WIIl'II

~n-+O snd Z~ ('sn bp rt'Jtftrt!l't! liS 0 I'onstllllt..Thl'n

whl're the \'"Iul'!l of G1 (r) I1rl' li!!tl'() in ll1hle 1 1111t!",ht'rl' r is to hI' idl'ntifipd with KA.

To iIIu!ltrl1tl' thl' p'f..l'l of 11 fillit(' Vl1hll' of A.801I1t' t~'pir"ll'nsl'S nrl' shown in fi~lIr~s ill lint! it,",herl' thl' al1lplitlldp nnd phns(' 01 G. I1re plott('c1as 11 fllnction of '10 for ko=5, kb=40, and \'IlriollS\'allll'S of~. It Ilppl'nrs thnt for thl'S(, conditioll!!the tolal sector IlnJtle 2~ npt>d not bl' ~renter t.hUllKbollt 50° in ordl'r to bt> fulh' l'ffl'('ti\"e.

In onlpr to dl'mon!!trntp tlip ..ffl'rt of tltl' ll('('toron tht> totftl fil'ld it is I'om·pni..nt to I'onsitl('r bothZ'. snd zr. small compnr(·t! with Z. Titus

Z-Z~_Z-Z;_~_.!.. e"110 - 11o = lin - 1\"

(16)

[ (.. 'Ud )"] 1.. .lIf. 1+,.-" ".... _ -I - - __<.'I. n: R. 4

11 _Z-Z; -, 11f'• ---, • Ut·

lin

wht'r(' U. is th(' t'Ontributioll from th.. ('ir('ulor8crp~n of radius a ond U. is th(' rontribution fromthl' Sl'rtor whirh ('xtl'nds from a to b. Th('poriion n. ('811 b(' writt('n

• Till. ,.,·flpld ..","Illon .... hr ..,It Irn

(1+"'!-) e'·'·... - .... 40Xros. (·Z;-Z) d~p (Ii)~p ~

wherl' tbl' r('('('i\'illt-; ont('nnll is IIssum..d to h.. inthe plane .=0. rJ ht' int..~rnl for n. ~i\· ..11 aho\'('is sufficil.'Jltl~· g('lI..rlll to d('t ..rmin(' th.. t'UN"t ofth(' sector us II fun(,tion of 1·11·\'lItion 111111 117.imuthangit'. Actulllly, th(' int ..grnl is unolo~oll!; to (lfI)for the t"irrular 8('r('('n wh('r.. t hI' limits of • ('xtNItIfrom -'" to "'. As b('for.., ollly first-ord('r pho!'oterms are rl'tain~d so t IlIIt thl' rl'('I,i\'inl! ant ..m",must b(' in th.. fur fi(·ld. 3 Tht' ('xt('lIsion to till!n~ar-fi~ld case has hp('n ('Ollsiclt'.r('d pr('\'iou!'ly b.\rWait (8/, In actual rOlJllJluni('ntion rirl'uil!' therereiving nntl'nna would alwRYs h.. in th.. fsr fidel.

To l'vsluott' th~ intl'l!rnl in (Ii) it ill ('on\'l'nil'nt.to use the approximation .7

I'OS .=1-'2

p=O out to p=a thf l!l('rl'l'n is ('ircular in shopI'.Th~ situation is iIIustrlltl'd ill fi~url' G. Th('surfar(' unp('dom'p o\'pr. t h(' arpo of t b(' s('('tor isX.. In tprllls of tilt' polllr ('oordinatl' !I~·!'tl'lJI

(p, .), thp ar('o of thl' s('('tor is dt'fillPd h~· -~I<

.<~2 IUld n<p<b.It. is ('on\'I'nil'nt to I'x"rl~s th(' flll'lor U "II lhl!

8Ulll of t.wo pllrts in th(· llIallll('r

for till' ~xpOlll'nt in th.. inh'!!rIlllll whill', ill Ih..int('grund, ('os • is r('plal('('11 hy unit.". This i,.valid sinr(' th(' principal I'ontribulions corrrsponclto smnll \'alul'S of.. An int~rt'sting l'I,t'(,k ont.his slat~llIent is gi\'pn hl'low.

FollowinJt tht' pror('dur(' us..d in th(' prl'\'ioussertion, 8 dimensionl~ funrtion G. is introc!u('t'db)' setting

Thl' intl'gral for G. llIa~' now be writtt'n ill tht'foml

i ia. e-,,(1 -.... '0' .G.=(2 )1" 3/~ "' f(z)dz (19)1r cos 0.. z

whl'rl'1 f~lll'l .....011/~ [ ]

F(z)=I_i exp -i'!.t' nt.-~IIo"'1 .... fo.I1I2 2

1·95

ground, Consrqut'ntly, it follows frolll (16). that

z-z; Z-Z; Z I ,.----- ~-"--''10 -= '10 = 'I" = l\' .

whirh is an ob\"ious grnrrRli1.ation of ()~).

To illustrfttr thl' azimuthnl \'nrintion of tl...fil'ld ",hl'n using a sl'ctor it is agllin dl'sirn [,11' towritp

With this infonnation it is ft simpll' IIIftttrr toromputl.' (1 +0> as a functioll of the azimuthallglt' & which is dt'finpd b,"

:\uml'ricnl \'lIlul'S of G.(~, ~) or (,'.(~) for vnriouspositiH' values of 6 IU(' gi\"l'l\ ill tnllirs 2 to t.4IIIclusi\"e. If nrftatin· valurs of 6 nrr rnrountrrrclit is uSl.'ful to note that

Thus

1+ n.+ n.=- 1+Z-Z; G.e-··14

'10

Thl'1I ngllin dl'lIotillg thl' 101111 width of thl' SI'('lorb~' 2.1, tht' amplilu«!rs of I +n.+n. nrl' showlI infigUTI'S 1\ anc\ 12 for :\ =3. B=O. ka=5. 6=21)°.IIl1d \"arious "olul's of !/t" from 11° 10 25°. IIIfigurl' 11. kb=40 whl'rl'as in fi~lIrl' 12. kb=2110.,\'1 ('XpN'II'r!, tlll' n1a:l:imum rl'SpollSI' ('orrl'spolllls10 smull \"Rlurs or &. In fll(·t, RS & illrrl'usC's Ihl'rl'spolI!'1' dl'rrrnsl'!! quill' signifil'nntly for thl'IRrgl'r Ill·rlor.

Tht' amplitude of this qUlllltity (e:'(prl'SS('d ill db)and the phase art' shown ill figures ~a and sb forka=5, kb=40, {j=0, and l\'=3. This would ror­respond to a rt'lath,t'I~' dr~' soil. It is c('rtailllyt'vident here thnt cOllsidrrllbl(' impro\'t'mrnt r('­sulls from the prescllc(' of th(' srctor. The cor­responding set of curves shown in figurt's 9a and9b are for a highly rondurting soil charact('riz('dby N= 10 and tJ=45°. The s('ctor screen b('rr hasa negligible effect on thl' pl'rformance of th(' !l\"S­

tern. In fact, there is en-n a slight d('gradntionfor the very low ~razing angles.

The marked unpro\'ement b~" usiag a lur~esector screen on a dr\" ground is indicated IIIfigure 10, Herr. ka=S:kb=200, N=3, and tJ=o.At low angles thc gain is grl'ntl'r thall 12 db PHilwith a total sector an~l(', 26. of 20°.

In the preceding discussion it has b('('n tll<'itl~'asswned that the r('c('i\"ill~ nntpnna is lorntrl! inthe vertiral plan(' whirh blSl'l'tS thr s(,l'tor. :\or­mal1~', this would b(' tht' optimulII 10l'ntion nndfor a fixed communicution link it would h(' COII­

sidered~ practicp to ori('nt thr Sl'ctor towardthp recelvin~ p.ntenna. Howr\"rr. thrrr IIIny hI'certain apphrations wher(' the rrrl·i\"illl:! nil t('1lI111is locatpd off tht' centt'r line. Thr forlllullls gi,'rllaooV(' are actualh' \'alid for this cns(' sinl'r 6. nlld6 2 may take an~' positi...e OT negl\th'l' '-n\ul'. Bow­ever. rather thun computing (nrel·tl~' from Ih('general fonnulas, it is desifllblr to rstublish !l01ll1'

simplp idt'ntities which enable the rt'sults in tllblrs2 to 8 to be used.

It 1II1l\" br rendil\" vl'rifi('d tllllt (;.(6" ~2). liSdefinrd b.y (18), has'thr following propl'rt,\"

5. Final Remarks

(24)

(23)

i'llli dZ. ==A; log, 21fc'

whert·

surfact' impl'danrc X, of tht' equinlenl grid al1llthe ground beneath, Thus

Z'- Z,Z-Z+Z.

Z == (iflGllll~ )112•

and d is the spacing bt'twepn the radial conductorsand c is thp radius of thl' wires. Such a formulllis strirtly \'Slid onl~' if (trlolow)'d< < I evrf\'wherrwithin the ground !I\'Stern. If tberl' are N radialrondurtors. it ran b'p sppn thnt d <'8n br rrplarl'r!hy 2'rp1N whpre N is usuBll~' of thp orl!rr of 100.• A¥ftllal.tf" from ~Irl'l. T. '.Burn. l.ralllO'.IOf)· 0' f:lfttrotnU'ftf'llr Th4'Ofy

Trrhnkttl t'nlyM'-")' 0' •"'nmll", ('o''''nl1",'o

In thp prespnl stud,', thr l'le('tril'al propt'rtirsof the ground art' asSumed to bp characterizedb~' a surfacp imppdan('l' which is a (complrx)constant Z oUlsidl' R 8urfnre S. Within S, thrimpedance Z' is allowed to bl' \'ariable. In tht'case of a radial 9t;re s~'stt'm rmlSnating from Q. itis appropriate to ust' fomlUlas which ha\"e brt'ndl'\'eloprd for thp surfarl' of a wire grid in tht'inlerfare of a cOlldurtillg half spare 1111. Ingpnpral thesr arp complil'nll'd. but rt'ct'nll~" soml'numeri('al results ha\'1' bppn obtained which shouldbe uSt'ful in this probll'lll.' At low radiofrl'qupn­(,it's for modpratt'I~' or \\"I'II-ronducting sotls itis II sutisfartory upproximntion to r('gartl thl'surface X' as thl' parallel combination of thl'

S

1-96

(29)

[ - b a ('f(r)nr=--:; g(?/)d!l,• • .. • -I

[

• ft

f(z)tiz=L; IJJ(a,HE•.• • J-I

With n prol'l'llnrl' for I','ulnntill~ thl' Frl'~11I'1

intl'¥:rnl, thl' rl'lIIuining prohlt'lII WIIS to ('olllJlulI'thr mtegral in (19). Thl' lJll'thod 1I111'd, Gaussiunquadrature, is desrribell brirfl~' brlow (17).

In q.uadrature mrthods R drfinilr int('grnl isaPllro~llJlat('d by R wl'iJ:!ht('l\ sum or pllrlicular"a urll or thl' ordilllltl' wilh thl' nb~I'il\l\US IlroPl'rh'dislribulrd in thr limits or illll'~rntion. • 'hus, .

whel'l'

Th(' ,absrisslls aJ. arr roots of thl' Ilt'grndrc pol~'­nOllllals, the wrIghts IlJ arl' hlJll'tiOll!\ or thl'!'I'roots, IIl1d E. is thr rrror tl'rm whil,h 1'llII, ill~l'nl'rIll, bl' mndl' urhitrllril." IImull with illl'rI'n!'in~1/. Th(' Gllussian root'l IIl1d wPights ur~' tnbulnh'llfor ,'nrious n for limits hl'tWI'l'II -I nnd I 1)\'Dn\'is and Rllhinowilz (lSI, hut otlll'r limits ('ni,bl' used by a rhange or \"ftrillblr as follows:

Flirt h('rmOfl' , in Ihl' (jnu!'!'illll Ilnlltlrntllrl' pro­('('uure, thr int('grnnd is npproxiUlntl'" by IIpol~'nomiftl or C2n -I) dr~rl'r whi('h hns I hI' ~111111'

ordinntl's liS thl' rllnl'lioll ror 1/ disl'rl'lI' IIhSI,j'l'lIl';.To obtain arrurury for G. ('II (19) WIlS writ \rn

i [iiO t-lIl1-'OfI~OIF(r)G.= - - dr

" 21f ('OS'/2 !/.-O 3 , r

iu f-lIl1-'UI~ol]o'(r)+ dr . ..

10 "z

i .. t-IIC1-,o.~olf'r) ]+ _, drU-3 "Z

(25)

(26)

(2s)

(2;)

and Gaussian qUlldrature was usrd with n=16 ill('I) (29) for ('nrh intrr\'ol of 5 for H. This work\\'88 rhr('k('d n~aillst (21) for !/.-0=0 l\IId ,'arions,'alu('s of ~I Rnd ~2. The answers agreed to th('fin digits asked for in lhl' rrsults.

7. References

I,t-II

f(z) = ---.::---: tit·o ,,21ft

6. Appendix

6.1. Evaluation of the Fresnel Intetral 6.:Z. Evaluation of G. by GaussianQuadrature

The intt'J!rnls orcllrrillJ! ill G (laa), (l3b), al\(I]o'(z) , (19) lire of the t.ype

Thesp Fresnel integrnl':: were 1'\"Il1uuted "." themethod proposed 1)\' BOl'rslllu (121. This ml'thodis based Oil thl' T illet hod of Lnllczos (131· TheFresnel intefrul defined h.\' Boersma is

The definition in (25) conforms to the onP. used ii,'Boersma (14\ in eq (26) if '

The '>Ower series in !, \'ulid for z O!: 4 is.. z -. I-i r 11 4 •

j(z)= -2-+t -U '1 i ~ (c.+ id.) (z) .

For values of the a~ulllellt O~z~4 in (26). f(r)is computed b~' a finitl' power series in z; for \'"Iursof thl' argul~lent z ~ 4, flz) is upproxilllated h.\' upolynomial In liz. For 1/= 12, the power seriesIn z valid for 0 :5z:i 4 is

j(z):u- 1Z .J~ ~ (a.+ib.) Gr

The numericnl vulurs or the roefficient!! a., b., ('.,and d. as developed by Boersma (14, 121 lire gi\'enin table 9, With these coeflicients t he Fresnelintegrals can be computed o\'rr thl' rllllge O:! z:5 ....in I!'enerlll, to eighl decilllul points. The suh­routine used in e"nlulIlilll!' the Frt'Snel intel,!rlllwas checked with tlu' tfthlt'S of Pearc" (151 andthose of Wijnguarden lind Seheen '(161. Thefornler tables, usin~ dl'fillilion (26), arl' neeurnteto six or seven diJrits dprll'IIdill~ on thl' sizl' of thellrgunlfmt while the hitler, uSlllg definition (2.'»,lire accurate to fi\'e digils.

/II G. H. Brown. It. F. LI'\\i~. and J. EJl!'I('in. GroundeYl'tcml' as a fori or in onl('nna l"fficil'nc~', Proc.IRE 21, i53-i8; (Junl' 193;).

12) F. R. Abbott, D«'Silln or ol'lilllllm burit'd R. F. groulldIIyell"ms. Proc. lin: f •• 841)..8:;2 (Jul~' 19:;2).

13J J. n. Wait allrl W. A. 1'011". 1111'111 reistanr(' of LFuniJ'lOlc a('rial.. Wirt·.. .,.~ F.lIllinei'r 32. 131-138(Ma~' 1955). (11l1'q (!I) •• '., .hould hI' I.. "'>

141 J. n. Wait I\nd W. A. Po!>". Thl' characIl'rlstil"!' of II

\'crtical .ntcnnll ,,-ith 1\ mdilll rOlldurtor Ilrounrle,·lItl"m. Appl. !'ci. H,'!u'lIrch IUlf, I;i-I\I.~ (1954'.

151 J. n. Wail, Efl'l'ct of thl' ground errt'l'n on the fit·ldndiatl'd from a monopoll', litE TrailS. 011 Antennasand Propagation AP-f, li~181 (April 19561.

16/ H. PII@:C and G. U. Montl'ath. Th(' "('rtil'lIl radiolionplltt('rn~ of ml'dilllll "'a"(' hrollrlcl\.!ltinp; III·rillls.Pror. lEE lOt. 2;9-29i (:\III~' In:;;",).

iiI (i. D. ~IOllh·l\lh. Thl' I'lll'rt of till' Irouml conl'tlllll~.lind of thl' I'arlh e~'.I('m. 011 till' p('rforllllllll'l' of II\'I'rtiral medium 'A-an' ol'rio!. :\IOllolrnph :'\0.:!i!llt, In.t. Ell'l't. Enlr., (I.omlon), "'1'. I-J,~

(Jlln. 1!l.,)81. (Thi. I'nl"'r I'ollinill. II "111110bl,' nndI'rilirftl .on·I'~· of prt'\·iou. work.)

1-97

18) J. It WRit. Th,' th"ory of RII nntrllllll onr .1Il ill­homoll:f'I'('OIl~ IIrOllllrl ,,1'1111', I'ror. l'}'III"o-illlll onEIt'ctrollulll:lIt·tic Th"or~' lUll I Alit ('1111""', <:0""11­hagen, JUI\(' 11162. (To bc publishl'd b}' l'crlI;:lmonPrt'll.~.l

Ill) J. R. Wait, Ellritatioll of ~Ilrfllrr "'R\"('S 011 cOlldllrt­ing. Iltmtifi('d, di,·lrrtrir-rl"d. nil" corruut.·dIlurfarr~. J. Hr••'urrh :,\IIS". 36;;"'3;";" (1"'1'. HI.';;",(In rq (i I) ("Iw) ~holl"l hI' (,,/w)l; 011 JIRII" 3i6,r, i~ d.'fi,,,,d h}' r,=r,+2h<:.)

(10) K. A, Nortoll, '1'111' l,rO"'ll!atioll of radio "'a,'~ o\"('rthe Ilurfacl' of thl' I'arth :lIld ill thl' upprr atmo~·phl'rl', Part I, Pror. IIlE %4. 136i-138i (UrI.1936); Part II. Proc. IIlE %5. 1:l03-1236 (!'rpl.193i).

Ill) J. fl. Wait. Oil th" th.'or}· of r"fI"ction from a "'ir""rid parallel to all intrrfucr b(·twCf'n hOmOI!('II"OU5medin, Part II, ApI'\. l'ci. Ht'l','arrh IBI 1. 3ii;""360(1959).

(l2J J. Borrllmu, Com"utatioll of Fr~Il('1 illtl'lIml,. !\Iath.of Com"utatioll H. ~.". G!l-i2, 380 (l!l601.

113) C. Lanez~, Applit'd a~.al}·~i~ (Prt'ntirl' Hall, EIIRlc."'ood ChfJs. N.J., I!l,,61.

(14) J. Boersma. Un a numt'rical m"thod for computatiollof FrCll'lt'1 intep;rals, Math. I 1111 t. , Ullh·. of Gronill­lIten, Rrport TW:l (1!J601. (Thr rlrri\"3tioll forf(r) alld cOII~tallt~a., b., 1'•• 'Uld t{. arr ~h'rll ill thisreft'renct'.)

(IS) T. pparcp}', Tablr of th,' Fr,.,.,,,,1 illlrllr:tl (Calllhri"II"Uni\". Press, 1956).

(16) Adrillll "all Wijnlltnllr,l"1I allli W. L. Srlll~'II, Ta"I.·of Fr''l!'1I'1 1lll"lIm's, Amst"rrlulII, Sord-l\oll'Uld.clll'l'ill!, !\Iij (l!I~!I).

(Ii) F. Kopal. SUIIII'riral all",,"si. (Joh" Wilt·,· .t :-OilS.1111'., ~,.". York. ~. Y., I!I:;.'",), .

(181 P. Da\"is Rlld 1'. Ita"illowill. AI"r;«,,- n,"1 ",.illh'sfor li"•••• inll If'':ldr,,' ur," of hi!:h ord"r, J. lI,.s,."rl'l,~ US 'I. 3.'"....3i (J'III, I !''''Jfi I.

7.1 Additional References

T. I.nrs"II, ~111111'rirnl ill\"l's' illltio" of till' "(Iui\"""',,,illlp,'dnllr,' or a wi", Irid ""rulh·1 III thl' illll'rfu,'"bt't "1"'11 two mrdin, J. 1I,'s.'arrh ~ liS II D (ltlulioI'ror.) ~o. I, pp. i-14 (JIIII.-FI'h. 1!16:!1.

!'. W. :\1 ""'}' nnrl It. J. J\ill~. TIIf' illlp"lIl1n~(' tlr " mOIll>­I)()I~ ",,"'nnll "'ith D rirrlll"r ~tlIIClllrlillll:."i.k lItrolll,,1."."'111 ,'n till' ~nrf"r" of R lo~",' h,,1( ~pllr". J. 1t.....·"r'·10~BS ISD (llatdio Prop.) ~o .. 2, "p. 183-188 (M"r.­Apr, 1\161).

A. ~. ~milh Rlld T. 1::, I'\"'RIII"', FiI·ld. in ""'ctricalhIlhorl ItrOllllrl ., ,trill.: An ':lIpl'rillll'lItnl stlldy. j,1t''C''nrch ~B!' 13D (nRdio Prop.) ~u. 2, rp. li'.-IRII(~'·Jl'.-lI~t. I!I.';!!'.

J. H. W"it. E"'rlroll"'lln"'i~ r,"Ii"lillll rrulIIl'\'li,,"ric',,1~trllrlllr,'" 1l','rl['"lIlIlI I'r'·... 1.1II"lulI "lid :" ,',,, Yorl<,1!1.·,!II. 11111'11111," n t1i'ru",illll of ~ll)t ru,lill'or~ Oil

\\"t'dg'-,. Rlld Imlr.pl,""",. \

REC

FIGt RE I. 1''''11',,1 ,lul"r t{,pnlr lor""t{ Olf' a errr"I"./11"01 3(',,,n trlllrll. " .•,If. ,. /r" nil on a IIolllog,nro".g,.ound.

.,0'

F'<:I'RE :!h. Til, plio., of 1+11. a. n fllllrl,oll of "'" ,,.,'11po.o""", lIa 10' nonrond"rlrn" ".o"n".

0'

Ie'

II

lo_aooII

le'

'00

c:II... - '0',J JO ~. ...

'"20 ..J: ..Go

•r~---:,~'-~'~'-~'~'-~.:-,-~IO::,-~,!=-,-~..=-.--:\,.=-.--:\,.::-,---tlO'

....

'0110

jo"OI'RE :In. 7'11, alllpl""t{r "f I + II. " .• n f,,,,rllon of tl. 1/",,11Jlo.amdt. kn for nonron""rl,nfl fI,o"nt{ illu31.alrng ,'',,Jf,r/ of ,II, rrrr"la. 3rrttn of ,at{, " .• n

ITht'.".ltna.C'Nn hI· ......'.I,·.1 Lll tilt" mOllln~Uotl of the- ""ff'hl" hr"h, "fU'" monO'Jnlf rftUlttn. "ani tM Ilr~'n("l' cd lhr ,roun'. "",,'n.1

i

1·98

FIGUD 3&. TAr omplitudr of I + II, 0,' 0 J,Ift~tio" oj rI. U'lthIHINIlNItr ka ond no"co"dlll:ting g,ound iUu,t,oti"g th,e,_ e/ lIar,e N.

'-0-.. ,10"

~ 10',.Cl. n'=- I

B••,-

0• ,. 11' P:I'

II II' 10

•111·10

"0.100-.0-

~10

~JO '- ......10 "-'."0

I

. .••

If'

12'

'0'•c:

+ r

b r..... 11'!G- ,.

I'

.,.-4'

-r

"0

"II'll'RE -Ill. 7"/a, nrllphl"'" "f 14 1/. tI .• " J,,,,rtlflft oj 01-.iII/I~Irnli"!1 III, 'JT..' "f Jinrl. 8.

.r, -os'

If' 35'

CO ,.-10'

II"

~00-

... ••II' "0-•:l rG-

,....·20.8'0

11'

• ... ...

rlc\'.£ -Ib, TII.e pllolt of 1+(I, OK a Junrlion oJ •• ,1I"o\rlll­i"" ,A, ,lf,rt of fi"it, fl.

FIGt'.£ 3IJ, TAr "AtU, of I +II, 0' 0 Jundio" of "'. fl'lthporo",dr' ka ona ntr-;cond"ding ".o""a iU",t,o'"" ,A,r§ed of 10rl1f N,

1-99

F,Ol'll: 5a. Tht alllplitud~ of I +S1. a,' d f/lnrl/on of ~.Jorhi,lal" conductin, ,round iUlla/ral,ng Ih~ ~ff~ct of lar,t ~,

'1'

.'...0 '0110

c: ,..,-• Il'

~ .....III I'..rco. ..

..r •

r

~

-I t10' .,.

}'IOl:III: 6, l'tr/ital tlulric dipolt localtd Ol'rr a cn",h",,.,tIUIIcirc/dar-.tclor .crttll which, il'tlf, is I"ing on a 1I0",0I/t­..,ou. ,round.

"'l"r--~-"""-""'-"T"'-""--T"""-T"""-r-.....,r-...,

!'IGI'IIE 5b, Tilt phair of J~ II. a' tI (''''rlron of ~.for hlf/"'''rond"rlin, ,round iU,,,'ralrnf/ '''~ tff~rl of lar(/. !\,

ItII

II

.,••~--L_"'_L-.~~"""'~'::--"'-_~""""'----!

+.•

FIOURI: ia. Tht amplltud, of G. a~ a Jllllclion of "'. 111".­,'rahn/7 tll~ effecl of jindt.6, •Th('ll(' cun'" are for a

circular Icrt'en of radius a and II .,clor "'hich ulendlfrom a to b).

+.

FHa'lIE iI>, Tht plla.~ of (;.110' a fll'lrlron of "'. 1/I"~lrtllr"f/'ht tff~ct of finllt ~,

!I

1-100

II

"to•• , Ie'•• ·eo ... ~,.l1'

IIIb.eo...,SO' "0'

! .. ...,c::

6"'- • 10'.. • ."a;- c:: .0'· 10' ....IS' ~ rr.....«10' f II'

• t·I' II'

-'I' I

" 'I' '" lt'

FIOUBE 8a. TIle amplitude of 1+O.+Ob 03 a l,m,IIo" 0/ ~.illuetrating tlae effut of finilt .1 for nO"t'ond.It'li"lI 9,0.m<1.

l;r:".:::::::: :"':.,~..."'fo':::":':·I::':...~~'r.~.::.~~r ~:::': h~~:~~: ~,be _bltwd Iorlll 01. drflIlar dIJt ...... _lor. I

••I"W\'IIE 8", Tit, pita•• 0.' I + II. + II" ,,- " .rllnr""" fl.' 01 ..

if",.trall"g lit, ,p,n 0/ fi"'" .1 /'" """,,,,,,1,,,11"11 IIm"".t,

+0

10'

rr luta!IIb ••o'I"S'

II' "'10

r.. e>..c::•. rl"c::

IS'•- ~

~... .1' altO'..«Z.. If'

,..I/"

I" II' II' .... l' ..

15

....,11I0'.0,..,.

II ""0

!-.d IS".d

~

FIOI'BE IIQ TIa, a",pfit"d, 0/ I + II.+ lib a. a 1"",lio" 0/ ~.lor Ia.glaf" conductl"g grou nd and largt ~ iII".tra',"g ,It,tl,et of fin it, ~,

hGI"IIE lib, TIa, pita., of J + II. + lib 113 a fu"cho" III •• f'"It'vltl" t'o"dllttl"g ground alld fargt ~ iII"."a,,", tit,tl,eI 01 fin ii' .1.

10

1-101

1I ..........-.,.........,-""T'"-r---r-T'---r-""T'"-..

.. , ,....., .........I. • • • .. ••

:.

c....• I.•·

F.ol·u 11. 'I'M a",,zit,,de of l+O.+O~ a. a f"nction oflAe a,i",,,,ltal a,,", • for "oncondueti", ,ro""d iUlllf7Glin,cite ,I. 01~iN +e/or kb-40.

....,".­,.",..,M

I .!::-~,.:---!.:-.~.!=.--!:r:--~I.=--+'::-~--:!::--=~

"'.FIOURZ 10. 7'Ae a",,zit,,1k of 1+O.+O~ GI a f,,,'ction· of

.". ill".f7Glin, IIw 'led of fini'~ A for no_ndwtin,gro..ncI.

..:

110· ,, ..... ,........-

..~.....~.:-.....~.~....-~.:-.....~.~.....~.

F,Gl·U 12. TAe a",,,IiI_ 0/1+0.+0." a/ufldion 01 lit,ali",,,tltGl a",I, • lor " ...co"dudi., rnncl illftltrali.,"" 'I. of firtilc +e/or kb-200.

11

1-102

1.12 Wait. J. R.. 1967, "Pattern of a Unear Antenna Erected Over a Tapered Ground Screen,"Canadian Journal 0/Physics. Vol. 45. pp. 3091-3101

(Reprint No. 327)

Reprinted with the permission of the National Research Council of Canada

1-103

PATIERN OF A LINEAR ANTENNA ERECTED OVER ATAPERED GROUND SCREEN'

JAMES R. WAITlul"'/_ Tckc_....iclJli"" SciftuIC"" A.o_,. ESSA. B..I4Ier. CoItwIMl"

Received June V. 1167

A theory is presented for the radiation from. thin wnical .ntenna locatedabove .nd .t the center 01 • sector ,round s)·stem. The formulation is carriedout for • ,eneral variation or the surf.ce impedance 01 the system. To facilitatediscupion. certain limitin, calleS are considered in _ detail. Of llpKial interestis the possibilit)" that the !owl.....1e rndi;uion pattern 01 .n h.f. antenna oyn adielectric-t)·pe Cround will be vastl~' improved b)' usine • ,round screen whosesurf.ce imped.nce varies eaponentl.Uy. in the radial direction. from the base01 the antenna.

I. INTRODUCTIO~

On a number of occasions it has ~n suggested that low-angle radiationfrom ground.based h.r. antennas can be enhanced by the use of large groundplanes (Wait 1956. 1963; Wilson 1961; Andersen 1963), To be effective. thesemust reduce significantly the surface impedance of the foreground out todistances from the antenna comparable with or greater than a Fresnel zone.Usually. in the analytical fonnulations of this problem. it has beeD assumed thatthe transmitting antenna is equivalent to a vertical electric dipole located onthe screen. In this paper. we wish to indicate the generalizations of the theon'required to account for an antenna which may be of both arbitrary len~th andarbitrary height above ground. At the same time. we shall also consider theground screen or earth mat to be tapered in the sense that. in general. itssurface impedance is not constant.

The geometry of the situation is indicated in Fig. 1 where we have usedcylindrical coordinates (P••• z). \\'ith the surface of the ground being z = 0and the linear antenna extending from z = hi to. - hi on the s axis. Without

To Receivin9Ant.nna

FIG. I. A line:lr antenna erected over a sector ,round s)'Item whose surface impedancediflers from the sun-oundinC around plane.

IThis work 'I\'a. can-ied out while the author 'I\'as a visitin, professor at Harvard UniYU'!'it)"Cambridce. M....chuletts.

CaMdlu J........ 01 I'IIYlin. v....._ U (\111)

3001

1-104

CANADIAN JOlJRl':AL OF PHYSICS. VOl.. 45. 1117

loss of generalit~·, the receiving antenna is located in the plane ~ = 0 at aradial distance Ro measured from the origin. The angle of radiation is denoted~. and is measured from the ground as indicated in Fig. 1. Within the circularportion of the ground screen (i.e., p < 4), the surface impedance is denotedZ.' (p), which we regard to be a function of p. The outer portion of the groundscreen (i.e., 4 < p < band - AI < .. < A,), the surface impedance is denotedZ,'CP, ~), which may be a function of both p and ... To facilitate subsequentdiscussion, we shall refer to the latter as the sector contribution. (In previousstudies, both Z.' and Z: were regarded as constants, in which case, the outerportion of the ground screen was truly a sector.)

The formulation of the present problem is really a atraightfoA'ard applica.tion of the Lorentz reciprocity theorem to the free.space region. > 0, whichis bounded by an inhomogeneous surface where the tangential fields satisfyimpedance boundary conditions. Using some of the formalism developed forsolving problems in network theory, such as Monteath·s (1951) extension of thecompensation theorem, ""e find an expression for the radiated field in terms ofthe specified antenna current and the tangential magnetic field on the surface• - O.

2. THE AZIMUTHALLY SYMMETRIC PROBLEM

To illustrate the purely theoretical aspect of the problem, ""e consider firstthe azimuthally symmetric situation where the surface impedance at the plane• _ 0 is Z' (p). The resulting magnetic field then has only a .. component whichwe denote H.'. On the other hand, if, instead, the plane. - 0 can be character­ized by a constant surface impedance Z. the corresponding field, of the sameantenna with the same current distribution, is readily computed and wedesignate this H •. Now, under the rather nonrestrictive assumption that thereceiving antenna is in the far field, we find the following:

(1) H: .. H. + k el(p~~:kRo) (l + R.)

X i- Z'(p') - Z H.'CP', 0)11(k,' cos ~.)p'dp·,• 'I.

where

R sin "'0 - (Z/,.).- sin "'. + (Z/".,

is a Fresnel reflection coefficient. Here, k - (1"")'101 - wle, '0 - (PO/IO)' ­120. ohms, and w is the angular frequency which enters into the implied timefactor exp(iwt). Then, of course, 1 1 is the Bessel function of order 1. Thederivation of (I) follows directly from the material in the Appendix of thepaper referenced above. It is rather important to note that H,,(p', 0) is thetangential magnetic field over the ground plane whose SIIrface impedance isZ'(p').

The usefulness of (I) lies in the fact that the integrand may be approxi­mated by assuming a value for H:(P',O) which need only be valid when

1-105

WAIT: PATTERN OF A LISF.AR ASTESSA 31J'J3

Z'(p') differs appreciably from Z. Thus, for example, if the ground screen is aperfectly conducting circular plate of radius a, we replace the upper limit ofthe integral by 0 and Z'(p') is zero. fn this casc, we may assume that H.' (p', 0)over the range of p' - 0 to 0 is the same as H.-(p', 0), which is the tangentialmagnetic field over a perfectly conducting ground plane of infinite extent.It is clear that this assumption involves the neglect of waves reflected fromthe edge of the screen, but othen\'ise it seems perfectly reasonable.

First of all, we have the following exact expression for the radiated field inthe absence of any ground system:

(2) H (P) ikcos ~.exp(-ikRo)• •z - 4 ..Ro

X fat l(h)(exp(ikh sin ~.) + R.exp(-iklzsin !fo)Jdll.J AI

This may be interpreted as the direct radiation of the linear antenna with aspecified current l(h) and its image-car~'ingcurrent l(h)R., For conveniencein what follows. we rewrite (2) in the form:

(3)

where

i cos ~o .H.(p, z) - 4..R

o(l + R.)exp( -skR.)[..F(!fo).

(4) F(~o) = 1 : R. i~t f(A)(exp(i*" sin fto) + R. exp(-ikk sin !fo)]clJI,

[(II) = 1""(h),

Here F(!/I.) is a dimensionless pattern function and f(lI) is a dimensionlesscurrent distribution function, while 1", is some suitably defineJ referencecurrent on the antenna.

The tangential magnetic field, under the perfect conductivity assumption,is

(5) H.-(P',O) _ - .!-, fAt l(h) exp(-ik,.) dh,8p AI 2..,.

where,. - [(p')1 + hl)t. An equivalent form of (5) is

(6) iH fit ,( 1 )H.-(P', 0) - 2" f(h)?, 1 +~ exp(-ik,.)dh."., '. ......We now use (I), in combination with (3) and (6), and restrict attention to

the perfectly conducting circular ground screen or plate of radius 0. Thus, wefind the following expr. wion for the total radiation field:

(7) H,'(P,,) - i[~~~. !/I. exp( -ikR.)(1 + R.)(F(~o) + OJ.

where

1-106

3094 CANADIAS JOl"RSAL OF PIIYSICS. \·OL. 45, 1"7

k' Z(8) 12 "'" - cos "'0 ;;;

J-[ il' (')' ( 1 ) ]X fen) L,- 1 +~ exp(-ilt,.)dn ll(kp' cos "'o)dp'.° I. '. ''''.

As an interesting check on this result, we may let a - CD, whence

(9) 121__ = It ~ ~ ~: f.:' f(h)exp( -ilth sin ",.)dh,

which follows from the Appendix of a previous paper (Wait 1967). Then,(7) reduces to

(10) H/(P,I)I__ .. iH", cos "'. ~P(-ikRo) Cit cos(kh sin "'o)f(h) dh,2. • JI.

which is the exact expression for ttae radiation field of the linear antenna over aperfectly conducting ground plane of infinite extent. In passing. we mentionthat the recovery of this exact result is a consequence of using the integralrelation (1) with the identity H.'(p'. 0) "'" H.-(p',O). It is important to notethat the latter is only an approximation when the screen is of finite size.

3. GENERAL FOR~IULATIO:"l

We now return to the Ileneral configuration indicated in Fig. 1. In this case,we no longer have azimuthal symmetry. This means that, in place of Cl), weuse

(11) H/ "'" H. + It exp~;:kRo) (1 + R.)

X J f Z'(p', .) - Z H,(P', 0) exp(iltp' ~os."'ocos .) p' cos 4l<!p'd.,• ~ fl

where the integration extends over the surface S of the ground screen, \\'hOl'Csurface impedance is Z'(p', .), being a function of both p' and •. We note herethat the observer is located at (P, 0, I).

Actually, (11) is an approximation in that the depolarization of the scatteredfield is neglected. In other words, it is assumed that the magnetic field has onlya • component even when the ground screen is not perfectly symmetrical.Furthermore, in order to bring (11) into a tractable form, the tangential fieldH/(p', 0) over the ground screen is assumed to be the same as H.-Cp', 0) overthe range of integration indicated in (11). With these simplifications, we use(11) to show that

(12) H/ (P, 0, ,) ~ iI", cos "';~~(- ikR.) (1 + R.)(F("'.) + 0. + 12.1,

where

1-107

WAIT: PATTERN OF A LISEAR ANTENNA 30!l5

(13) a. ...~ f' Z:(p') - Z[cos !/to J. ". f Alf(h) J4- (I +~)

A, '. Ik,.

X exp(-ik'.)dh]JI(kp' COS !/to)dp'

and

(14) 0, ..~ f'~ f41 Z.'(p',.) - Zcos !/to J. 211'1 -6, ".

X[ ff. f(h) 14 (I + ~kl)exP(-ik'r)dh]f. '. 1 , r

X exp(ikP' cos !/t. cos .) cos I/Idl/ldp'.It is evident that O. represents the contribution from the circular portion of theground screen, whereas O. represents the contribution frolll the sector portionof the screen.

The results given by (12), (13), and (14), while formidable in appe.uance,are in a form suitable for specific applications. For example, if the groundscreen consists of a radial wire system, the surface impedances Z.' and Z.' areexpressible in terms of the physical p.1rameters, such as the number of wires,their spacing, and their radii. We shall not enter into this aspect or the problemhere since it has been discussed previously (Wait 1959). Another factor is thecurrent distribution on the antenna. Obviously, this plays a role and cannotbe normalized out of the problem. However. for sufficiently thin linear antennas,the sinusoidal current assumption may be madf'.

4. SO~IE SIMPLIFICATIONS

Considerable simplification results if tJle length of the antenna is smal!compared "'ith a ,,·a\·elength. Then the quantity r, may be replaced by aconstant r. which is given by

'. - [(p')I + h.'). where h• ... (hi + h,)/2.

Then (12) is written in t1le simpler form:

(15)

where

(16)

and

H: ~ i1". cos !/t~ek(-ikR.) (1 + R.)lt l·· !(h)dhIG.(ho) + Ii. + n.l.11' • ~

G (L ) exp(ilth. sin "'.) + R. exp( -Uhf sin ",.)• "' - 1 + R. '

- It J' 'L ) (p'>'( IJ1 (L' Z.' - Zdn. - --J-. exp(-I""t --r I +~ I"'P cos "'.) p',cos.,.. ,'_t ,. u:r ".

1-108

3096 C.'NADIAS JOl'RSAL OF PIIVSICS. \'OL. 45. 1"1

(18)

'k J'. f6. (D')' ( 1 )(17) fi... - ~ exp(-ik,o) 1;- 1 + -.-2w cos "'0 _'_ .-..,6. '0 ~k,o

( Z'-Z)X exp(ikP' cos ell cos "'0) cos ell • 7/0 d c/ldP'.

It is also worthwhile to note that (15) can be expressed in the familiar form

H. c ;:'texp( -ikRo) cos "'oW',

where

and

j ''''1'... I. f(h)dh ...", i ,,·

I(h)dh,.,(19) W' .. (1 + R,>IGo(IIo) + fie + fl.1I2.If now iho« I, the expression for the pattern factor W' reduces to that givenpreviously (Wait 1963) where it was assumed, at the outset, that the trans­mitting antenna was a source dipole located at the center of the circular screen.In (16) and (17), this amounts to replacing '0 by p' which is a valid approxi­mation for finite values of hoprovided kp' » kho.

5. SOME ArrUCATIOt\S A!'\() CONCLUSIONS

In order to illustrate the general applicability of the preceding results,several simple situations will be considered where the relevant formulas forthe pattern functions are exprl'ssible in closed form. First of all, we considerthe pattern function W' for a dipole located on and at the center of a circularscreen of radius a. From (19) and making use of (16), the appropriate form is

(20)

where

(21)

and

11" = (l + R,>I1 + 0.112,

u. c -~ J'. F(kp')exp(-ikp') (I +~kl) J1(kp' cos "'o)dp',cos rO ° ~ P

F(kp') = IZ - Z:(p'»/...

As indicated before, 0. is the fractional correction to the pattern factor whichaccounts for the presence of the ground screen whose surface impedanceZ.' (p') differs from the surface impedance Z of the otherwise homogeneoushalf-space, When this impedance contrast F(kp') is zero, then, of course, n.vanishes. In an earlier paper (Wait 1963) some consideration was given to theevaluation of the integral in (21) for F(kp') 0= canst. and CJ finite. Even in thisrelatively simple situation numerical integration was required, although, forsufficiently large values of ka, useful approximations could be made to yieldresults in terms of Fresnel integrals.

'-'09

WAIT: PATTER~ OF A L1I'\EAR ASTEI'\NA 3007

It is rather interesting to note that (21) may be evaluated in closed formif F(ltp') is allowed to vary exponentially such that F(x) = Fo exp( -bx),where % ... ltp'. This can be regarded as a special case of a circular screenwhose effectiveness is maximum very near the antenna. Then, we find withoutdifficulty that

(22) D. = - c:~J f:· exp(- (b + i)x)J.(x cos ~o)d~ e II to [ -lb .. .:)x1~-I

-iib

~~--l)Jl(X cos ~o)dx] .

When exp[ -b(ka)]« 1, it is evident that the upper limit of these two integralsis effectively .... The integrals are then of the standard types (Gradshteyn andRyzhik 1965):

(23)

and

f- (a'+8')I_ a(24) J

oe-~x-1JJ(lJx)dx = fJ

which are valid for Re a > 0 when fJ is real. Using (23) and (24), we find that,for ka "" ... , equation (22) is given by

(25)

where

- Fo (l + S.>IS. - (l - ib»)n.... - --..-- SC05·~o •

S• ... (1 - ib)2 - ,os'~oJt.

If b now ~ecomes much less than sin' ~o, it is seen that (25) reduces to

(26) f,. ~ F./(sin ~o),

which is the expected value for radiation over an infinite ground plane ofconstant surface impedance Zo'.

On the other hand, if 6 becomes sufficiently large, n. vanishes and theresultant pattern corresponds to that for a dipole located on a ground planeof surface impedance Z. To illustrate the behavior of the fields for this case, thepattern function I(cos ./to) W'I is plotted (in Fig. 2(4), (6» as a function of ~o

for the case where Zo'(O) - 0 and a pure dielectric ground is chosen such that

Z/"o .... ((c,/lt) - COS~O)I(I./I,)

and 1./10 - 3 and 10. The latter are two typical values of the relative dielectricconstant of dry ground. The values of 6 on the curves are a measure of theextent of the circular screen. For example, the radian distance 1/6 is theelectrical length from the dipole to a point where (Z - Z.(,'»)/"o is lie timesits value at p' - O.

('261 I.S

1:.,'-:0u.

Fo

:::

1·110

e""~'-'*\J ~..,.u~ fo'T~\"e ...

( Z/(:» .b&Ir +:>

3098 CANADIAN JOL'RNAL OF PH\·SICS. VOL. 4$. IlIlI7

01

-~OS

"is~ t4

OJ

G1

II

0III 15 2D l~ 3D 0 III I~ ZO 15 3D

+OIDettwII +oIDe.,.....1

(0) (b)

FIG. 2. The pattern of a ground-based dipole antenna .·ilb a symmetrical ground ~ystem

with a radially tapered surface impedance.

It is not surprisin~ that the cun.·es ill Fil:. 2(n), (b) show that the 10w.alll:ll'radiation is much enhanced when b is chosen to be sufficiently small. Also it issignificant that the pattern does not have any lobe structure. This feature i!"to be contrasted with the case for the abruptly truncated ground screen(Wait 1963).

To give some insight into the sector contribution n., we note first that (Ii)may be written in the form:

(2i) Iii. =

where

It---cos 1/10 f · (')' ( I )exp(-ilt,o)~ I + ~It

•._ '0 I '0

X A(kp' cos 1/10. to" to,)]l(p')dp',

I(28) A(y, to" to,) .. -2.

.-1

and we have assumed that

(29)

being a product of two functions M(p') and N(.). Clearly, if N(.) .. const.and to, .. to, - .-, then A(y, to" to,) 00: J,(y) X const. and the expression forn. has the required form for a circular ground system whose surface impedancevaries only with p'.

To illustrate in a qualitative manner the influence of a • dependence inZ.'. we choose

1-111

(30)

WAIT: PATTERN OF A LISEAR ASTEJI'JI'A 30!I'J

where p is a SOlie factor and ...... is the direction where the sector has amaximum value of N(.). Then, because we restrict attention to values of a andb which are large compared with a wavelenRth, it is permissible to replacecos. by 1 - (.'/2) in the exponential in the integrand of (28) but replaceit by unity elsewhere. At the same time, we permit AI and A, to be sufficientlylarge that exp( - pAl) and exp( - PAt) are both «1. Then, it is clear that

A(y, A.. At) ~ AI.

where

(31) Al ... -21 . S.... expl-P(. - ••)'Jexp (i,(l - h ')Jd ••In _

and thus

e'·'• f. (itl>",!P),) 1(33) AJ ... - (211"')' e exp - 2pTry if _ i(2P/y)1 In .

If P is now sufficiently small.

(34),.,.

e f.(211")')": e ,

which is to be compared with the asymptotic approximation

(35)

which is valid for,» 1. Thus, "'e recO"'er the expected value for a circularscreen except for the factor ie-If. which leads to a rapidly varying factor in theintegrand for at' and is of no significance for low-angle radiation over largescreens (Wait 1963).

To illustrate the inftuence of the azimuthal tapering of the ground screen,we show, in Fig. 3, a plot of the function AI(p)/A.(O) which is the ratio of theright-hand side of (33) and (34). For this example. y - 10 and .. takes thevalues 0, 5, 10, and 20°. These results indicate that the sector width parameterp plays an important role. Also any asymmetry of the propagation path.relative to the direction of the sector, will degrade the radiated field.

It should be stressed that the curves in Fig. 3 apply only to the relativecontribution from a small annular rinR of the ground system. Without detailed,numerical evaluations, it is not possible to draw further conclusions about the

1-112

3100 CASADIAJI: JOCRSAL OF PH\·SIC5. VOL. .5, 11MI7

o 0.8

~'a3: 0.1

P

FIG. 3. The influence of azimuthal tapering of the surface impedance 0{ the sector portionof the IfDIlnd oystem.

relative merit of azimuthal versus radial tapering of the ground system. It isevident that the whole subject warrants hlrther study.

Finally, we should like to indicate that a formal extension of the theorycan be made to account for the departure of the field H.'(P',O) from itsassumed value H.(p',O). First of all, we observe that the surface-waveattenuation over the screen can be considered if the function F(kp') in (21) isdefined by

F(kp') = Z - Z:(p') U'(p'),",

where

W(p') - H.(P', O)/H.(P', 0)

is the attenuation function (Wait 1963). Formally, (20) and (22) still hold ifF. and b are defined in accordance ·~:th the substitution F(s) - Foexp( -b%),where s - kp'. However, more work is :-eeded to fully develop this approach.

ACKNOWLEDGMENTS

Appreciation is expressed to Kenneth P. Spies for his assistance with boththe numerical and the analytical aspects, and to Mrs. Eileen Brackett for herhelp in the preparation of the manuscript.

REFERENCESANDEUEN, J. B. 1963. The radialion field from. vertical dipole of an inhomoeeneous

(I'0Il00. tIl Electrornaenetic theory and aatennas, .iU4 6, Eo C. Jordan (pqamonP,.. (bford), p. 1113.

1-113

WAIT: r.\TTF.R~ OF 1\ L'~F."R ASTF:NSA :\101

GIlAD5HTHS, I. S. and RYZHIK, I. M. 19G1, Tablet of integral~. series. and prOllucts(formulas 6.611:1 and 6.623:3 ",'ith • - II (Academic Press. London).

MONTEATH, G. D. 1951. Proc. In~t. Elec. Enll:rs. (London), 98. 23.WAtT, J. R. 1956. IRE Inst. Radio Engrs. Trans. Antennas Propagation. 4. li9.-- 1959. Appl. Sci. Res. Sect. B. 7. 355.-- 1963. The theory 01 an antenna over an inhomogeneous ground plane, in Electro-

malrnetic theory and antennas. edited 6, E. C. Jordan (Pergamon Press. Oxford).p.l0i9.

-- 1007. Radio Sci. 2,995.WILSON. A. C. 1961. J. Res. Natl. Bur. Stds. D65, 641.

1-114

1.13 Wait, J. R., September 1967, "On the Theory of Radiation from a Raised Electric DipoleOver an Inhomogeneous Ground Plane," Radio Science, Vol. 2 (new series),pp. 997-1004

(Reprint No. 330)

Reprinted with the permission of the American Geophysical Union

1-115

RADIO SCIENCE, Yolo 2 CHe. $erieal. No.9. s.......... 1967

On the Theory of Radiation From a Raised ElectricDipole Over an Inhomogeneous Ground Plane

Jam-.s R. Wait

In.titute for Telecommunication Science. and Aeronomy, ESSA, Boulder. Colo. 80302. U.S.A.

CR~~~i-" f~brua". 21. 1967,

Th~ inllu~n~~ of a finile pound p1an~ on Ih~ fi~ld "r a raiaed .1~('lri~ dipol~ ia run.i.r~d rro", ananaJY'ic:al atandpoinl. The _lhod for ...aJua'inll ,"" in'~p.l. it diKU.aed brWn) .nd aum....~mv·toti~ result. are iii-en whi~h iJIuatral~ 1M !i;:-:"inll behawior for IllUUnd e«>rnn. nr la'll" dia",..I..,. IIit ahown th.I the resultinll pallem "")' 1M' ..'Ul4..'" inl..rpr..I" in I.rIIla of Ih.. di...rl r.diali..n ..r .h..aourn dipole. ita i..... in an infinil. puullCl .,gill!••nd <'Gftlributiuna from tM ~dll" ..r I,", Il,..und_no

1. Introduction

The possibililY thai 10w·anl1e radialion from vertical antennas can be improved by the use ofextended ground .ystems has been discussed occasionall)' in the past. While the imJlrovement isvery modest for ground screens of wavelentEth dimensions (Wail and Pope. 1954; Palte and Mon·teath. 1955), the increase in low·angle radialion is substantial when the radial wire~ or mesh areextended to many wavelengths (Wait. 19631. The u~e of sector·shaped screen~ has al~o been con·sidered. both from the theoretical and the experimental standpoints (Wail and Wallers. 1963;Gustafson et al.. 1966; Bernard et al.. 19(6). In fa<'t. the a!Ueement belween Ihf" "lJ"f"rYf"d radia·tion patterns and those based on the surface impedance model is sufficientll encourattinjt to pursuefurther this approach.

In this paper. we wish to generalize the theory 10 accounl for the finile heii!hl of Ihe anlennalocated over the [aOund system. In the previous formulation. the dipole was localed on and at thegeometrical center of the sector. This extension of the analysis is considered worthwhile in viewof the need to choose the optimum relation between ground·system size and antenna hei~l. How.ever. here we will nol dwell on the eniDneerin, economice of the situation, but will dire<'t ourattention to the method of calculation. At the same lime. lOme aepeete of the asymplutiC' behaviorof very lar,e screens will be pointed out. The resulting formulas lead to lOme interesting limitingcases which provide physical insi~t.

In what follows. we will deal only with the far·field radiation pattern which is assumed to bevertically polarized. This is dearly a special case of the leneral mutual impedance formulation(Wail, 1963). The situation is iUustrated in fipre l.

2. Formulatian

With respect to a cylindrical coordinate system (P, •• z). the earth's surface i. z-O and thedipole Q is located at z., It on the axis. As indicated in fi,ure 1. the circular portion of the groundecreen of .urface ampedance Z; is bounded byp== a. while the tector of surface impedance Z. i.bounded by p - 6 and • - + ~I and - ~" In the absence of any pound .y.tem. the surface imped.ance of the earth i. Z and is ..sumed conltant. Apart from the finite value of II. the Ileumetry isidentical to that used in the quoted references (Wail. 1963: Wait and Walters, 1963; Bernard el al.,1966; Gustaf~l)n et al.• 1966). Howeyer. in the ,eneral mutual impedance formulation. there il one im·ponant modi6cation. Thi. i. easily seen by noting that the ellprellion for the tangential magnelic

997

1-116

998 James R. Wa;'

I

F,GURE I. alr"ir,,1 rlrr'rir d"."lr I"nllrtl ",...r" r"",bln""""c;rculor-.rc'o' Ie,un .-hid. ;11,11. i.t IYI~ on a 110"''''prwoIII ,HM.

'I

field at T of a lOurce dipole at Q is of the form

where r. =(p' + 1a')I/I, Ie = fIJ/c = 2,,/(wavelength), and where I. is the current and I" is the length o(the dipole. As usual. the time factor is exp (i""l. The form of (I) indicate!! that, whenever the factor[1 + (ilepl-IJ exp (- ilep) appears in the previous formulations. we should insert. in its place. thefactor (plr,)Z(I + (iler.)-I ) exp (- iler,).

Keeping in mind the necessary modifications for a finite value of Ia. we find the expression (orthe radiation field to be

where

where

and

E ip.ofIJ{I. -lirA I .,. rr/,z a--- e • cos .......2"R.

11" = (1 + R,) IGa(/a) + 0~2.

R _ sin .". - (ZI.".)r- sin ..+(Z/.".)·r+urA .....+ Rrr-urA ... '.

G,(Ia)= 1+Rr

•

0-0.+0..

Ie 10

II'(I~ (Z'-Z)n.- -- e-Itr, - I + -:-- Ja(kp cos rile> _0_- dpCOJ" ~o ~ .ler ".

iii 1· f'" p' ( I) (Zi.-Z)o.=- e-Itr, - 1+-.- ea--.-.. co. ~ --- df/Hlp,2" cos tit. ~o .--6, ~ ./':r ".

(2)

(3)

(4)

(5)

(6)

(7)

(8)

where ".-(p.J••,l/t-1201r. The result. pnn by (2H8. reduce to the corre.ponding forms givenpreviou.ly (\\'ait. 1963: Wait and Walten. 19(3) when Ia il set equal to aero. for 0-0. the patternfunction reduce. to the familiar form

where Rr i. the appropriate frelnel coeflicient to account for reftection from the homoleneous ftat

1-117

eanh oC .urface impedance Z. Thus. the dimensionless parameter {)( =0 .. + 0,,1 may be identifiedu the contribution Crom the combined 1I'0und .ystem. Just as in the earlier work. the crux of theproblem il the evuuation oC the intell'u exprelsions (or n. and n..

3. The Contribution from the Circular Screen

Fint of all, we Ihall consider the intell'u n. which is the contribution from the circular portionoC the lfOund .ystem. To limplify the disCUlsion. it is allIumed that the surface impedance Z~ isconltant over the area of the screen. Then. after a trivial chanle oC variable. (7) may be rewrittenm the Corm

Z-Z'O.=--! e-1W/·C(lra)

". (9)

whereC(lro) eW_'4_ i...·_,,_2- (I +_~I-:::__)

cos 1/10 • ,,2+11' i("z+W)'/2

and where H -/rho When the radius of the Icreen becomes infinite. (10) may be evaluated in dosedCorm (u indicated in the appendix) to give

This means that, in this limitin. cue.

Z-Z~ r 'H .....n.=~ ..

". Ian l#Io

Ill)

(l2)

Then, at the lame time. if the lfOund Icreen i. lufficiently weU conductin. luch that IZ~ZI-e I.

Thil combine. with (3) to giYe

I-R,O••--e-........ ••.

1+R,

.,·1...... =COl (kIr lin ..)

(13)

()4)

which il the required pattern Cactor for a Yertical electric dipole at hei&ht h OYer a perfectly con·ductiftl plane. Thil merely demonltrates that the ,eneru lurface impedance formulation leadsIG conliltent re.uhl when the extent of the Icreen is unlimited.

In order to develop a luitable asymptotic formula for larse yal... of Ira, we combine (9). nO).and (11) to let

where S - lin til. and C- cos til.. If ka .. I. and .ince .r > lia oyer the ranle of intell'ation. we may

1·118

1000 Jo...., I. Woit

replace the Bessel function by the first term of its asymptotic expansion.' Thus. (151 is expressiblein the form

(J -; f'Xp (-2;.1'(;1],1.1l(16)

For furtber analysis, (16) is written

(l1J

wherez -Z' ~-IHS

ll·-~--e ". S'

and F(z) -x-III esp (-i(r+IfI)"'+ u ].

It i, clear that the riaht·hand side of (17) .ani,hes when the tp'Ound system is of infinite estent.Thus, when Ira is not infinite. the two intep'a1s may be interpreted as the influence of the finitesize of the circular pound system.

Funher insight into the problem i, achieved by intep'ating the riaht·hand side of (17) succes·sively by parts. This leads euily to an espansion of the type

(lB)

where Ft-l(lra)-d-F(z)/dx-I,••e for n-l. 2. 3•... Nand FIOl(ka)=F(lra). For sufficientlylarge ka. the remainder RN can be made arbitrarily small if N is chOlen judiciously.

When tbe leading two terms of (18) are retained and the remainder R.o; i. neglected we easilyfind tbat

wbere

and

• [ ~-IU+C'IIr.. ( i(l + ip) )]}-, O+C)(ka)'/I 1+ 2Ira(l+C) •

fIJ- [(ka)'+ 1fI]III-ka

(Ira

p-2ka [(ka)I+IfIJIII

(19)

If H is sufficiently smaD. fIJ and p can be replaced by zero, and the asymptotic development in (19)is equiyalent to the previous formulation (Wait, 1963; Wait and Walten, 1963) where H=O at theoutset.

•,......... J,tOC,.{w!l).._ he-a.,., ................_.., ••,cI·... I,c'·... _.

1·119

Unfortunately. in mosl cases of practical interest. the asymptotic series of the t)'pe given by(18) does not converge wen. The difficulty is that (I_C)(kO)"1 is not a large parameter for nearpaing aDJIes (i.e.• Cal) even when ko is large. This suuests that we return to (16) and expressit approximately in terms of Fresnel intep'als which are wen tabulated. Taking a clue from thenature of the ..ymptotic form liven by (19). we simplify the intepand of (16) by employing theapproximation

exp [- ;(%'+ /P)III:t ixC] a exp (- icIJ) exp [- u(J + C)]

which. of course. is exact if either % -Ito or if H = O. Without dilficulty. it is then found that

1001

(20)

where c1J- [(lta)1+/p]"I-lta a /P/(2lto) = (lch )'/(2Ira).Equation (20) is in a form suitable for numerical evaluation since tables of Fresnel integrals

are available for all values of the argument which. in this ('ase. is the parameter 2(1ea/17')1/1 sin (aIIo/2)

or 2(lto/,,)l1I cos (""'2).' When the arlUment is sufficiently large. we may use the asymptotic ap·proximation

I" ,. II Ie - ~ dt a - i - e-I./I).·• 1rU

where neglected terms contain hilEher invene powel' of IL

Employing this form. (20) simplifies considerably to

where

(Z-Z~) e-hr/4e-" I [e-IIh.l"t••• 1I1

fl.-Ua1a- -- ---.• 2". (217' cos' "'-)111 (lto)lll sint (""'2)

Z-Z' e-u:· ..• ..a,.al •. and cIJ a (kIa)IJ(2Ita).". sm ...

(21)

To live further insight into this result. we ..sume that IZ;/ZI C 1. corresponding to a highly con·ductin. screen. Then from (3). we find that the correspondiq pattern function F' (with no sectorscreen present) is

, ,(l-Rr ) e-··· J1 . {e-IIh......lIiofZl. e-IIh....I..III}F a cos (Ith sm ..) - 4 (2-'0 cos' "'-)111 s.n ... sinl (""'2) & cost (""'2) .

(22)

Here. the fint term cos (ill sin ..) is the pallern for an infinite FOund plane. while the secondterm is the correction for the finite size of the actual circular screen. The result .. liven is only..lid when lea sin' (",-/2) .. 1. (lea)'" (kIa)'. and... indicated above. IZ;/ZI C 1.

The fint term in the curly bracket in (22) represents the contribution from the front edge of

'0. _ - _ /.·.·_...-1. - ClIo" -'11-51.'1_ Cl., ... 5'.1_........~ '~'_l

1-120

1002 Jame. R. Wail

the screen whereas the second term may be identified with a re-fteetinn from the hac'k ed~p. AI,tu­aUy, for angles near grazing. the laller is relatively insignificant. Thus, if lItii <II 1. ill atlt.lition tuko"" • I, eq (22) may be further approximated b)'

(231

While this is a greatly oversimplified formuh. it does indicale Ihat Ihe finite ltwund plane willonly appear "infinite" if (ko) 1/,,,,, is sufficienlly lar~e. An)' furlher quantitative discussions neces·,itate the evaluation of the Fresnel integral form given by (20).

4. The Contribution From the Sector

A few remarks will now be made about the method of handling the contribution n" of thesector screen. The double integral to contend with is given by (8). The approximate method forcarrying out the tIJ integration is identical to that used before (Wait, 1963; Wait and Walters, 19(3).In the cue where both ka and kb. 1 and when Z' is a constant, the result may be written

where

where

and

(24)

(25)

(26)

The Fresnel integral F.(z), appearing here, approaches one if 6, and 6 1 are sufficiently large.The method for evaluating the integral in (25) is identical to that used before (Wait and Wahers,

1963). The only difference is the additional factor exp (- iell(s))in the integrand. In fact, if ko • (kh)',

this exponential factor may be replaced by one over the range of integration in (25). In this case,the previous reswts (Wait and Walten, 1963) for the contribution n. and the resulting tabulatedintegrals may be taken over without change. The pattern function including the effect of the circularscreen may then be obtained without difficulty from (3) and (6).

5. Final .emarks

In this paper. we h..e indicated the necessary extensions of the theory of the "finite groundplane effect" to account for a raised antenna. While we have considered a vertical electric dipolewith its free·space pattern described by cos ... it is clear that a vertical antenna of finite lengthmay. in most cues. be treated by simply replacing cos •• by the appropriate free.space patternfunction. Also. it should be stressed that the discussion given above refers to the far·field patternof the composite radiating system. From the receiving standpoint. this means that the pallerns socalculated are only valid for a downcoming plane·wave incident at a grazing angle cII•. Some of thecomplications which arise when the incident wave emanates from a test transmitter in the "nearfield" is discussed in the references quoted (Wait, 1963; Wail and Walten. 19(3).

1·121

On the The«y of Rocliation ftom 0 Roited Electric Dipole Ove, on Inhomove_ Gtound '10M

The work reported in this paper was carri~d out under th~ .ponsorship nf th~ Advanced R~·

.earch Projects Agency, Wa.hinJlon, V.C., under ARPA Ord~r No. )83. Th~ ~xtf'n!li..n ..f thetheory di.cu••ed here was prfJpfJs~d by Abraham Waldman of the Mitre ClIrJllIratillll. Bf'tlfllrd.M.... I thank K. P. Spies for hi. helpful comments.

6. Appendix. Evaluation of the Infinite Integral G(oc)

The integral given by (10) for Ikol ... OIl may be written in the equivalent form

1003

(A))

where e-"= k/lkl. If the air i. allowed to have a vanishingly .mall conductivity,l> i. an arbitrarily.mall positive con.tant. FoUowing an integration by parts, (AI) is transformed to

(A2)

where C-=cos 1/10. We now ob.ene thai

a at aax Ix.I.tCx»=- aCax J.(Cx) = ac (CJ,(Cx)J,

which permits (A2) to be expre••ed by

= e"".S - I exp (- iNS).

The infinite integral occurring in (A3) is weD known.'

7. References

(A3)

(A4)

(AS)

Bernard. G. D.. W. E. Gu.tulOn. and W. M. ChaM! IF~b. 19661. HF IJ'O'Ind .,lIe_: rnulta of a numeric" an..y.i•• Repl.No. 1359. U. S. NaY, Electronic. Lab.• San Dicp. Calif.

Gu.tulOn. W. E.. "'. M. Chue. and N. H. Balli (Jan. 19661. Ground .,lIem etrect on HF antenn. propqation. R~pl. No.1346. U. S. Na", Electronic. Lab.. San Diep. Calif.

Pace. H.• and G. D. Moniealh 119551. The "enic.. r.dialion p.uem. of medium .a"e broadc••linr;leri.... Proc. In.l.EIec. [ap•. 102. No. S. 279-297.

•.......-........... I .... 2.s..•.m.wll~ •• I.iIlt.5.~,..... t.M.Il,..'ltU~T....oI~.5moo............. 'A<..- ........... '.N.U

1·122

1004 Jo..... R. WOil

Pean:ey:T. (l9S6I. Tables or Ihe FreinellnleJfal.Calllhrid~.. rniver~ily !'ren. Camhrid~... l\ta.~.1.

Wail. J. R. (1963). The Iheory or an anlenna over an inhllmll~..neou. Jfound plane. in Elerlramallneli,' Thellry and Anlenna•.E. C. Jordan led.). PI'. I079-1098lPer,amnn Pre••. Odllrdl.

Wail, J. R.• and W. A. Pope 11954). The characleri"ir. of a vf'niral anlenna wilh a radialrllndurlllr Ilrllund .yltem. Apl.l.Sci. Rea. B4. 177-195.

Wail, J. R., and L. C. Wallera (15 Apr. 1(63), InRuen..e of a seclor lIOund acreen on Ihe field af a vertical anlenna. NBSMOllOlfaph No. 60.

8. Additional References

Wail. J. R. (1967). Panern of a linear anlenna nver a la!'f'r..d Jfllund ~('re..n. Can. J. Ph". 45. lin 1...... 1.Witaon. A. C. 119611. Meaauremenla of low an,le radiali"n (rllm a m"nllp"le.J. Rea. NBS 65UtRadill Prill'. I. No. 6.MI-f>45.

(Paper 2-9-269)

1-123

1-124

1.14 Wait, J. R., OCtober 1969, "Impedance Characteristics of Electric Dipoles Over aConducting Half-Space," Radio Science, Vol. 4, pp. 971-975

(Reprint No. 388)

Reprinted with the permission of the American Geophysical Union

1-125

Radio Scitlll,ce, Volume 4, Number 10, pates 971-97S, October 1969

Impedance characteristics of electric dipolesover a conducting half-space

James R. Wait

ESSA Research lAboratories, Boulder, Colorado 80302

(Received June 13, 1969.)

The input resistance of a Hertzian electric dipole located over a bomogeneous conductingbalf-space is considered. It is shown that the functional dependence of the input resistance OD

the 'loss tangent' of the baU-space is coosistent witb electrostatic concepts, provided the fre·quency i~ sufficiently low. On tbe ether band, at tbe bigber frequencies, the results are com·patiblo with the surface impedance formulations based on the compensation theorem. The re­sults have possible application to remote sensing of planetary surfaces from an elevated source.

INTRODUcnONWhen an antenna is located near or on the earth's

surface, a substantial fraction of the input power is'wasted' in the ground. A measure of the perform­ance is the ratio of the radiation resistance to the lossresistance. The loss resistance is proportional to theenergy dissipated in the ground. To improve the per­formance. the antenna may be raised to a sufficientheight above the earth to ensure that the reaction ofthe ground is negligible. For low-frequency opera­tion, this is not very practical because the heightmust be comparable or somewhat greater than theoperating wavelength. Another approach is to em­ploy a metallic ground screen at the base of the an­tenna [Wait and Pope. 1955]. For VLF transmittingantennas. this takes the form of an extensive radialwire configuration, which may extend over severalsquare kilometers. Nevertheless, the radiation effi­ciencies may still be less than 50%.

Another important aspect of the situation is theinverse problem where the data on the measured in­put impedance are used to estimate the electricalcharacteristics of the ground [Keller and Frisch­knecht. 1965]. Such an approach has promise in re­mote-sensing applications for lunar exploration.

In this paper, we wish to review briefly the rele­vant electromagnetic theory of the dipole impedanceproblem. Various approximations and interrelationsin available formulations are discussed. We hope asa result to have provided a clearer understanding ofthe physical bases for remote sensing of the environ-

ment for dipole antennas located in the vicinity ofconducting surfaces.

We consider an elementary Hertzian electric di­pole located at a height Zo over a homogeneous dis­sipative half-space whose conductivity is " and di­electric constant is (. H we superimpose the solutionsfor vertical and horizontal dipoles in an appropriatemanner. the results for an arbitrary inclination of thedipole are obtained without further analysis. There­fore, in what follows we consider only the purelyvertical or purely horizontal orientations. The for­mulation of the problem is very similar to the oneused previously [Wait. 1962] in a difterent context.

VERTICAL DIPOLE FORMAL SOLUTION

With regard to a cylindrical coordinate system(p, <1>, z), the vertical dipole of length th, carryinga current I. is located at Z = Zo on the Z axis over adissipative half-space which occupies the regionz < O. The situation is illustrated in Figure 1.Sommerfeld [1949] gave the exact formal solutionmany years ago. For this problem, the fields in theinsulating upper half-space can be obtained from anelectric Henz vector which has only a z componentgiven by

II .., ..!..!!. {exp (-'YoR)4Jrieow R

+ 1- R()')'l. exp [-Uo(% + Zo»)Jo(}'p) d'A} (I)o Uo

where

Uo - ().' + 'Ye'i".971

1·126

972 JAMES R. WAIT

z

TZo

rCL>,.

I

Fig. la. Vertical electric dipole (VEO)over a homogeneous conducting balf-space.

cr, -,/£0

Fig. lb. Horizontal electric dipole(HEO) over a homogeneous half-space.

(2)

VEOK-IO

(5)

After performing the derivative operations and pro­ceeding to the indicated limits, we obtain

10

8Z = - (~)I 1- Ro.) x' e-·" tA (4)4r'toW 0 110

where CI = 2<:0. It is convenient to normalize this bywriting 8 = R.,T, where R., = 20k2(ds)2, the realpart of Zo, is the free-space radiation resistance ofthe dipole. (Here we assume that (PG!~)1/2 =120"..) Thus we may write

8Z 3 1 1- N"llo - II. x·if .. 1-2

71k

.,2 + .- exp (-alia) tAa 0 IV 110 III 110

A simple yet revealing limiting case is to considerthe static or dc limit. Then III -+ 0 and, as a conse­quence, Uo = "1 = A. Thus

Ro.) .. (N"llo - III)(N2

110 + III) -:

III .. 0.' + "I1"i'",

"II 0= [11&oW(erl + Itlw)t'·, N .. "111"10,

R .. [p' + (: - %0)"]11'

Here we have designated the dielectric constant andthe magnetic permeability of the upper half-space tobe ~ and "." respectively. To satisfy radiation condi­tions at infinity, the real parts of Uo and u, are de­fined to be positive for A ranging from 0 to co on thereal axis. The harmonic time dependence, accordingto the factor exp (;"'1), is understood.

To perform a complete calculation of the self-im­pedance of the source dipole, the current distributionon the dipole must be determined. This aspect of theproblem may, however, be deferred if we confine ourattention to the change of the impedance resultingfrom the presence of the lower dissipative half­space. Thus for electrically short antennas it is per­missible to retain the dipole approximation. By defi­nition, the change 8Z is given by 8Z = Z - Zo,where Z is the self-impedance of the dipolein the presence of the half-space and Zo is theimpedance of the same dipole located in free space.Thus, Zo =lim (:0 -+ co) Z.

According to the 'emf' method, we can calculatethe impedance increment 8Z from the formula

8Z" lim [_ 8£.ds]'_'. J.-0

where

8£, = (k2 + o"/Oz")(II - II)

o(3)

II .. lim (II)o '.-_

Fig. 2. The. input resistance of a Iltrtical electric dipole(VEO) 10 the p"s~nc~ of a conductinl balf·space.

1·127

(6)

If we write N2 = K - ip wbere K =cd~ is tberelative dielectric cODStant and plK = 0'1Icl III is the'loss tangent' of the lower balf-space, we obtain forthe real part BR of the impedance increment

DIPOLE IMPEDANCE 973

THE HORIZONTAL ELECI1UC DIPOLE

The formulation for the borizontal electric dipole(HED) is very similar to the case of the VED dis­cussed above. The only slight additional complica­tion is that the Hertz vector for the problem isrequired to have both a vertical and a borizontal com­ponent. Then following a straightforward applicationof the emf method, we find that the impedance incre­ment az, resulting from the presence of the conduct­ing half-space, is

Fi.. 3. 1be input resistance of • horizontal electric dipole(HED) in the pr",fU:1 of. COIIduetiol baH·space.

(8)

(9)

>.exp (-auo) - fA

uo

In proceeding to the static limit (Le., ... -. 0), wenote that (Uo - U1)/(1Io + "1) -.0, but (uo - "1)/k~ -. (N~ - 1)/(2 A.). Thus, in this limit

8Z 3 N' - I i- ,_.1Ro = f 4k3 N'l + 1 0 ~ e fA

13 N2 -t"" I 2(2Jczot N 2 + 1

which is identical to (6) except for the factor 1/2.Using numerical integration data from Jlolg~r and

Nob/~ (1963), the function R/R~, for the HED, isplotted in Figure 3 as a function of p for K = 10 andfor various values of the parameter ka or 4"'Zo/A.o.(The approach of these authors is similar to that ofWair [1962).) The similarity of these curves andthose for the VED in Figure 2 is striking. In bothcases, there is a maximum in the energy absorptionwhere p = K + 1, as predicted by the simple staticapproximation.

To illustrate the dependence on the relative dielec­tric constant K, the normalized input resistance R/Rois plotted in Figure 4 as a function of p, for a rangeof K values for two fixed values of the dipole height.Although the curves have the same general shape,the maxima in R are shifted to the left as K is re­duced. In all cases, the general behavior is in accord­ance with the static approximation given by (9).A significant point is that, for a fixed value of Zo/AD, asingle measured value of R/Ro will, in general, leadto two possible sets of values of K and p.

HEDK=IO105

0105a::.....a:

10

10'10"2

6R 3 21'Ro .. (~t (K + 1)2 + pi (7)

For a fixed value of kZo and K, it is evident that BRbas a maximum value when p = K + 1.

To illustrate the quasi-static bebavior, the functionRIRo, based on (S), is shown plotted in Figure 2as a function of p for K = 10. This is mainly a meas­ure of the power absorbed in the balf-space for avertical electric dipole (VED) placed immediatelyabove the surface. As indicated, for small values ofthe dipole beight (i.e., 4"Zo/A. « 1), the input resist­ance cbange 8R (= R - Ro) greatly exceeds thefree-space radiation resistance Ro of the dipole. Also,in accordance with the simple static limit, the maxi­mum value of R occurs, for a value of p = K + I =11, for the example sbown. However, as the param­eter 4.Zo/A. becomes comparable with unity, thecurves lose their cbaracteristic shape and a clear-cutmaximum is no longer evident. The behavior of theimpedance increment in tbis range is discussed below.

1-128

974 JAMES R. WAlT

(16)

(IS)

(12)

AZ/Ro .. (l/ N)/(1ea)

where

!(x) .. 3[x- l (l + ix)e-i. - EI(-ix»)

This result shows that Re AZ or AR varies inverselywith N or p1/2. This is in accordance with the curvesin Figure 2 if p is sufficiently large. On the basis of(16), we might be led to the (erroneous) conclusionthat AR or SR increase indefinitely with decreasingloss tangent piK. As indicated by the curves in Fig­ure 2, this does Dot happen. It is important that (16)not be used outside its range of validity; namely, IN:

where

or

where

Having dispensed with the perfectly conductingcase, we now define the impedance increment AZresulting from the finite conductivity of the half-spaceas follows:

U c: Z - ZC-) .. 'Z - ,Zl-) (II)

o = 1-~ exp (-auo) d).

o Uo Uo + I(k/ N)

- _lIill.1N) EI[ - ik(l + N'"1)cr)

U",", 3 1- ).1 exp (-auo) d). (13)R;; =""Ni? 0 [uo + I(k/ N»)uo

Noting that N = [(11"1 + i Cl .,)/i co .,]1/2 has an argu­ment between 0 and orr/4 (for nonvanishing 11"1 and(1), we can express (13) in the form

AZ = _3_ [1 (I + 11ea~-i"Ro Nk';;,s ~

- :a II-i.. + kl(1 - -!;s)o] (14)

1- -i.EI[-ix] ... - 11- d)'

• Y

is the exponential integral as conventionally defined.Now, if INI is sufficiently large, (14) can be furtherapproximated by

U 3 1-).1 ()- c: I ';l exp -auo d).

Ro k 0 ~Uo + UI Uo

Now, for sufficiently high conductivity and/or highfrequency, "1 = (.\.2 - N 2k2 )l/2 can be replaced byiNk over the significant range of the integrand. Then,(12) reduces to

30::

010.....0::

10

10-1

102 I 102 104

p=60CT),O

PiS. 4. The input resistance of RED ahowiDS the inftueDceof the relative dielectric constant eX = _/..) of the balf·

apace.

'Z - 'X-) - - (ds)',o 1- ).IUO-I exp (-auo) d>.4trik °

- - ~::~: (I + ika) exp (- ika) (10)

Normalizing by Ro and considering the real part Re&Z'-, or~R'-), we find that

'R'-) 3 .T - (leat (SID (lea) - (lea) cos lea]

where, as usual 0 =2:0. Here we confinn that BR ' .. I I

Ro - 1 as :.0/114 -+ 0 corresponding to the expecteddoubling of the input resistance (i.e., R -+ 2Ro), fora vertical electric dipole located on a perfectly con­ducting surface.

ASYMFI'OTIC APPROXIMATIONS

The results we have just discussed are dominatedby the electrostatic coupling of the dipoles to the dis­sipative or lossy dielectric half-space. Cenainly, therewill always be a sufficiently low frequency where thechange of the input impedance due to the half-spacecan be computed on the basis of a static theory. Thequestion then arises as to how this approach to theproblem can be reconciled with the usual methodsthat involve the surface impedance description of thehalf-space. To shed some light on this question, wereturn to the integral formula given by (4) above.

rtrst of all, we consider the limit where the half­space is perfectly conducting. Thus, if N2 - C1C, wehave

1-129

should be large compared with unity. This is arather important point since the compensation theo­rem approach, if improperly applied, may also leadto an incorrect conclusion. For example, on the basisof the compensation theorem, in its general form, theresult for the change of the input impedance 13. ofthe VED resulting from the finite conductivity of thehalf-space is [Wait and Pope, 1955)

1 1-u" J;! 0 E_H.I-

12ffP dp (17)

Here, 102 is the current at the terminals of the electric

dipole, E_ is the resulting tangential electric field inthe radial direction p, whereas H.lot>l is the tangentialmagnetic field in the azimuthal direction ., for anassumed perfectly ~onducting half-space. If (17)were applied with the proper value for E", it wouldlead to an exact formula for 13.. However, to facili­tate its application, E_ is usually first approximatedby assuming that E_ is proportional to H., which inturn is replaced by H.' ot>I. In fact, it is an extremelyinteresting exercise to show that, on using the ap­proximation

E_ .. '10(1/ MH.I-I (18)

( 17) reduces precisely to (16). Furthermore, the de­tailed analysis of Sommnfeld and Renner (1942)gives an identical result for Re 13. or ~ if :heircomplicated 'second order' correction is ignored.

A similar argument applies to the horizontal elec­tric dipole (lIED), where the integral formula (8)and the assumption INI » 1 reduces to the simpleresult

~ :::: I N~ (1 - £;-... (19)

This again illustrates the inverse dependence on INI,which is in accord with the curves in Figures 3 and4, for sufficiently large values of p.

CONCLUDING REMAIlKS

The results given in this paper should clarify thevarious mechanisms responsible for energy loss froman electric dipole source located above a conductingground. As indicated, at sufficiently low frequenciesthe coupling between the antenna and the environ­ment (i.e., the half-space) is electrostatic in nature.This is related to the so-called E-field loss discussedin the context of high-power VLF transmitting an­teDDas [Watt, 1967; Wait, 1963) operating overpoorly conducting soils. 00 the other hand, the

DIPOLE IMPEDANCE 975

higher frequency approximation to the exact integralformulas leads to results that are normally associatedwith the compensation theorem approach. In thismethod, the Poynting vector or some variant of it isintegrated over the whole ground plane, and in thisway the ground reaction on the antenna is estimated.Actually, there is no contradiction between theseseemingly difterent approaches if we keep in mindthe nature of the approximations and the ranges oftheir validity. Of course, in the case of the half-spaceproblem, it is possible to work with the exact integralformula, and the numerical data so obtained are validfor the whole range of the parameters. However, inmany cases of practical interest, the environmentalconfiguration is more complicated than a homoge­neous half-space. Then, a quasi-static assumption or asurface-impedance description is useful. We have at­tempted to give some indication of the meaning ofthese limiting situations and to remove some apparentcontradictions that have been alluded to from timeto time.

Finally, we stress that the dipole formulations con­sidered here are only valid for wire antennas that aresmall compared with both the wavelength and theheight above the half-space. Relaxing either or bothof these restrictions leads to a difterent class ofproblems.

Ackflowledgmentr. I am I"Iteful for the valuable com­ments from Mr. Lewis Vosler and for the assistance ofMrs. Eileen Brackett ill the preparation of this paper.

This work was supported by the Air Force CambridaeR_reh Laboratories, Bedford, MUSlchusetts, UDder c0n­

tract PRO.cP-69-824.

REFERENCES

Keller, G. V.. and F. Frilchknecht (1965), Eketrict" Meth·ods Ifl GeophysiCtlI Prospectiflg, 278, Perpmon, Ne.·York.

Sommerfeld, A. N. (1949), P""W DI6enlllilli Eqlllllions.270, Academic, New York.

Sommerfeld, A. N., and F. ReDner (1942), Sb'abJUDpeIIer·lie WId Erdablorption bei DipolanteDDeD, AIIII. Phys.•41(1),1-36.

VOller, L E., ad J. L Noble (1963), Curves of JfOUIIdproximity loss for dipole antenllas, NBS Tech. Note 175.

Wail, J. R. (1962), Possible iDfuenc:e of the ionoIpbere onthe impeclulce of a JIOUDd·buec! anteDna, J. Ru. NBS,66D(5), 56J-569.

Wait, J. R. (1963), A DOte on E.field and H·6eld to.. forJI'OUud·based antellJW, PrOt:. 1EEE, 51(2), 366.

Wait, J. R., and W. A. Pope (1955), Input raiItaDc:e of LFunipole aerials, Wlreleu Eflg., 32, 131-138.

Watt, A. D. (1967). JlLF Radio En';n",riflg. 73, Perpmon.New York.

1-130

1.15 Wait, J. R., 30 OCtober 1969, "Surface Wave Effects with Large Antenna Earth Screens,"Electronics Leiters, Vol. 5, pp. 552-553

(Reprint No. 389)

Reprinted with the permission of the lEE

1-131

SURFACE·WAVE EFFECTS WITH LARGEANTENNA EARTH SCREENS

The radialion field E. for a lime factor ellPUwl), al aninfinilc distance R and elevalion anale'" is cllprasible inthe formz

(6)

•D ~ - C~~fc-IO·H'"tlr.pl (I + f,) J,(lcp cos.;,.)dp (~)

o

Z, - PIr/J,..hen: 3, is a dimensionln< reactancc parameler. A loadaJ'Pf01timation is r., ~ Idi~l In Id.12trc)) .. $ .. "'here dIS lhespacinl bel..ftfI lhe .. ircs in lhe mesh. ,\0 is lhe free-space..·...len..h. r is the .. ire rad,us, lind S. is a correction factor..hich is ncaliaib\e if d;Ao is suflicicnll~' small.'

As indicaled by cqn. ~. Ihc inlclr.nd, .part from t"'= factorJ.. is propordonal 10 lhe I.ntenl;a' maJRCtic field .1 •dlSlance p from lhe dipole. If herc 1If(lr.pl is replaced It)' un,I)."C ha>-e lhe assumption thaI the lanlCnlia' maanclic fieldis lhe same as if lhe surflcC ..erc rcrfccII)' conduClinlthrou8h.OUI. Clearly, if lhe reaClancc parameler 3. "cn: clI'ecti"clyzero, Ihis ..·ould be a ,oad Ippro.imalion c>en for a ":rccn

lhe around-..a~ field of lhe d,pole is .Iso modified by lhefaClor I + D, ..herc Ihis is 10 be c>·alualcd .s ... - O. InfaCl, .s ..c .. ill sec belo... I .. D is csM:nlially indcJlmdcnt of" for small .nlles. Hcre D is 10 be oblained fromZ

jI< (z·)Z, - -T ;; . F - (Z - Z'l/'/Oo .nd where J, is lhe

Bascl functIon of order one "'hi!c II·. is .n '.lIcnualionfunction' ..hich accounts for the modificalion of lhe tanaen·tiel maanctic field by lhe finilc surface impedance Z· of lhearth 1CIWIl. In obtaininl cqn. ~, "'C haoe net\eClcd the cIl'ectsof bact rdIIctions from lhe edac of lhe earth system (i.c..1 , - b). Other invesliptioas haoe Shg,.l1 Ihis is a ..-elljustifiod USUftlIIlion.J,_

N_ the modified surf.ce impedance Z' will be imqincdto be composed of the parallel combinalion of the aroundimpedance Z (evalualed .1 arazilll inc:idcncc) and the clJcctioeimpedance Z. of the wire arid."! Thus

Z· - ZZ,J(Z + Z.l . . . . . . . . . . (7)

..here Z - ~ - til - K -')1 is lhe surf.ce impedance oflhe around for pazinl incidence. If the wire'arid spacilll ismuch less than ...~YC\enllh and if ohmic losses in the wires.re nca1ilible, "'C Iulo-,~ lhat Z, is purely inductive. Thus weSCI

• • . . . . (IIE _~/ds W'~COS"2" R

.t is 'Olllled oul IMI ...rlee", cOlWlUC1ln•••"h "'1't'1'"...., nol be tM opt.IIlum if 10..•..... 'adeallOG n 10 be IN"·IIUMd 'or ....... u'nnll CUffCllt. Thll IS • conwqucftC't' 01lhe IUrI," ••\'t: CIIC.t. 'or I" anlt KNIll WI'h 1'".11 ....ve·live nactaIltCC.

Eatended arth scnem .n: lOlIlCIimes UIed for imprO¥inlIow-anlle radialion from h.r.•ntennas. They usually IHC lheform of • radial wire or • rnctaIlic mesh which is plac:cd onthe arth's lUriacc in lhe immediate vicinily of the anlenna.'In the theorcIicallralmenl for lhese systems, il is convenienl10 describe lhe surface or lhe earth screen by an cRective1Uri_ impedance Z', which, in acnerel, is dill'crenl from lheIllrf_ impedance Z of the IfOUIld beyond the screen. Atlint 11-,__Id Ihink thallhe idlalsystem would be 10ha... Z· - 0 correspondinl 10 • screen of effecti>-ely perfectcondllClivity. It is the purpose of this Jetter to show lhal Ihisis not the cue.

The liluation is illustraled in Fi,. I. A ftftical electricdipole it located on • f1a1 around. The effecti... surface im­pedance of lhe around system is • constanl Z' OUIIO a radius bfrom the buc of the antenna. The I"IUftd beyond radius bis taken to have a _I surf_ impcdancc Z, and "C_tMtthesystem is rotalionally symmetric. As indicaled,, is lhe radial cIistancc to an arbilrary point 011 lhe arth'saurfacc from the dipole.

Hen ZI'Ie it the normalitcd surface impedance 'Jf lhe 'un­modified' around· Jr the Ialler is hofnoIeIlCOUS .nd Iossless,.... haoe

when R. - lsin" - (Z/'leII/lsin'" + (ZI'lell, 'Ie - 12Ow,and 0 is • 'modificalion' which vanishes if lhe earth screen is.bsent (i,c. Z· - Z or if b - 0), In the Iimilinl cue of no18IlItlClWll,

~o

°__----~-.o

11.10

~o

o~

Fig. Z ·AIf,.....,IO" '"''rtlO''' W, 'Of • p.rr,.II, mllurt,.,."".e,--0_'".. __ ........ELECTIIONICS UnfllS 30111 Oe,ob" 1969 Vol 5 No. 22

iOIoe

(4)

, . • . , . . (2)W' - (I + R.)(I + 0)/2

ZI'Ie - (I/K)l1I - (cosz"l/KII

where lAO - 4" x 10' " Ir. - 2trJ(..a¥Clcnllh) and Ids is lhecurrenl _I of lhe dipole. Here W' is a pattem function__Iitcd 10 thaI il is unily noer a perfectly conductin,around plane or infinilc ClllCllI (i.c. Z' - Z - 0). No.. ,followinl .n early clcrivalion,z

when K is lhe relalioe permillivily or the dicllctric around.The c:orrespondilll resull for W is CUCl.

The modificalion of lhe lUl'face impcclance ,n a circularrqion of radiUS b. from Z 10 Z', results In improYln, Ihc'CUI Ilecll' by • factor I + O. Also, il can be sho.. nJ lhal

&12

1-132

tI

of finite elllellt. H01O't\'CI'. If the reactance or the screen is anappreo:iaiJle rraction or the impedance or the pound. it isllOl justified to tet WI ~ I in the inlqrand or cqn. ,. Thisract is c1earl)' iIIustralcd in I 1. where the ampliludc andphase or WI are tho..n plOIlCG ii' a runction It, ror a losslftshomcJIenlOUS earth wilh a relalive pcrmillivlt)' K - 10. For

10 the surrace effecl would nol be prcdlClcd. This ract couldbe bpi in mind in lhe clcsi", or praehe.1 Clrth screens rorhr. anlcnna. over poorl)' eonductin, ,,"ound.

"",, w,.,,,\

\\

\\ ,

...

CH 0-3

-------..., ..",

'\ w,"\

\\\

\\,,...

11"0 ','0·2lIIt.tOO

2

°O.!-"-'-.o....I~-'-~.......,IO~'-'--'-~-'-L.J.~

'0·...1•.... I .".·"lfj.~" IfIJeINialiolt ".", dIM ro file ",..anc-., -.._ W.-I _

6,Fil·' D~. f1tI rItI.wrh· _

I .. ish 10 thuk K. P. $pia ror bill11i11a1a ill ...~"I cakulaU-.

I. e. WAn

C...r"'i~ lru,i,,,,, lor b_d1ill £II.irtl1tlrWll,,,t ~16Wtr1U"i.rrliry IIIC.....atnJdrr. Colo. IOJOZ. USA

Ril_I C'OLUl'I.'. I .• ud lWCUI. , .•. (Ia): '''- ...,.I~·

Hdl. I"'~~. 13.1'\. 21Giwi _..r..m,2 WAIt.,.•,: ...... .....,ofu...............a•• nra......

fJl....•...--LC.IEd.I:·~..., _·J =..~.~~.~~~.., dw"""'" _ of dw_,_ OD dw """-01• ...-. .1'roc. 'Z£. Iftsa. IISC 292-_

4 oun..-..... L.~ ud MLLL N. II.:~ __

~t.~~'t'C''''''~.~~.USN_~, .AftON..... T.: ....1.,...~ of -""-..ID'~"""_ OIlo••JuI, ••,

1M of .. - I. it .......t ....t die inductive portionofr kI dow. putiaIlJ lnPl*! aurf__w 10

~ die aurfllZ. T1Ila -.f_....ve .«act ror.. -0·2i1_ ,or ~.N_•........ .. _ W, II lall)' lfOIlI llIIil)'..... of _ of ip lndicaIIIl. AI ,....... oul by....... ....., .-.:e IlIoukI for • IoU)'............,.. ClIIIIidIr dIiI .

.... of I + D. ,. .. - 0'1 IIld O' 2, islllownill t.~. for i6 - 100 aDd K - 10......... ..". ... 1M _If pa dil' ....lil)' wilh_s ea..... w,-I.

....,. III Fit- 4, ........ hoW II + D I .... wilh..._ ......,.. ,or6- )anclIO.Itt-IOO•.. ..... kI .nt. In 1Ilit-.II + III •• _ of hoW I11III __ isIII ...... 1M 1IId.1 11II __ illdic:ate IMI........... or I, wIlich is of die onIIr of O' I ............... W.-I plI ...pIiDll, ..........lduc

ILICfIlOtllCl unalll .. 0.... ,... VI' 5 N. :u

1·133

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1.16 Wait, J. R., and K. P. Spies, 2 OCtober 1969, "Fields of Electric Dipole on RadiallyInhomogeneous Ground Surface," Electronics Letters, Vol. 5, pp. 478-479

(Reprint No. 394)

Reprinted with the permission of the lEE

1·135

Reprinted from Ek"'",lfi..s UIt"J. Vol. 5. No. 20. 2nd Octo~, 1%9

Fig.'• ............'y

P

Fig. ,. AaulMflwrf«e-im".._••••Iion

;,... "I " -I).'.•• ",:.wC'I11 n.l.••

I 4·92 2052 4·93 20·43 4,94 20,)

4 4·95 2015 4·96 19·86 4·98 19'57 5·00 1918 5'01 18·79 5'03 18·1

10 5·05 17'~

JlOciance conlraSI function. whkh. b~ delinuilln. vanishes ifP > o. In this inte.,.1 representalion for the penurbed ntdia­tion field. 1V'(pI is the unknown allenuation function for thelanlCnlill ma,...tic field of the SOur.e dim1e in lhe lqion0< p .-: n. This function "'IS sho.. n to salisfy a Vollerra·inl....1equation of lhe form.

(J') lIZ r' "t. - ),1""\)')1V'(x) ~ H'h) +.:.. Ff."1 - -_. . d.·2.. -0 1..1' .1'11'/2 .

(51

,. p 0 0'. ..•••• - 10. ,_ - JO

.• = lip. and I' - - U./2K7./'1o,2. lIe,e II"h) is lhe ...ellkno...n allenualion function for propaption 10 a 'numericaldislance' p 0YC1 an homoacneous IIaI lurface. In eqn. 6, erfcis lhe complemcnl of lhe error function. Methods of lhesolution of eqn. 5 for the allenuation function W'(.., aredncribed elsewhere.'

As mentioned above, lhe specificalion of lhe functionalform of lhe lurfaee-impedance conl..st funclion F(..). where.. - Itp, will determine the form of the resultin, allalualionfunction W'(..), For present purposes. we will consider

Z - Z'(..) - Ze-N . C7I

..here Z corresponds to the appropriate value for a homo­,ellO"," lonlcu around of relalive JlOrmillivity .,1.0'Tho lituation is iIIuslrated in fiB. la. In this case. we c"-

Z/'1o = (.o/.,I"ltl - .0/.,)1/2 . (8'

which is the e~act form for a wnically pollrised plane wawat lEralin, inciclcnce. The e~ponenlill variation indicated byeqn. 7 is an idealisation for a around S'.Teen ..·hich is taperedsmoolhly from the base of the anlenna. Here "'e do not dwellon the ph)'Sicalru'iSlbililY of such a around s)"stem. allhoulEh"e could point out Ihlt a ..dial...·ire system will have asmoolh.y decrasina etJeclive imnedance as one recedes from,he~ of ,he a,,'_ a'onlt ,he around s.mac:c. For oureumple. lhe dimensionless pantmeler h is a measure of lhen1pidily of the imJlOdance taper. For eumrk, h - 0 wouldcorrespond to a perfectly conduetinlt ,round plane. while" - -r- cormrnnds to a honlOlCneous eanh of surfa.cimrcdance Z Ihroul!hout.

Once ..e ha\C the numerical vllues for the allenultionfunction II"tlip). lhe pallem function 1'/.1/'0) can be oblainedd'rft:t1y from eqn. 3. follgwin,the numer'cal intCJration ofeqn 4. In this QIe. the upper limit a of lhe inl....tiun

lable I

c"""roncJs physi.ally 10 the outer e",emuy 01 the IfOUndsyslem

As ,ndlCaled b)' eqns. lind 3. ,he comrie. flClor t I .. U.)'s Ihe nt"O of lhe field .. ith and .. ilht\ut the tarCfCd,"'und s~·..em. As il 'urns out. thl~ faCiO' is • slo.. I)· ~ar)'inl!

fun,""n of the anlEle "'0, at least f,or lhe low nnsle~ of interestI" 10l\s·d,slance h.f. rrora,.""n To illuSlrale Ih,s roin' .....sh,,"" the .mrliludo and rhlse of (I , U•• In T.ble I. for alyrICal ~elec1ion of the relevanl ra..me'er~.

1 he ncar consllncy of I ... U. as a function "I "'0 justifin Iheusc of Ih,s facior as a quan"II"'e measure of the efTec".enessor lhe ,round l)"Stcm In fact ..... can a.sen thaI the etJecti\"Cmomonl of lhe source dipole IS modlflCd b)' lhe factor I + U•.

0.

N

.:.

An i,nc,ra1oCClualion method is employed to calculill~ Iht'fiekls ror ao"'..n.1c rachalion over. plane boundJ.'~ .. llh .iIIn

=='i::~~.~':~.; it~'::::; l:siS.::~~~~~)'h:............ r.<lOf whicll depcncls _ bolh I.... II..... Md I.... O..on'of the in s_.

There is an intrilJUin, possihility that the 10w-an,1e radiali"nfrom h.r. antennas _y be enMrKed by modifyin, the fore·pound. I - J Physiallly, this Qn be achieved by usin, anextended ..d;'l·wire or a mesh pound syslem. In the pm"""theoretiCIIl trutments of the subject, it has been assumed_lIy tMt the I"OUnd I)'Stem is etJectiwly a perfect eon·duetor owr its I~I atent whether it is elise: or sector shaped.­In this letter, we wish to ronsider __ impliClltions of taper·iIla the pound l)'Stem. In this cue. the etJcctiw surfaceimpeclance is a smooth function of the rad;'1 distance from thebase of the antenna. Analytic:alty. Ihis is a considerably moreinvohed probIcm tMn for a constant or _0 lurface im­pedance.

For simplicity. we assume azimuthal symmetry. and "ealso represent the source as a ,round·based Hertzian eleclricdipole. The situation is iIIust..ted in fi,. IA. ..here the

FIELDS OF ELECTRIC DIPOLEON RADIALLY INHOMOGENEOUSGROUND SURFACE

surface impeclance of the around beyond a specified ntd,us nis a constant Z. Within the pound system (i.e. 0 < p -: 01.the lurface impeclance Z'(P) is taken to be a function of p,the radial distance from the source dipole. As indicaled. "'0II the a. IUbtencIed by the obscrwer'l direction and thepound plane.

In lhe ablencc of lhe pound 1)'Slem li.e. /I - 0 orZ'(p) - Z).lhe far.field ..d;'tion paUerR 1'(1/10) of lhe ,round·bMed dipole on the hornoIa- lIat eanh is

1'(';'> - 'I + R.I(cos ..)/2

R _ sin .. - (Z/'/o)

• sin "" + (2/'/0)

and '/0 - lzo". Here, of course. It. is the app,opn.leFresnel-reflection c:oefIIt:Ienl for a normalised lurface 'm·pedance ZI.".. It wu shown before' thaI the correspond,n.pattern function is

1'("'0) - 10 + R.l(1 + n.)I(cos ",-'/2

where n. is the all imponant cornction factor ..!lith accounlsfor the preRllCe of the around 1)'StCRl. The Ialler is

n __ -"- [W'(IcP'F(lcp)e-J". -...)( (I + J:,,)J1("PCOS"'oldP (4)

where It - 2tr/fwawlenllh), J - Bessel function 01 lhelint kind. and Fflt,,) - (Z - Z'(PI"'lo is the IUrface·lm·

1-136

• ~,1f1 - 10, .. <' ,-• Of/" - J, ...... S'

15th August 1969J. S. WAITK. P. SPIES

£SSA Rn~Qrrh LDborQ'or;~s

Bould". Colo. 80J01, USA

This useful concept was also oorne out in an earlier study forcons'ant-impedance ground screens.'

For the tapered ellronential model considered here, someconcrete numerical valucs are iIIustr:llcd in FillS. 2n and bfor relative permillivities of 10 and 3, respectively. Ineach case, the abscissa is the taper rarameter b, while theordinates arc the amplitude and the rha~c of I + Un' In cachca~, three values of the ground-systcm radius 0 are indicatcd.[Here, ko = 2"Q/Cwavelenllh)).

From the curves in Figs. 20 and b, il is evident that, if thetaper parameter is sufficiently small (i.e. almost constantsurface impedance), the results are equivalent to a perfcctlycooductin. circular screen of radius Q Iyin. on a dielcctrichalf-space. This fac' is confirmed by comparin. the resultshere with those obtained previously' for such a situation. Asthe taper parameter b is increased. it is evident that the groundsystem is less efl'edive in the sense that I + U. is approachingI. Also. we sec that the dependence on the finite value of "0is less as b increases. In fact, if b > 0·1 the discontinuity inthe surface impedance at p = Q is not si.nificant.

The method employed here can be applied to more realisticsituations which involve elevated antennas of finite lenglhercded over inductive wire-grid systems. The incorporationof ohmic losses in ,he ground is also straightforward. Workon the subject continues.

ReftmKCSI "'An, ,.•. : 'ElI'eel of the ~round .elftn on the field radialed from a

monorofe·.IItE: T'.IIJ .. 1956, AP-4, rr. 179·/1'2·...'''lUl". G. D.: 'The ell'eel of Ihe ,round eon<13n" .nd of an

e.'lh ')"Inn on tht f'C'dorm:ancC' of • ~C'"icnl mr!Jium·'ul\·c .cnar,I"or. IF.F.. I'I~A, 1000l', rr. 2'1: ~Uf

) ('0(. It. , .• and rt'.lI!. w. L.: ·F.n~(,'1 or lo"~' r~rlh on .nlenn3. 'Olin',J. 1Irs. """r. /Iu,. 51.IId.. 1964, ..\). rr. 2~1-:~~

4 "'An, ,. It.• and "·ALtl.S. L. c: 'Innuence or. sector cround ~crcC'"on the field of • verlleal .nlenn.·. N.lIon.1 Buruu or SI.ndard,Mono,r.ph 60, Ar'i! 1961

~ \10 An. J, •. : 'The theor)' or an :lnlCnn3 O\'cr un inhnmo~C'ncou\,round rl.ne·, ill ' ...DA.... r. r. IEd.l: 'Uottrom'cnellc theory .nd.n'tnnu' (Per,amon, 196Jl. rr 1019-tO'lA

6 "'All. J.....nd SP,U. K. P.: 'inlecrai equ.lion arrlfo.eh 10 Ihend.alion r,om • ,..,rue.1 anlenna over .n ",homo,eneou. ,rounQplant'. lI.d,o Sri., 1969, (10 be rubh.hedt

7 KPoIG"l, P., .nd 'MODAl, •. u. c 'Innuenee or the ,round nea,'ran.millln,.nd rete""', .erial'on Ihe " ...n,th of medium-frequencysky ....ve.·. I"or. 1££, 1969, 116, pro 911-919

30

25

30

.. ka"OO

-::'~-30........."'''It.

~ __"It.~ph_

- -9::..........,ka-'OO ........... ....

..._ ka.30-...._, ,oa- .:::~-.:: _..... phase....__-.a..ka...OO "\r~ ........ ........

Ol"-l~""''''''''-'---'I...-''''''..&....l~,",",,''---'''''''''''O

C>003 0-0' 0'03 0" 0·3taper parameter b

a

2

O..............~o..L.L- .............L....I........................- .......-...JO0003 00' 003 0,' 0'3

taper parameter bb

a6t---- ___<:-

Fig. 2 Mod/tintion of .f(KtiV. dipal. mom.nt ,.su"ing f,om.uumld su,f«.·im/lftlanc. va,iation

1-137

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1.17 Wait, J. R., November 1969, "Transient Response of a Dipole Over a Circular GroundScreen," IEEE Transactions on Antenna and Propagation, Vol. AP-17, pp. 806·809

(Preprint No. 396)

Reprinted with the permission of the IEEE

1·139

Reprin'ed by permis,",nfr- IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION. Vol AP·17. No.6. Nowlnller 1969

pp.I06·I09Copyriaht 1969. by t.... Institute of EIeClrical and EIeclronia Enpneeu, Inc.

Printed in the U.s.A.

(:;)

(I)

(2)

(3)

(.._ )(Z.' -Z)• J, -; CO!' lI-. --".- tip

Jr' ~ (I + R.)[G.(II) + 0]/2

R. _ lin #. - (Z/".).in #. + (Z/".)

exp [i..(II/c) lin "'.] + R.exp [-i..(II/r.).in "'.]G.(I&) - (4)

I +R.

wben

I' [ _, ( r )0--- ellp (-i..../c) - 1+:-c_#. _ r,' ...r,

earth, i.oo Z". This, in dect. is "YU'I that the elIective lurfareimpedant'e for the eatire pound plane is Z" for 0 < ~ < a, amiZ for ~ > a.

For t'Ollvenience, ..e ..ill discllSl the problem initially f"r a timefaet« ellp (i,",). The cnrrent moment of the dipole j,. deliAnatetlP.<w}, and tile~tic field in the Jar .one " ... the form

H ) iwP.(w) . R / ) II").(1' - 2trcR. ellp (- .... c CO!' lI-. (..

...here R. is the dilotance to t~ ol...rver, "'. the anllllt' ~nhtt'lIllecl

b)' R.and t~ earth'llurface, and c the vel.,..it~·of Iilthl in" vacnnm.II" ill a pattern function which i~ norm.lizetl to be unity if thcdipole heilht 1& trere aero, and if the entire lround pl"ne Vie. eperfectl,· eonductinl.

An approximate formula for Jr' is live" ... folloa'!l [4]:

Ab,trllCt-Tbe tiIIIe-domaiD relpoDSe of a nrtkal electric dipoleiocated ewer a circaJar IrOUlld screeD jl CODIidered. Tbe rele•."ttiIIIe-bumoaic IOlutioD is used u a ltartiDc poillt. B1 maIliDc aa_bar of approzimatioDl a nry IiIIIple fonaaJa is oblailled forIbe tr...ieat reapoIlIe of Ibe far lIeld wbea Ibe dipole curreDtm_eat is a nmp fllDCtioD of tiIIIe. It II Ibowa daat for earl, times,Ibe panem rnpoue appean al If Ibe IfOUDd screea were of illlDitemeat, wbile at .ery Ioq times, Ibe respoue appraadaes tbatapected for a bomoceaeoul flat poIIDd.

Transient Relponse of • Dipole oYer • Circular GroundScreeD

FOlUlULAT'ON OP TIlE TUIE-IlAa.oNIC PIIO.LEIl

With ~t to a cylindrical coordinate .,stem (~,.,r) theearth'l lurface ie at • - 0, and the Inuree electric dipole is Iocat..1al • - 1& on the uill. All indiC'lted in Fill. I, the cimder llfOuntl..roen ie toea,ed on the lurface r - 0 a"d ill bounded II,· ~ - a.In Ihe absence of the lIlfOu"d .ywlem the lurface impeda"ce j,o Z,which doe< nol Vir)' wilh ~. On the other hand, the lurface im­pedance of the pou"d _n, in para1lel with the homopneou<

INTaODUCfIOX

Low"'lle ndistion from a dipole antenna oYer a finitel,· eon­ductiq around may be improvetl b)' the IN of a melallic IreMlnd1eJ'ellfl. The limited li.e of any practical Iystem doOIl not permitone to naIiIe faJly t .... ideal pattern for a perfectly eond~ti"lIl

lround plane. Quantitati.. Itudiel of thilo important elIect hanMen t .....ub;ect of ..riol" tht!Cln!tic,,1 a"eI e.perimental invllltip.tions [1}-[3]. Without ellcepti"n the an"l,- han been canieclout for tim&-barmonic: fields. In this eommunication we wish todevelop a tranlient lOIution. To limplify the diK..ion, a relalivelyaimple model is adopted, and the lOuree dipole _t is _umetl 'to be a Itep function of time.

1-140

(:O.u.UN'CAT'ONS 807

,

--(13)

(15)

( I!I)

"")A (/) - .4,(1) + G(I)

1 f-.II"0(1) - 2- -:- ellp (- iwAS/c) ellp Ii"") d..... _....

• i..ellp~d...(a + i ..)(6 + i ..)

Tit. it,t.....lion ill readily carried IIlIt wilh the f..I"""illl ....lIlt:

IA.(I) - --- I[{J exp (-fJl) - a up (-al) ]u(')

2(fJ -e)+ [(Jexp (-fJl') - anp (-al')]u(f) I (14)

when r - , - (2I&/c) ain #•. The terms multiplied by .. (I) IIl1d• (I') _ proportional &0 the direet IIiIR&! and the sipal ftfIectedfrom the pound plane, ....-.inly. If fJ and a 8ft allowed to lip"

praach aero, i\ an be .en that

COIllIidered to be perfeclly conduelillll:. n,Wl W' in the illlf!lrallci01 (12) is to be repIaeed by (7), liyinl

1f-I.~ (I) - A.(I) - - -2 lJ + ellp [-i..(2I&/c) aill #.]12.. _

A.(I) - 1.[-! II +up [ -i..(2A/c) ,in #.]1 ~ exp (i"') d..2.. _ 2 ...

where

.11" _ ~~f- -L-. [I + __._c__ ]S + 4cC _,' + II' i .. (p· + "')'"

• Cll" [-i.. (p' + "')"'lc)Jtl .."clr) tip (1i'

..·ilh 4 • Zl.... 8 - ain oJ·.. IIl1d c: - 1'... oj •• • ·nr the raml' l<II".'"fun('tion ,,(/) • ",(t), , ... _pu"'" f1811en. is thC'1

II"' - l't'" (""'Sic) + III" (It\)

From the nature of this Jimitinl proeelIS it is clear that &he con&Ourin (15) ill &0 be indented below any sinplariliell which OfflIr nil

lhe- reel alii&. III &be followinl, it ill undeftlood that thito convention/aoIdo.

In I'" limilinl raM 01 a pt'rl.eI.I,· ('CJlldllelillll: 1I:M....d "Ialle Ihl'ftMIiated fi.ld ill wen to be Iwo allJlf'rimrc-d ~lC~p fllncli..n~. TI..t'ClrrIi'opUndinl anurre dipole mnm.1I1 i!I a ramp fllll('lilln, i.r.,,,(11 - 1.. (1). n,i.~ i. a eOnftlli.II1 1'·I&e of IlIIlIfee for an illi.ialIltaHI)· 01 tI.. lranaient -"'"'!Ie of • II:n.tIInl Il('n!ell clleill'Cl h~' adipule.

. [I + i ..(,,: A')II'] (i~)iii I exp [-i..(,/d (I - C)]

- iflll,,[-i..(,,:rll' +CI]I.XII[-i.....(,)]rI, (21)

SoNK S,MI'LlneATIONIl

In order ttt pnX'f!f!l1 wilh Ihe Ilf!n.rlll _, a IIlllnltt'r nf .itlll,lili'-I1­I iCJ"~ a.. nnw n,,,,le. Finat. it ill ",.........1 Ihnl tI,(' RrnUllI1 ...·r...·11 i.......t"I'tl'· l'OlIdlletilll (i.e.• Z: - 0). 'fllll' (2) i.. eXllrt'!<"ihle ill litesimpler furm.

1'0 facilitate the inlqraliot,ll ill Oil, &he follo..·illll: aoymll'nli('1_ for tbe 0-.1 fUII('liot, are emplO)·etl:

(..lc)J ("" Clc) 0< - [2..:.."CJ"• [up (i"" Clc) - i ellp (-i.." C/c)]. (;.'0)

The~_ 01 ""inK this h~lo-f""I'lf!nl"· approlli...nti.." lin­

d~ in &he Appendill. nil... (17) redll,," In

.Ir np ( - iwA8/c) S.1 I I 1- I "i.. ::l< - ~ -... A '2..r)'11 Co;; • ""~; -:+~,

(10)

(II)

i.. ellp (i"')

)ellp (-al) - ellp ( -/II)

,,(1 • ..(I)II-a Ilia)

where a and fJ are eonstanta, and where .. (/) • 1 for I > 0 IIl1du(l) - 0 for I < o. As will be ,ho""11, thj" douhle ellllOllcllli,,'fllnn hu ~in mathemalieal advahtall"". The Fourie. I..,~~formof the current lIIClfM"t is liyell by

PII. ,. Vertk'al .~rtc: dlpok! 0 ..... e1~....und .........".

w...... - (,..1..)'" • 12(). and r, • (-" + A')'''. TI.. ,"III,aPen by (2)-(5) reduee to the forma 11ft11 pmriousl,· [I]. [2],where A••&eq.... w 1efO. While (2) • approximale it doe!' 1Wd.....&0 nrioua eornc& Jimitiaaa forma. For eample, II • - 0, t1_ n_W.,llIId

2W' - (l + R.)O,(A)

- up ["'(A/c) ain #,] + R. up [-i.(A/c) !lin #,] (ea)

wben R• • &be appropriate F_I ftfteetion fOelIicient for obliquetefIec&ion fram a lat earib whole aurf.. impedance • Z. n ..odJer JiaJi&itIc _ • when the pound ICreeft ndill. beeomesinfinitely Jarp. For uample, If IZ.'I « Z, we find ht [4), in thelimit. - -,

o - (ZI..) up [-i..(A/c) ain #,]Il1in h- (Ib)

Tbit comb_ with (2) and (3) &0 live

u" • COlI [ ..(Ale) ai" #,] (7)

whieh ill &he expected ,.Uern for • vertical _trie dipole Oftr •11.& perfectly condlletinll IfOUnd pIlIne.

T.ANeU:NT AWALTllIS

In order &0 approach the &rallllient prublem, it is ima«i"ed thaiat time I - 0 the dp • auddenly ,.",ted 011. TMn lhe eu".e,,'mome"t.

p.(..) -f.,,(I) ellp (-i"') til - _.!-(_I_. - -~). (8bl_ fJ-a a+,.. fJ+ ...

By superpositioll tI.. transient "'puRlf! A.(I) 01 &he radialionfield ill then obtained by iDvertinl &he &ranaform

H.(..) • J~ A.(I) up (-i"') til (9)

or, __ aplicitly, fl"Olll

At(I) - i;[ H.(..) up (i"') d....

By u&iJiainC &he abill rule, it ill _y &0 obtain

At(1) - 2!.- COlI #-.4 (I _~ +hin #.)2~. c c

w....

A (I) - i;.J:: W'(..) up [-"'(A/c) ain #.]

• (a + i..)(fJ + i ..) d... (I:?)

The function A (I) ill deipated III the pattern -JNIIl!'f!. In order&0 _ &he ......'1 01 this quantil)·, the whole ,"",lid pia... j,.

1-141

lEU: TR"""",TllJS. lJS ASTE""". A..... I'ROrAGATIO", NOVEllaER 1969

Fill. 2. Tlmo-ray In""I,,,·,.''''n or 10'.1 &ranol..n' olcllal

ltD::' t. : IAIII:=t± -----• I f I-~-~

T-FI,. 3. Tr.n....n' .....pon.. of _...und·!Nuom <1,,,,,1,, wllh d ........ orncon.

,.. - Ila - IIIS)C + [1., - illS)' -1II'8')'"IS-'. (:.'6)

All all@nlllive form or 1:.1:)) "'hirl, i, ,"illllJle fur lllllllCriMlI.....rk i:<

, II ..... A - Z,.. II d""_ co lie ./I &h·..h· • con""n' .Iolrh ,. In·""""nden, or r"",,_,. Til.. I. c&.ktl, 11d onl, f... _ .... <lit'I..,••k·Uk......uncl III. •

.... (1 - CO) - 2".1<1 - IIISIC - [1., - liS)' - II'] - O. 12:.)

1'1.. relevlUlt root is

_~ __1_1"""-C' ...!..8 + /A (2e)'''' • ,,,"

• I d ..[, _ !! (I _ C)] (29)[I - ("/e)(1 - C) ]", " e

wllt'l'e

GU) -

/A 8 1 { . [1 2(1 - C)o]- 8 + /A (2(1 - .e)]III; - - c:e

+ i} v [t -; (1 - C)l (30)

I(T) • (I/w)[ arealn (1 - 2/T) + ../2] (321.nd

.·I,ieh i. tht' @~I...clt'tl value (or lloe pallern r"clor ror • dip.le 011tM 811rl_ or • lint imp@<I,,"<'@ ..urface or illfi"ile extent. n,e situa­ti,," i.. i1Ju..tralt'tl ill Fil. 3 ..Ioere A U) is .blclM!tl u a (unction ort Ioe IillM! paruneler.

A (I)] ,_ - I - [A/IS + /AI]· 8/(S + /A)

A con.,ict.rable Inalytiral limplific.lio'l ,. obt.ined when tltedipole is located on tiM! acnen. Then (27) redu_ to

1'1"... it rail he IIftn Rt tiM! i"ilial i"stlUll I - 0 tloe reopon!lt' i.••ullil JIUlIil in Ilep. nlis ill (ullu"'ed by • lint'n.ly t!t!CreMillll lilt',,,1.....ieh IJrlillll .1 T - I. It thell a"Uen, oR to • constant value inacconJante ...it" tiM! limil:

T - (11/0)/(1 - C) ~ (a/o)/(S'/2) ~ (a/a)/(J/-.'/2).

n .. rarlor SI[2(1 - C) ]111 in (30) nlay be replaced by unit)'.i"te it ill .......(1). _umetl t"at I - C« I. 11M! lllIultant pattern-JIOIIlIe ror t"is lilftil inl aituation iI tMn liven by 'be foUoorinaremultabl,)' limplellllult:

A(I) • v(I) - [/A/(S + /A)Jf(T)u(T - 1) (31)

CoNCL""0N

While II,C prt'<ent lUll I,..", is h..ortl on • hillhly it!e.lizt'tl mOllel,tiM! ullos do indieat@ t"at tiM! finit@ @lll@nt or " ."ound eereell .'ilIn iry Iilt'lificanlly tiM! pulR Ihlp@ or tiM! rlldialt'tl fit'ld. T ..ilI mlYIll! .n import.nt factor in timine lyIt@mll wlM!re some reature oftI... radi.led pulR shape illaged u • point to measnre tiM! traveltime. FurtMr work on this luhj@ct ahould consider tiM! rrequencyc"'"enclenCle or tloe ,round..url.l't' ImpedlUlc@ ."t! liM! eqniv.lelltillll'l!t1"nre or tIM! wire mesh which mMes up tiM! lrountl ar.n'C!n.AI.... raisrtl ant@nnu of finile lenltth will lead to more romplielllt'tlillieiral evaluation but, in principle. till! p~nt results ean btl..laplt'tl if lu~itiOl\ is ,IRed.

ArrE!wllt

III JIl!rrurntillll II.. i"lt'lIir.tinn nver w ill (I!I). tiM! Re.o<wl Inllcli""WIL. ~pl,,(,@<) b,· Ih@ lirsl It'rm in il .. lL'ym!'lolic expansioll. The.il(llifirtlll.... or Ihi. ~Ie!, .'iII he e1i...".....1hrit'll)·.

\\t' lIrt! cull....M1@c1 ... ith the t'VlluatiulI or tiM! f"II..... ;1I1l ....mi...illl«'lrnl:

11- @~p r-i...,/e) ( e ).11) -2" _ Ce· 1+;;:,. iJ,I-e/c) exp (i"") dw. 13-'1\

12·1)! I".' + 111')11I _ ,..C +~ - , _0e e e

8/A 1 1" 1- ----- --8 + /A 12c)'"" • ,''1

,,·I.i,·h is eqlliyalent to

Gill -

"'1",... it i.. implicitly _"11",,1 th,,' the llOrlllI\Iil~I '....I",·e 11\\­I..~I,,"rt.' /A is illdelM!lIdclIl ur tilt' r""I"I'''''y,' II .nll 1M! IOftll that tll@illlrj[rRllt! ill (2:1) ill H1'u ru. v"h~ ur" > ,... where

(,,' +111')'" " lIS.1,,) .---- - -+-.

e e e

h. ,,~.. or th@ appro~imatiOllS ill th@ o.igillal deriyalion [4]. th@faclu. C'" ean btl replaced by unily. Am, linee __ cIea1illl withIarp lIereenl whare (000/c)>> 1. th@ finat aquare bracbt term ill11l@ inteer-lld of (21) may btl repllC@d by unity. On tha otllt'.halld. in the _nd 8qu_ braeket, tll@ t@rm multiplied by -i callbtl ignored Iince the ph_ ill .aryilll rapidly. ItlJ indiealed hefore[4]. thil aeetected term ill related to the wa.. rellecled front th@bade ed«e of the acreen. With t~ .implificatlona, it ill fCMlIld '''at(21) red_to

.... ex!' (-i.JtS/e) S/A 1 1- 1 1i.. ~ - 8 + /A (:.I.e)'" • ,,," (i..)'"

• up [-i..I"/e) (/ - CI]@xp[-i"'I")]d". (Zl)

UMill1l th~ form rot tll@ int","d ill (19), 'h@ iJlI@lrIliUII OV@....call IIuW lie readily @/ft!Cted. ThUi

G(I) __ ' S/A -'''-1- I ..[1- 1"/rl II-C) -.(,,)] tI" (23)S+/A (2c)'lIw • ,,'"[1- 1"/el (I-C) -.1,,»)'"

I=---.,....,.......,......-:-..;..,.----:-=-=- d" vI,.. - a). (27)[I - (lIe) I,,' + A')'II + ,.c/e - IIS/e'J"

•·..r 11l@ raNd diJ'OIe li.e., 111 > 01. rmlllt'r I"@flu..li..n or Ihi.. illit'lliral~ not -em JlOI'i"ihle.

TIl@ time-n&Y inl@rpreIAli..1I or lilt' _nil. i.o illdicllt'tl in Fi~. 2.TIM! dipnle at T .. excil@(1 at 1 - O. U tiM! puillls M .nd N 're Oiltilt' r.ommon _vefront. theil, nbviou,.ly. the ."ound rellt'clt'tl li~.1•• ,... will arriv. at the oboft-y@r II li",t' 1 -.Is Ialer, "ICh th"l

a • TN - TAl - I,..' + A')'" - ,..c + lIS. l:m,Ti,e .um total or tt- ra,..., inl~"tetl rrom tiM! @(I~ or lI,e rftlloul to illfinily. lives th@ c"nlrihlllinll G/Il to 11ot' Iran.;"111 J'OlIlO4!at 1I.e dipol@.

1-142

(351

CO....VNICATION.

NOlinl that iJ,IZ) - ',liZ,. the Fourier inlqra!lab_ [3J canbe .-d to tho.. that

,I') - ~, {~(7)' -(, -;)']'" -(, -;1(7)'-(, - ~)']-,nl ~[-I, -;1 +7]. (34)

For tbe timea IIIIcb tbat " - (,fell' - ClI« (,,Ie) U -CI. itau be _n tJaat

11[' - (,lei (I - CI J,(1) :0< .ca"(~).nc' - (../e)(J - C) J'"

While this is au admittedly enade .pproamatiot•• it is qualit.l ivel}·aatidaetory, for this NMOft. and bftauae it fadlital. tbe lUt.quet.tintearatioG _ ,. ID fact. &he ...ult lEi"" by (35) conw.ponda tothe form .-d in the _in text ..hi..h is a.-d on the Iimiu/ll ...YJnptotic form for the &-I fuMt"" J,(..,c/e) ill tbe freq~pl_.

lAMa n. WAIFESSA n-reb Leba.UB, Dept. of Com_Boulder. Colo. 80302

nEnu:"~

(II J. R. W.1l .nd L. r. w " ... Inll n.... 01. _Of' ....und """,,non &llr ft.l<. 01 .....&1 n nn.:· IIS Mono 110. April Ill. lUI'''.

121 (;. 1> • .".n.rd. W. E. GUII.I...... and W. )1. l·ha.... "HF aroundISIIem ""U'.I or • namerlta' .nahlll:· t·. s. S.,·s Eloc\ronlaI..bl.. S.n Dleao. r.m Rrl" 1359:'.._rl: 191'06.

131 W. E. G,,"'.IlOn. W. )1. Cha.....nd N. II. B.IU. "Ground I.·_m·11"". on H F .n'mna ,",,,,,"'io..:· (. s. S.,·, E_1Ik'a I.a.....San 01.11". ("alif. Ht"'l 13"'•. Janu ,· ,W.r.

f4f J j( ".fl. "(In 1ft.. lit.....' <t' r." 'I'Om • rrIJ4I'd .."""ric 4''''''''o¥t'r an 'nha"'-t'nt'Ou" .round plan",." Bollio St', .. "01. 2. PII, Wi­1004. ~"...mbrr Hili;

'51 n. A. l·.n."....11 .nd R )I. FOI'''r. Fou,j" In",.." 'n, P,."irol",,''''ltion. S ...· lork. Amrri","o T""'phon" .net TrhllraPbCo., 1031. pal... _., ...d " •.7.

1·143

809

1·144

1.18 Wait, J. R., and K. P. Spies, January 1970, "Integral Equation Approach to the Radiationfrom a Vertical Antenna Over an Inhomogeneous Ground Plane," Radio Science,Vol. 5, pp. 73-79

(Reprint No. 397)

Reprinted with the permission of the American Geophysical Union

1-145

RJulio Science. Volume .5, Number I, pales 73-79, ""nary 1970

Integral equation approach to the radiation from a v~rtical

anteDDa over an inhomogeneous ground plane

JtIIPIU R. Wait and Kenneth P. Spies

as" R'H~"Lilbo,lIIorie#. Bo"ld". Colo,ado '0302

(Received July 31,1969.)

The fields of a dipole IOUra localwd over a variable im~daDee ,urfaee .re considered. Amelhod of obtaininl solulions of lhe inl~,ral equalion for Ihe lanaenlial field is oullinwd.Numerical resulls are liven for Ihe r~sullinl radialion pallems for an ~llpoMnlialJy larercdvari'lion of lhe surface im~dance. II is shown Ihal lbe la~riDI ha. tbe adv.nlap Df reducin,!he lobe struclure of the low'Inlle ndialion.

INTRODUcnON

The use of extended II'ound screens to enhancelow-angle radiation from HF antennas is now com­mon. To be effective the ground screen should re­duce the effective surface impedance of the fore­ground c.~ct to distances comparable with a Fresnelzone. The utility of the ,eneral analytical formula­tion [Wait. 1963) has been demonstrated on a num­ber of recent occasions [Bernard etal.• 1966; Gus­tafson et aI•• 1966; Collin and Zucker. 1969).

In this note, we wish to discuss the evaluation ofthe radiated fields for a situation where the surfaceimpedance of the ground is tapered as a function ofdistance from the base of the antenna. For simplicityhere, we assume azimuthal symmetry, and also werepresent the source as a ground-based Heruianelectric dipole. This situation is illustrated in FigureI, where the surface impedance of the ground be.yond a specified radius Po is a constant Z. Within theground system (i.e., 0 < , < Po), the surface im­pedance Z'(,) is taken to be a function of, that isthe radial distance from the source dipole. As indiocated, ",. is the angle subtended by the observer'sdirection and the ground plane.

Now, in the absence of the ground system (i.e.. p~

= 0 or Z'(p) = Z), the far-field radiation pallernP(!#I.) of the ground-based dipole on the homogene­ous ht et:rtb is given by

P(~.) - II + R.I (cos ~.)/2

R lin ~ft - CZ/"ft).-lin 'I. + (Z/"o)

CI)

(2)

73

and .,.. = 12(),.. Here, of course, R,. is the appro­priate Fresnel reflection coefficient for a normalizedsurface impedance ZI.",•. It was shown earlier thatthe corresponding pattern function is given by

I'(~.) - 1(1 + R.)(I + 0.)1 (c:os ~.)/2 (3)

where n. is the all-important correction factor thataccounts for the presence of the ground system. Theground system factor is given by

n. - __le_L'· W'(kp)F(lep)f>-il.cos'l••

.(I + ,:,) JI(le, C:OSJ'Io) dp (4)

where Ie = 2"./(wavelength), where J1 is the Besselfunclion of the first kind, and where FClep) = IZ ­Z'(p) J/"" is the surface impedance contrast functionwhich by definition vanishes if p > Pt,. In this integralrepresentation for the perturbed radiation field,W'(p) is the unknown attenuation function for thetangential magnetic field of the source dipole in theregion 0 < , < Po. This function was shown to satisfy

2

F11: I. Venical dipole located cenlfall)" 0\'tI' a radially varyiqimpedance surface.

1-146

7. WAn" AND SPIES

a Volterra integral equation o( the (orm

W'(x) _ W( ) + (,x)"2 1' F(y) ",ex - y) W'()') d

x 211' • lY(x _ y»"2 y

(5)

where

W(x) _ I - 1(1I'p)"2~-' erfc: ( lp"2) (6)

and p = -(u/2) (Z/'/lI)2. Here, W(x) is the weU­bOWD attenuation function for propagation to a'numerical distance' p over an homogeneous Oatlurface. ID (6), erfc is the complement of the enorfunction.

i-- , ." • [ ( i )"' ]Q~ + f.': a. x = a. + ~ o. - 211' a.-, x""

(12)

where

r(~)r(n - ; + 9r(n ~ 2) a.-.

(13)

Finally, we equate coefficients of like powen o( X'/·

on the two sides of ( 12) to get

SOLVING lHE INTEGRAL EQUATION

We now outJine a series method of solving (5)for W'(x) for small values of the argument x. SinceW(x) bas an expansion of the form

and

0.' = o. _ (.l)"2 ~2. 11.-. (n = I. 2. 3, ... ) (14)

.W(x) - L O.,x"'2.-.

which is effectively a recurrence formula for the(7) evaluation of Q'. in terms or Q'n. 0'10 ••• , ft'. ,.

When these results are applied to the case whereit seems plausible to assume that W'(x) has a simil:lrexpansion

. .L o~ X"'2 - L o.,x"'2••0 ••0

Noting the integral in (to) is x -/2 times the betafunction B[(p + 1}/2, (n - p + 1}/2J and usingthe standard relation

Upon substituting the appropriate expansions into thein!egral equation we obtain, on interchanging theorder of summation and integration,

(15)Z'(x) - Z Z _e.-= --~

'IQ '10

(where b is a nonnegati\ie constant). the series (orU"(x) is given by

W'!.,,) = I - \ y; b~x'f2 + 1". y; b'~x'l2L

+ y·sb'E')(' - I. y; b'~x,n - Ib'E'x"

+ my; ('ib'~ - S6b'fJ."OJ'

+ iHb'('x' + .. , (16)

where t = (;/2)1/1 Z/'IO'When x is not sufficiently small, series ISis not

useful for evaluating IV'(x), and we must r~sort tonumerical methods of solving integral equation 5.The technique outlined below is an adaptation of onedescribed by Wagnf'r (1953J nnd yields IV'(x) for aprescribed set o( x values, XIo X:. x~, •••. The fir~l

step is to compute W'(x,) and W'(x:) using seri~s

16; the solution then proceeds inductively, IV'(x.)being computed when ",'(XI)' IV'(x:),···, W'(x...dare known. From integral equation S. we writ~

In.\".) ... W(.".) + (;;;)'"

·{f' K(.t•• ")W'()") dy + {' Kt"•. y)W'(y) dy

+ ... + r~. K(.t •• y)If"(y) dY} (17)

(9)

(8)

(10)

(II)

-W'(x) - L o~ x"".-0

fCII)f(l:)B(II,o) - nil + v)

1· ,.-lIn( )'.·.-1112 d.,. x-y yo

(I )"2 • • •

- 2x L L L r,a;., 0.••w .-0 ••0 ,-0

where the coefficients Q'.. are to be determined. Inaddition, we assume the known impedance function(Z'(x) - Z)/Y/f' can be expanded as

r:( ) Z'(x) - Z i--,. .12r'X - - - - ~).,x

'10 .-Q

(t0) becomes

1-147

FCJ·) ",(Jt - ).)K(Jt • .1) - l.Y(x _ .1») iii (18)

Now assume that W'(y) may be approximated in theinterval (X~h XJ) (; = 3,4•••.• n) by a quadraticpolynomial fitted at XH. X~h and XJ; thus

(27)

(20)

INTEGRAL EQUATIONS FOR RADIATION OVER INHOMOGENEOUS GROUND "

(21 )-(25) were approximated by a 12-point Gauss­ian quadrature formula. To remove infinite singulari­ties in the integrand. the integration vari:lble W:lSfirst changed to (J by the relation y = XI sin' 0 in' 1(.1'.) and by X. - y = (x. - X... I) sin' (J in I.{x.)

and i(x.). In ' 1(.1'.) and '::(.1.). W'{y) was eval­uated at the Gaussian abscissas by means of series16.

Computing time is a rapidly increasing function ofthe number of points at which W'(x) is evaluated(about 5 minutes was required for 60 points on aCDC 3800 computer); thus it is desirable to spacethe points Xh .1'2, Xlh ••• as widely as the demands of

(19) accuracy will permit. One must, of course. take XI

and x:: to be small enough so that W'(x,) and W'{x:)may be computed from ( 16) ; some numerical experi­ments then indicated that adequ:lte accuracy was ob­tained by taking the interval X, - Xi I to be roughlyx,/IO. Further confidence in the results was gainedby applying the numerical technique outlined aboveto the integral equation for JJ'(x):

('x)'" Z f.' wey)W(x) = I - 211' ;;; 0 V'(x _ ),)Jin dy

when:

+ (y - .1',_,)(.1 - .1';0') W'(x;)(x, - x,_,)(Jt, - .1',_,)

which is just the well-known Lagrange interpolationpolynomial of degree 2. When the appropriate poly­DOmialS are substituted into (17). ODe may solve theresulting equatiOD for W'(x.) aDd write

(iJt )'11 i-

W{x.) + 2 ~ I ;(x.)W'(,x.) _ 11" ,-,

I _ (~:)'" I(x.)

where

~II)

o-

,.,.C'II)

o- 80 -

3:

O. 03·

-o__ - __ _ C'

"" ".. ----- _ - .3CD: ,,:J --,.,. ---," , ~3 _JlCI-- 0.0;i_-- --- 5:

, " , -- U ---- OOL c/ "",II'"...... _---- -:- - .s=.' .... - .. ------ -----.. - - - 0 ~

10 20 o.ool 30

'040 C'

.35lo

~c.:::.;;;;;olollloG:"""'.&.I:.l:==::..:...L..II...l....I..I""':I 0 '&.30

(21 )

(22)-3:

(23)

(25) -3:

(24)

(26a)

(26b)

K(x•• .1) W'cy) dy

J.(x.) - [' K(x•• .1) W'()·) dy

1..

I.(x.) - ..

+ B,CY)W'(x,_,) + C,(,Y)W'(x,») dy

U - 3. 4•..•• " - I)

I.(x.) - J.:~, K(x•• .1)[ A.CY) W'{x••,)

+ B.U·)W'(x._,») dy

1(.1'.) - J.:~, K(x•• y)C.(y) dy

C,(,Y) _ (.1 - Xi_,)(I' - .1'._.) (26c)(X; - X,.,)(X; - X,_I)

aDd K(x•• y) is obtained from (18). The intcgr:lIs

xf 'I!. 2n. b. The Illenu:lIion funclion for In e~ronenli:ll

\·.;il'iC'n of the surface im!X'd:Jncz. The :Ih<ci«:1 i< the norm;tl·ized dis'ance.lt - 2.p/\\,a\c1cnl!thl for the d.('Ole.

1-148

76 WAlT AND SPIES

and where Z corresponds to the appropriate value fora homogeneous lossless ground of relative dielectricconstant c,/fO. In this case, we choose

To start the calculation, W (0.10) and W (0.11) werecomputed from (6); then a formula analogous to(20) was used to evaluate W(x) at a sequcnce ofpoints ending at x = 30. These results were com­pared with values obtained from (6); at no point didthe mapitude of the relative error exceed 3 X 1()-6.

THE ATIENUATION FUNcnON

As mentioned above, the specification of the func­tional form of the surface impedance contrast func­tiOD F(x) wiD determine the form of the resultingattenuation function W'(x). Using the numericalmethods outlined above, some represeotative resultsare obtained for the case where

which is the exact form for a vertically polarizedplane wave at grazing incidence. The cxponentialvariation indicated by (28) is an idealization for aground screen that is tapered smoothly from the baseof the antenna. Here, we do not dwell on the physicalrealizability of such a ground system, although wecould point out that a radial wire system will have asmoothly decreasing effective impedance as one re­cedes from the base of tbe antenna along the groundsurface. For our example, the dimensionless parame­ter b is a measure of the rapidity of the impedancetaper. For example, b = 0 would correspond to aperfectly conducting ground plane, whereas b = co

corresponds to a homogeneous earth of surface im­pedance Z throughout.

Specific values of the amplitude and phase lag ofW'(x) are sbown plotted in Figure 2 for c,/to = 10and for t,/fO = 3. The abscissa here is the electricaldistance x measured from the base of the antennaalong the ground (i.e., x =2"p/(wavelength»). Vari­ous values of b are indicated on the curves. As ex-

(28)z - Z'(x) - a-I.

o.s

-.-B......Q. 0.6

c~

~

'0Q. 0.4

€g I€o • 10

Xo • 1000.2 €IJI€o=3

Xo'IOO

010 30 0 10 20 30

0/0 0/0 (Degrees)

0.8

......:l......

0.6a..c~

~-0a.. 0.4

Filo Jer, b. The far·fleld radiation pattun of the dillOle as a function or the ele\'ltion anllle •• for In ullOnenti31ly tapen:d aroundsystem that is truncated at ~. - 100 (i.e., 2rp. - 100 ,u\-elensths). Various values of b are shown.

1-149

INTEGRAL EQUATIONS FOR RADIATION OVER INHOMOGENEOUS GROUND 77

peeted, the magnitude of the attenuation functionW'(z) is considerably less than unity for the lowerdielectric constant of the ground. We stress here thatthe curves shown in Figurc 2 apply ani}' to a puredielectric ground. This would be a reasonable repre­sentation lor a real physical situation of operation atHF over relatively dry ground or thick ice la}'ers

RADIATION PATTERNS \~'t'~e"Once we have the numerical values for the aUen­

uation function W'(kp), the pattern function P("'o)can be obtained directly from (3) following thenumerical integration of (4). In this case, the upperlimit pO of the integration corresponds physically totbe outer extremity of the ground system.

By use of the numerical results for W'(z) shownin Figure 2, the pattern function P("'o) is shown inFigure 3a and 3b for e,,/eo = 10 and 3, respecth'el}',for Zo =kpO =100. It is not surprising that the low­angle portion of the pattern is greatly enhanced when

b is decreased to small values. In fact. for tbe caseb = 0.003, the ground system is ,·cry similar in ef­fectiveness to a perfectly conduct in!! circular disk ofradius pn. This fact is confirmed by cOl11p:tring theresults for b =0.003 in Figures 3a and 3b with thecun'es given previously for a pcrfectly conductingcircular disk of the same radius [Wait arid Walters.1963) and for the same values of the relative dielec­tric constant of the ground.

Fa smaUer values of b, the curves in Figures 30and 3b are smooth and monotonically increasingfunctions of the grazing angle "'0' This can be attrib­uted to the fact that, for the value of .10 (or kpo)

chosen, the surface impedance contrast functionF(x) becomes negligible before x reaches .In. Thus,there is no appreciable scattering from the edge ofthe ground system at p = po). For some practical ap­plications, this may be a desirable characteristic forthe radiated fields.

The inftuence of varying the extent of the ground

3010 20

"'0 (Degrees)

1.0

3010 20

"'0 (Degrees)

1.0

Q.8 0.8

~ ~a:- M a.. 0.6

c: c:... ...41 cv- -- £91£0 =10 - £9/£0 =30 0a.. 0.4 a.. 0.4b·a-OJ b·O.OI

Fia. 40. b. The far-6c1d pattern for an upoaentially tapered around s)"Stem sholAina the effect or different truncatioD distancesfor b .. 0.01.

1-150

78 WAIT AND SPIES

os 0.8

OL.l..l...:....l~ J..,.I,.,. ~.........

o 10 20 30

1/10 (Degrees)

30. , ...

10 20

1/10 (Degrees)

- OS - 0.6~ ..:-- -Q. Cl.

c c~

0.4 €1J1€o - 10 ~ 0.4~ lI)- -b- 0.1 -0~Q.

€Q 1£0' 3

02 0.2 b' 0·1

Fi•. So, b. The far·field paltern for an nponentially tapered lP'ollnd system showin. the etrect of dill'erent truncatioa distancesfor b - O.t.

Integrol Equotion

Kirchhoff Approximation -

----...­,-,,,,"0.1,,,,,

/,,,I

I,I

II,

II

II

II

I.0.2

0.8

0.6~

Cl..

~ 0.4 ­!!-~

O~...l...I..~~...l...I....I-J,..~'-l...1....i--1-~L...l...I...J.,.I.,~

o 10 20 301/10 (Degrees)

Fi,. 6. The far·field rallern u5inll the int~gr:l1 solution rorW'(x) compaud with the KirchhofT-t)'rc: apl'follimation which

selS ",'(x) - I in the field intearal.

system is illustrated in Figures 4a and 4h for .,/(..= 10 and 3, respectively, for a fixed value of b( = 0.01). 1r is not surprising th:ll for these cases thelow-angle patiern is again improved as x" is in­creased. Also, there is some evidence that the scat­tered wave from the edge of the ground system i'\producing some minor wobbles in the pallerns. Simi­lar results are shown in Figures Sa and 5b. where thevalue of b is increased by a factor of 10. Here. theinfluence OD the finite value of XI) is much smallerbecause the surface impedance contrast has de­creased to a negligible value before the 'edge' of theground system is reached.

In a previous study (Wait. 1967] of tapered groundsystems, the attenuation function W'( lip) was ap­proximated by unity in the field integral given by (4)

in the present paper. In effect, this is a physic:l\ opticsor Kirchhoff type of approximation. It has the mcritof leading to a relatively simple closed-form exprcs­sion for the resulting pattern functions. Certainly. fora truncated ground system of low surface impedancc.the approximation is very good (i.e., where to /;,Po « I in the present context). Also. if the groundsystem taper is sufficiently rapid, the contribulionfrom the iDtegral in (4), where W'(p) dillers appre-

1-151

IIIITEGRAL EQUATIONS FOR RADIATION OVER INHOMOGENEOUS GROUND 79

ciably from unity. would be expected to be negligible,even when the upper limit po is effectively infinite.This fact is confirmed by comparing the computedpattern function P(l{Io) for the case (l) where theintegral equation is used to obtain W'(p) and (2)where W'(p) is replaced by unity. The comparativeresults are shown in Figure 6 for .,1.... where thevalue of Zo is effectively infinite. As we see from theplotted curves, the Kirchhoff-type approximation isoverly optimistic for small values of b when thetapered ground system is effectively infinite in extent.

CONCLUDING REMARKS

The results given in this paper apply to a ratheridealized radiating system. For example, the sourceis a Hertzian ground-based dipole and the groundsystem is represented by a surface impedance bound­ary that is allowed to taper smoothly from the sourceto the extremity of the system. To consider a morerealistic system. we should take into account theohmic losses in the ~ound, the inductive reactanceof the radial wire ground system, and the finiteheight and length of the source antenna. The inclu­sion of such factors constitutes a straightforward ex­tension of the present formulation.

In general, we feel that the intentional tapering of

the ground system can lead to major improvementsin the low-angle radiation characteristics of HFanten!!!!s.

Adnowlrdgmrnt. The continued interest in this work ofthe contract mODitor Philipp Blacksmith is much appreciated.

The ~search reported here was surroned by Air ForceCambrid,e Researcb Laboratories. Bedford. Massachusells.UDder contract PRO-CP-69·824.

REFERENCES

Bernard. G. D., W. Eo Gustafson, and W. M. Chase (1966),HF Jl"ouDd systems; ~sults of a numerical analysis, U. S.NallY EI,ct,onic8lAb. R,p. 1359. San DielD, Calif.

Collin, R. E.• aDd F. J. Zucker (editon) (1969), AntrnnaTh,ory. vol. 2, McGraw·HiII, New York.

Gustafson, W. E., W. M. Chase, and N. H. B:l1Ii (1966).Ground system effect on HF antenna propaption, U. S.NallY Elrct,on;c8 lAb. R,p. 1346, San Diego, Calif.

Wa~ner, C. (1953), On the numerical solution of Volterraintegral equ3tions, 1. Math. Phys. 32. 289-301.

Wait, J. R. (1963). The theory of an aDtenna over an in·homo,eneous ground rlane, iD EI,ct,omagn,t;c Th,ory.'and Ant,nnas. edited by E. C. Jordan, pp. 1079-1098.Per~3mon, New York.

Wail, J. R. (1967), Pallern of a linear nntenna erected overa tarered ground screen, Cnn. 1. Ph"s., 45(9). 3091-3101.

Wait. J. R., and L C. Walters (1963). Inftuence of a sector,round screen on the field of a vertical 3IItenna, NBSMonog,aph 60.

1-152

1.19 Wait, J. R. t and K. P. Spies. July 1970, "On the Radiation from a Vertical Dipole with anInductive Wire-Grid Ground System," IEEE Transactions on Antennas and Propagation,Vol. AP-18. pp. 558-560, (and an extended report version AFCRL-69-0404)

(Reprint No. 411)

Reprinted with the permission of the IEEE

1-153

It.prl..led loy "......,10.. / ...... IEEE TIlANSAC110NS ON ANTENNAS AND PIlOPAGATIONVol. AP·II. No.... Jul, 1970

pp.55&-560Copynpl 1970. b, ".I",lilule of EJeclricaland EJec:1nIftia ......... Inc.

PIUtn"ED IN THE U.s.A.

On the RacUation from a Vertical Dipole with an IDducti.eWire-Grid Grouod System

AIll."....-TII. far Ield of ...rtical electric dlpol. 011 • lectiollll1J,Iaomo,ea.u poad ,laD. Ia couldered. Tile epedIc model uedIa • dlelectric-like .,..ad .laicla Ia modlled b, DlIiDC III Iadacti.ewire crid or m.... eer.ea Ia • "cioll lanoudillc tile dipole. Attell­tiDa Ia f__ 011 til. modIiCltioa of tile r.diatioa ,.tten r..a1tiDcfroDl til. "....c. of tile Iadacti.. eroud .cren. It Ia d.Dloll­Itrated tlaat tIae 10....&1. ndiatioa me, b• .,••U, eDIwIc.4 b, •poDDd _ wlaida IZtIDdI oat Ie IS or mor....eI.......

JJw.oDVcnON

The performance of HF .ntennu is adye,..I)· influenced by thep,..nce of • poorly coad~tia, ,round. Unfortunatel)', for one_a or aaotber. it ie _etimes required to operate. commuDic..lion ayateln witb at I_t ODe terminal oyer ice, frvlen eround, oroiber dielectric-lib meteriel. In aucb _ the Iow...le radial ionie .,.atJy red~. One re",edy ill to raiae the antenna to a hei,htof _raJ _veJellltlaa. Wlaile tbis may luffice for eertAin applie.­lio_, it ..uaU7 ie _Dlpanied by deep nulls in tbe vertical radiationpattern, DOt to mention tbe increued C08t or the lupport Itructure.The otber remedy ie to e",plo)' a wire ,rid or meah eround ",atIMneatb tIae antenna. Not only doea tbis at,bilize Ibe impedance

IIUucrtpI .....ftCI ..p...ber 211. Ite.: ,ulled Pebnaa17 4. 1.70.

1-154

ANALYTICAL MODEL

1\" • Jr - (l + R.)/2 - lin 1I,/[lin II. + (Z/",l]. 131

fartor of I + n. A~. it can bt' Iho...n [2] th.t the cround wavefi...01 01 th.. dipol.. ill alllG modifil'rl hy the faNor 1 + n, ...here thi.is to he evaluat ..d as ~. -0. In fart, u "'e ,,·illae. belo..·• I + nis _ntiall~' indt'JICndenl 01 "'. lor .mall analftl while, of co_,th.. pattern f..tor II" itae'f vari linearly ...ith II•.

The ...orkinc formul. for 0 hich fonna the buis of tbe p,..ntpaper is civen by [2J

..·h.,. p. -(i"_/2)IZ'/",I' an,l F - IZ - Z·)/",. lIere. J, i..1M B_1 funrlioll 01 ord..r nile ...hil"lI'. is a "lIoum) ....v.. DIII!III1"­tionlullrtion" ...hirh .rf'Ount~ lor the modifi..alion ollhe t.II"ellti••n'AIMlir fi..ld b~' the finile "uri..... im)ll'llall"" Z' 01 the Crollllolo.,..n. Oft.n, 1\'. i. "'1,1......1 hy unit}· "'hieh i~ j""tifi..,I. "I .."lIr....il Z· - II and if bafk rl'lI.rlin'lII lrum th...,llEp "I th.. a'''",,'' ."n,,·11.,.. ilIlO~ (5). 011. m.~· ol"'f'rve that 1\'. i_ el""..,y rel.t ..d 10 til..,,'...'m. di.-re",ion fum'tion [Ii].

Artllall~', (6) "'u deriv..d Oil Ih. b",.i, thlll II, i. a 'lIIall I'"r"ml!lrr,hut ,hi,. L'''umJ,tinl1 i~ unt nvt'r'~' rr~'ri("i,,'(' "lUI ill f"rt ~. rnl1 hr..<..,111/,10:10' nr~n "'ilh 1I1',Iij:ihll' ",ror. AI-o. i, .1""tI,1 hI! "'f'lIli,,,...,1th"l th.. atl.nualinn funrli"n 11'. i. u,u:lII~' " ..rive.I on the b:l.i.Ih:lt 'Z·/... I« I bill, in far', Ihi, i. oVI',I.,· r"",irliv•. II, ill I"rt,Ihp \·"h." nl Z· i. I'nl,,'n~·PtI ,.· ..i,·.. i- al'prn",i"l .. Inr a,ll7.ina i,,-ri.I N·. 'I', joe ,·"lillfl\·t-n _h(tll Z',,,, i .. nut !'liliAn [!i]. lit nn.," c'n..:f',

lor 1, "1 l'Ul)-..... "'ill "'''umr thai Ih" 10/l1l1l1n.' nr.. vAli."Furlh.rmore. the mndili..d surl"... iml't'.ln".... z· ... i11 he imnaill".ltob.. roml'<....d of Ih. 1,,,r,,U..1 mmhill"lin" III th.. a"''''"' im.."tlA"rt'Z tenlu.lt'd al arazi"c ill..i"'..IIrt') a"d th....n.clive illl.....lll"''''Z. 01 Ih.. "'ire arid. Thu.~

11',(1<_) - 1 - ilrp)'11 eX» (-p) erfc (ip'll) (7)

-~I b I

(21n" • (l + R,W + III /2

Tbe lituation .. illustrated in FiC' 1. A vertiCAl electric dipoleis located on e.t around. The ellective .urf.re imredan.... of thel'Ound .yatem is • coNlt.nt Z' oul to • r.di..... II frnm th.. h...... nlthe .ntenn•. The around heyond rldiUll II is taken to have. f'Ol••tanleurlace impedance Z. In the pment formul.tion, ..........ume th"tthe .yatem is rot.tionally .ymmetric. Tbe colllequcnee5 01 th~

_umption are dise_d later on.The radiation field E. lor. limf' factnr exl' li...l) , .1 • di.t.nre Ii

and elevation angle 'I•• is expl'e!lllibl.. i,. the form

E i_/d, "llp l-ikRl-~lI"h~05~.) It

wh..re ... - 4r X 11I-'.1< • 2r/h,·"v..If'llalh) ••Ilcl/d~ i. Ihr ,,"rrNIImoment of Ihe dipole. Here. 11" is a "attern 'um·tio" "nrlll"li,rclluch that it is unil)' onr • perfect I}' eOlldurtill([ arou"d 1,la"t' 01infinileextelll (i.e .• Z· • Z .0). No.... lonoa·illa.n ....r"· cI..ri,·.tio"[2]. the first-order iteratioll 01 all illlt'([r.1 equalio" ~·i..I;I,

where H, - ["in II. - (Z/",) ]/[,ill II. + IZ/~,l]. 'I, - 1211•. ",,,In ill a "modific.tio,," ...hirh vDni.h.... il th.. arollnd 'rr...·11 i.. "10 ,,,(i.e., Z· • Z or if II -0). III the limitillC cue 01 110 c,nu",1 .rr ".

Here. Z/'" is tbe norm.lized ~urfl"" imredance of th." ulln.odifi..J"lround. If tbe latter is homog..neous .nd 1011I1_. "e have

of the antenna but, if it is luRiciellll}' exlended, the vertic.1 radilliollpattern will 8pproacb the ideal ellJlft'ted for. perfertJ}· conductingcround pl.ne. There is now ••ubst.ntial liter.ture [I }-[S] onthe lubjeet of extended cround _III conrine botb theoreticaland experi_nt.1 iny.ticatiolL~.

The cenen! theory of the radialion field bas been diselll'Rd ratherextensively. Tbe method IIIed is bued on tbe rompensation thcoremwhich is c"'I)' rel.ted to tbe Lorentz reeiprocit}· rel.tions forvector electrolllllllletic fielw.. In the .ctu.1 ealculalion.~. it i~ .....u"I1~·_umed that the current di~trihutioll over the arou",1 ~..r..t'llis identieaJ tll that for the tame .ntenna loc.ted over • perlert I}'

conduetine cround _no If tbe wire spacinC in the cround 't'reenor meah .. 'ufticientl)· amaJl thit m.)' be .n exrellent approximation.However, in the ceneral CIIlt', th.. currentll ellciled on th.. I[round_n will be .ttenuated or modified due to the nollzero v,lue ofthe ellective ,urr.re impedance. In the p,..nt rommunication,we examine this q_tion. To facilitate tbe dise.-ion, .n idealizedmodel is _pIoyed.

Z/'" • II/K)'''[I - (cos'll.)/K)''' Z· • ZZ./tZ + Z.) fill

M' ::0< h(l0/3) • ~. X 3.16.

K' =- h(3/2"') • II. X 2.12

wbile if K - 10,

wbere (Z/",) =- (I/K)"'[l- (I/K))'II••nd~.ilthecr••illa.II'I..in radians. If, for eumple, K - 3. "'e have

Z•• i.".

.. hr,.. Z • ",K-'"'' - Ii-I)'II, i- th. ~url"f'l' i"'I.....I."'.....I th..~r...",d fn, Ilra,i,,!: inri.I.." ..... II thl' ..·irp ari.....n'·;lIa i. III1I1'It 10....Ihall a .·avel"'llllh and il ohmir 1....<4'. ill thl' ,,·i..... are "t'Jtlill:ihlr,...t! k"o... thaI Z. is purel)' illllut'livt'. Thu...... ""t

...h..re • i•• dim..".iolll..., ,.."..Ianr.. l"ran'I!I ..,. A lEo",1 .1'1""';'mal ion '" • - Id/At) loa. [d/(2trt)] + So, ...h..rt' d i. Ih" ~I'ncill!:

1...1..·..11 th...·i.... in th. m..h. At i. Ih.. f,......"'''' ......·.v..I.."alh. t

i. Ih.....ire r.diu.'••nd S. i•• rllrrerti"" fartn, "'hirh i. "I'j:lij:ih'eil d:At i!' .umrit'n"}· "mall [I]. [a). For I"_,,t l"'l)IO!Cf'-, .. r ,. ill110' <1..·..11 on th.. ccomel ri..a'I,:.r.lllete", of tht' "'ire crid. bUI "im..ly"I...ilr the enf't'tive value 01 •.

w. ~I",,' th.I, in ,h.. prpct'1I1 "otr. Ih.. arou",1 j,. a"um...' tnI... " In••I tli..I...tri... Wltil.. Ih.. Ih....r~· h". 10....11 dl'vrlo 1 In' ".f'"lIr'~1 .Ii i,,,"h·t- .,nn1l4I, it .. illl1'liruo"" ,1", .li"wlI..:",iun if til n,ln'I\'f'

<lirl""Ir,r f'O""a"I i.. lilt.." tn "1' ",nl. Th;.. 11/'1:1""1 III ,·oll,hll'li.."t'II,r."I' i. "alid "·h.,, ./1\_« I...·h..re • i. the ('f)"durlivily 01lhe ,rou"d.

151Jr =-11./(Z/",)

where K j,. the rel.tive diel.elrir ro""I,nl o( Ihe di.,....I,ir a"""I<1Tbe eorrespondi!'!I: rault lor II' i. euet. If 110... "'e (OCUlI our .11 ..11"011on low eruine .neln, we see that

At. typic.lanele 013· we have I\' .0.111 .IId 0.165,10' II' -:I.n<l 10, IWJlet'tiveh·. This "cuthark" en...t. ,,·h.rt! II' v"ri... h"..~r1~·with ".. at low anclt'll. is. t'OrtIC'IUCllre olloc.tina thl! .11'"'''''' ""rr• dielectrie-type cround.

The modification of tbe .urlaee impedanc.. in • eirrul:u '.11;0"of rldilll b, from Z to Z', raulla in improvinl th.. futback h~ a

1-155

I£EE ftANaACTJONI ON ANTENNAI AND PROPAOATJOII. IULY 1970

PIa. I. MoeIIDcatloo. 10 declbrl•. of radiation lIeld of dipole _ullin.maa p...oce of ....uod ocreeo of radl... t.

A numeriral ".Iual ion or (G) "L~ r.rried ou, fnr " 0.;.1.. r""c..of Ihe p.rameler.l "'., trll, Ii. , .nd I. The r."c.. or inl~r.lln" "·Ltauitably aubdivided, .nd • G.\JllIIiA" quadr.lu" " ...~ ."I,It..<I 10eaeh aubinlerv." The del.iled I.hulalioIL~ or Ih.. "."..r" f"..'nrIf" .nd the .II-imporlftnl "ftramel..r I + II .....vail"hl 1-.." h.r..(7]. Here. we ..ill pn!llent *,me of Ihe raullo in Cr.l.hi 1form ."ddile.. the ph)'lie.1 ~ilnillra"re of Ihe r.di.lio" "allern.

In Fi,. 2 we iII..lr.le Ihe mlllltnilud. of I + II AI' • fun..tinn 01Ihe ltr.linlt .ne1e "'•. The ordi"at .. ,,·hi..h i. 20 Inlt.. 1 I + III ,.the decibel ralio of Ihe radial ion field ... ith .nd "·ith,,u' Ih.. ,ruu"d.)'Item. The radi.. b of the Iround .('reen is ll00l2w)loo or .huut16 w.velenetha .nd the rel.live diele('Irir <'On~t."t of Ih.. (1< 1 \lround is 3. V.rio... values of I, the norm.lized ....'een ,." .are abown. Nole the dependen('e on Ihe Ir.zinll an,l.. "'. a"d ,h..norm.lied _tanee I of the ...ire _h. Artuall~·.I - 0 <'Or""VO".loto a perfeetly eonduetinel<'reen of radius b Ii.e., the ,,·ire ",.(',n, dis elfeelively aero). The other two eurve- (i.e., for I - 0.1.".1 (12'COrNIpond to typie.1 v.lue- or the no,mftlized rea"I.".... All th..eu",_ in Filt. 2 iIIUlI"ale Ihat I + 1/ io .I""oa..hi"lt " "·..II·....'i,, 1Iimil ... "'. lends 10 aero. The n'&lnilud. or" Ihe or,lt,,:.... (or II, ..email values of "" is a meMure or ,ne abilily of Ih. Ilroun.1 ...r "to enhan('e both the 10...·.lIIle Iky wave radi.'io" ."d Ih.. ltrou".1...ft from the dipole. Immediatel)', "'e ... Ih", Ih. Iwe' Ilro","l I

CoNCLt'IION"

JlEFERESCEI

J. II. WAITIi:, r. SrIEO

&~~,\ Ueseal'l'h lAbs.Houleler. Colo. llCmY.!

~~reen i.~ nol ne..._ril)· II... O"e ,,·hi.·h io perfeclly eonductilllt.More ...iII be ~.id .boul Ihill.ter.

A partirul.rl~· inl lilll 'e.lu" or Fill. 2 i. Ihe mann..r in..hi"h II + n I varies ilh ""for the dillerenl value-or there.('Ian...I••r.meler I. In leneral, " .• _ lhal ... "'. i. in..re....... from I('ro,I I + n I de..reues 10 a minimum, then riHI &I"in to aubllidiarym"ximum, .nd tMn Olldllale- .·ith a ~in\JllOidal·like ripple. AsindiUled before (5), for the ea... I - 0 Ihe "att.rn e.n be interpreted...." i"Ieri""n.... bel"''''" Ih(' dire", r",li,,"i,n, or Ihe dil.n'. ""dIh.. t1irha..lio" from Ihe edlt.. of Ih"llrnu"d '''Fe('". The .il""li"" i•.imilar for nonzero value- of I, bUI no,,· Ihe fillt null in the pallernO<'''Ur.l .t lo...er .nllle!<. This is a <'OftlIequenre of the redured ph­v..ln"il)· of Ihe"lround w.ve" ell..iled b)· lhe dipole a10DI the llIOu"dtlt'r~f'n.

The dl1..ndenre of Ihe Ilround ureen modifi".Iion '.etor II + II I0" the exlenl or Ihe lround I<'reen is .ho..n in Fi" 3 for K - 3a"d for the ..ni..hillll~· ~malilruilll .nlliet'. Various values of the...a"Ianrl' p.r.mei..r I .re aOOwn. The d.....ndenee on A:b for theI - 0 ...... io 'ull~' enlll'Urll'nl ...ilh Ihl' ellperled PIOIJOrlion"Ji'~' ofII In ftrb)"'. A aimil.r trend i.~ obierYed for olher v.lu... of I u"10 .bout 0.1. For I.,.er values of I, there ia. marked dqradation,parlieularl)' for the I....r kb values.

Olh.r ('.I('ul.liollll (7) .ho... that for I vallJell of ord.r of 0.1Ih. lo,,··alllie radi.tion i.. maximized. This enhaneement etan be.ttrihult'rl to the trapped .lIri e ...ave "h""eler or the ltrullnd"·"'·e l'x..it('d over .n imped." hou"daf)' Ih.1 hL< ." ."I"eciahleilltlu..tiv. l'OmIJOn.."t. In f.el. Ih.. malt"illltl" of the Ilrollllll ...ave""".."u"Iin,, f""..Iio,," I", m,,~' .........1 lI"it~· if Ihe "nml'ric'".Ii-'s".... ""r"n,..I..r I" I i. ill Ih.. r""ltl' frum I to IU ..hil(' al Ihe-"n... lim...'11" > o. Th Mlltlitinno "'" me' ov.r • porlion orIhe di.lanre r.llle in the p nl ..Iculalion~.

Thl' "",ulls li"(',, ill Ihi. ""I.. iIIu'lra'e Ih.. r",h..r .ilt"i(j,·ft"ln\ll.IiIi.."lin". for IOn indu..liv....tn... Ilr"III111 ". "f Ihf' ra,liftlio"rrum \IF 1"1..,,nll.". It io h.lir,·..d I".. 1, "1 ,·al,'ulali"". arr Ihrlire. tIl .hn,.· Ih••ITI'.·I .., " IIUIl.-i·lrll"I 1 sur1ft,·.. "'''''1' ""'il,~1 n..Ihr Itrollllli ."....". While Ih.. nllld.,\ ..11IL.i.t. or a" Izinllllh31 'ylll·m..lri ".I..m. the r",,"lt. "an h.. apl'li".1 "'ilh "..me ronlidl'n... 10....." _h",....1 Ilrou..d orreen. if th" s....tnr is dire.'ted 1""'1\1.1 the",,'t'ivi"l!: 1",i"I, or roul'«'. Ih.. ""Ilulnr ,,·i,I," ..f Ih......."" ,hnul,1lit- !lulfirif'n"~' IfI~(, to flllrulII'u,""'~ ftt Ir:l.o,;l out" .'"rrsnrl J.nllr. ftS

in.hrlOlf'C1 ill !lOme d..l"il.IOC(',..h"... (5).Th.. n.ajnr ...."..hl.io" from Ih.. 1' "1 "·,,rk i- thllt • l..rr"c'I~'

..n"tlu"li"lt ltrou",1 ...teen m,,~' ""' ' Ihe h..sl ,rllllllli "rr('t''' irlu,,-.ftn«lfl radiftlinh j" to !lfl maxirni7.(ltl. "'f' rtnphn.o,;i:u·, Iu.,,,·~vf",

Ihsl Ihi" rn""lu"inn must'" I.m!,..re,1 h~' Ih.. r..aliz.,i"" Ihal Ih"\·..rI i..,,1 r.di.lio" "al'('IIIS.re I......Jt'Cirahl(' illh.. I",,-·.ulll. radi" , ill".. n.uift\ized.

III J R. ".".od w. A, rop<'. "lnplI' r••I•••n.... of LF unlpolr ar'I.'...·..·..,1,.. E",.... \01. 32. PP 131-138. M.) 19:1:1.121 J K \\"alt. "T~f' Utfl)1)" of an &"'4"nn& 0\-'" an tnhomornt'oul

.round pia,",:' an £I,rtromo:rn,.ur Throrl/ ."" Aft,,.,,nOJ. E C.Jnrdan. F.d. O.ro,d. Enslan I'('rs.mun. ,.y. 10711-109•. IlIG3.

131 \\ E Guo••rlOn. W, M ('ha••. anrl 1". " lIalll, "Uround .)·....m~~:~~~':.~. ~:n ·l;::;:'·6~f"ft~~,:n.~~.~. r9r.~a,·)· EI.-ctronlcs Lah.

loll J ~1. lIorn. "A numf'rtcal anal).I. of IIF ... 'c-ndrd·IIC"r'or .rnundI,_.pmoe:· ", !t. ~.\') Elr<'tronl ..." J..ab. (C.-n,rr), "an Uholto. CaUl.•Itrp. I no. Jan. IIIG; ..

•!l1 J It \\" 31t. "("harartrr'&"'" n' .nl,·nn:l. nn"' tn".)' "n" h. In ..'n,,,.. ,,n71l,. ..,I'. It F. (.'uilin and t· J ZIu-kf\r. t:,,- !'litO" \ ..rk Mrliru"·

IGI ~tui.>~~1~i ..~' .:!ndC"IS21, CnnlA' Tit,. "/a.mo n .."rsiftn "·vftCfion.:\:,." \'or'" A,.df'mlr P"n. 11\(,1

I;, J. II "ait and 1\. P ""."1. ··aldiatlon from. nrllral dlP.!c'l.. '1I1h~:ul:~~o~I~;;.~lp;r~i.~ ~~:!',,~:i. '~~l~~:i; 2~:~'oo:I~' AIi Cit I,O~··

K-3

kb -100

kb

DIsctJlllIION OF NUIIERICAL RUULTS

----... \"', \,

'\ 0.1 \

" \0.2 \ \, ~

" \/1, /J\ ,' ...

DeDeDdeDCle on wlrt..mesb .cref'Ddl~mf'~r'or,·.rlous ft&C'taattlof wire meall for" - 3.

GI 10...•5O

c:; •~

&

4~

2

0O'

Fl•. 3.

18

GI... 16.5O

c:;+ 14

12

1-156 II

AFCRL-69-0404 4070000 -05117 8000

RADIATION FROM A VERTICAL DIPOLE WITH AN INDUCTIVEWIRE-GRID GROUND SYSTEM

by

James R. Wait and Kenneth P. SpieaEnvtronmental Science Servtce. Adminiatration Reaearch Laboratorie.

Boulder, Colorado 8030Z

Contract No. PRO.CP-69-8Z4

Project No. 5635Taak No. 563506Work Unit No. 56350601

Scientific Report No. 47

September Z3, 1969

Diatribution of thl. document ia unlimited. It may be releaaed to theClearingbouse, Department of Commerce, for aale to the general public.

Contract Monitor: Philipp BlackamithMicrowave Phyaica Laboratory

Preparedfor

AIR FORCE CAMBRIDGE RESEARCH LABORATORIESOFFICE OF AEROSPACE RESEARCH

UNITED STATES AIR FORCEBEDFORD, MASSACHUSETTS 01730

1-157

RADIATION FROM A VERTICAL DIPOLE WITH AN INDUCTIVEWIRE-GRID GROUND SYSTEM

James R. Wait and Kenneth P. SpiesESSA Reaearch Laboratories, Boulder. Colo. 8030Z

Abatract

The far field of a vertical electric dipole on a aectionally homo­

geneous ground plane is considered. The specUic model used is a

dielectric-like ground which is modUied by USing an in,,' ~ctive wire

grid or meah screen in a region surrounding the dipole. Attention is

focussed on the modification of the radiation pattern resulting from

the presence of the inductive ground screen. It is demonstrated that

the low -angle radiation may be greatly enhanced by a groun~ screen

which extends out to 15 or more wavelengths. However. some rather

curious e({ects are nO'ed, such;u pronounced lobes in the vertical

radiation pattern when the reactance of the meah acreen ia of the

order of 0.1 times tile characteristic impedance of free space.

1-158

-l-

INTRODUCTION

The performance of HF antennas is adversely inUuenced by the

presence of a poorly conductinc Iround. Unfortunately. for one

reason or anothflr, it is sometimes required to operate a communi­

cation system with at l..st one terminal over ice, frozen ground. or

other dielectric-like material. In such cases, the low-angle radiation

is Ireatly reduced. One remedy is to raise the antenna to a height

of several wavelenlths. While this may suffice for certain appli­

cations, it usually is accompanied by deep nulls in the vertical

radiation pattern, not to mention the increased cost of the support

structure. The other remedy is to employ a wire Irid or mesh

Iround mat beneath the antenna. Not only does this stabilize the

impedance of the antenna but, if it is sufficiently extended, the

vertical radiation pattern will approaCh the ideal expected for a per­

fectly conductinl ground plane. There is now a substantial literature

(l.SJ on the subject of extended ground screens coverinj both theo­

retical and experimental investigations.

The leneral theory of the radiation field has been discussed

rather extensively. The method used is based on the compensation

theorem which is closely related to the Lorentz reciprocity relations

for vector electromalnetic fields. In the actual calculations, it is

ueually aeeumed that the current distribution over the Iround ecreen

is identical to that for the eame antenna located over a perfectly con­

ducCinl Iround ecreen. U the wire spacing in the ground screen or

mesh is eufficiently small this may be an excellent approximation,

However, in the leneral caee, the currents excited on the ground

screen will be attenuated or modHied due to ti.e non-zero value of the

effective surface impedance. In the present paper, we plan to ex­

amlne th,S question. To facilitate the discussion, an idealized

model is employed.

1-159

-3-

THE ANALYTICAL MODEL

The situation iB illustrated in Fil. 1. A vertical electric dipole

is located on a nat ground. The effective surface impedance of the

ground system is a conscant Z' out to a radius b from the base of

the antenna. The ground beyond radius b is caken to bave a conscant

surface impedance Z. In the present formulation. we as sume that

the system iB rotationally symmetric. The consequences of this

assumption are discussed later on.

The radiation field E. for a time factor exp liwtl. at a

distance R and elevation angle "0' is expressible in the form

Ei ~ w Ids

o

z"-ik R

W· (COS" )!....--o R

(1)

k = Z"/(wavelengthl. and Ids is tbe-7

.. " • 10 •where ~o

current moment of the dipole. Here. W' is a pattern function

normalized such that it iB unity over a perfectly conducting ground

plane of infinite extent Ii. e •• Z· = Z = 0). Now. foUowinl an early

de rivation IZ].

(ZI

where R = Isin" - (z/." ))/lsin" t (z/." I). ." = lZO", andv 0 0 0 0 0

n is a "modification" wbich vanishes if the Iround screen is absent

Ii. e., Z· = Z 01' if b - 0). In the limitinl case of no Iround screen.

W' =W = (ltR)/Z = sin"/lsin" t (Z/.,,))v 000(31

Here, Z/."o is tbe normalized surface impedance of the "unmodified"

ground. If the latter is homogeneous andlossless, we have

(41

1-160

(€I)

·4-

where K is the relative dielectric Constant of the dielectric ground.

The corresponding result for W is ellact. U no·"e focus our

attention on low grazina anales. we aee that

W .. ~ /(Z/r, ) (5)o 0

wbere (Z/"o) .. (l/K)t(l - (I/K)l'. and "0 ia the IIrazinl angle

in radiana. U. for examp~e. K = l. we bave

W .. ~o (l/Z') "0' 2..1Z

wbile. if K = 10.

W .. ~o (lO/l) = "0 x 3.1&

At a typical anile of 30

we have W " 0.111 and 0.1&5 for K " 3

and lO. respectively. Tbis "cut-back" effect,where W varies linearly

with ~o at low anales.is a con.equence of locatinll tbe antenna over a

dielectric-type IIround.

The modification of the surface Impedance in a circular region

of radiua b. from Z to Z'. reSults in improving the "cut.back"

by a factor of 1 + O. Alao. it can be Shown (Z) tbat the around wave

field of tbe dipole ia aho modified by the factor 1 + n. wbere this

is to be evaluated aa ., - O. In fact. aa We will aee below. 1 + no

ia esaentially independent of "0 for small angle a While. of courSe.

tbe pattern factor W· .tself varies linearly with "0'The working formula lor n which forma the basia of the present

paper ia given byb

n", _.!..L. Se-i kp W Ikp)(l +cos"o g-

owhere • -p ,

W Ikp) = 1 - dllp)' e erIc Ii p )I

1-161

(7)

-5.

i k "( Z',,\zP = - T T) and F ~ (Z-Z'llr

oHere. J1 i. the

Be••el funchon 0 of order one while W i. a "ground wave attenu-I

ation function" which accounts for the modification of the tangential

magnetic field by the finite .urface impedance Z' of the around

.creen. Often. W i. replaced by unity which is ju.tified. of course.g

if Z· = 0 and if back renection. from the edge of the ground .creen

are ignored (5). Some .implification. and limiting ca.os of (01 are

di.cuued in the Appendix.

Actually. (01 wa. derived on the ba.i. that 410

i. a .mall

parameter. but thi. a ••umption i. not overly renrictive and. in

fact. tl can be u.ed up to 300

or .0 with nealiaible error. Also.o

it .hould be mentioned that the attenuation function Wa

ia u.ually

derived on the ba.i. that Iz'/., I « 1 but. in fact. thi. i. overlyo

re.trictive. If. in fact. the value of Z· i. employed which i.

appropriate for gradng incidence. Wg i. valid even when z'/Tlo

ia

not .mall (5-7). In any ca.e. for pre.ent purpo.... we will a ••ume

that the formula. are valid. Furthermore. the modified .urface im­

pedance Z· will be imagined to be compo.ed of the parallel combina­

tion of the around impedance Z (ev.\luated at gra&ing incidencel and

the effective impe4allce Z of the wire grid. Thu••a

Z' Z Z I(z + Z Ia a

(81

wbere Z = Tl K-f (l - K-ll i. ia the .urface impedance of theo

ground for gra&ing incidence. If tbe wire grid .pacing i. much leu

than a wavelenatb and if obmic lo••e. in the wire. are negligible. we

know that Z ia purely inductive. Thu•• we .etaZ i Tl &a 0

1-162

-6-

where 6 i. a dimen.ionle.. reactance parameter. A good approxi­

mation i. 6 • (d/l. ) 1011 [d/(Z. c)] + 5 where d 18 the .pacingo , cbetween the wire. in the me.h, >'0 18 the free-.pace wavelength.

c 18 the wire radiu., and 5 i. a correction factor which i.c

negligible if d/>.o 1••u!!icienUy .mall [1, 5]. For pr..ent purpo....

we will not dwell on the geometrical parameter. ot th. wire grid. but

.imply .pecUy the effective value of 6.

We .tre•• that, in the pre.ent paper, the ground 18 •••umed to

be a 1o••le.. dielectric. While the theory ha. been developed for

a .eneral di..ipative ground, it .implifi.. the di.cu..ion if the

relative dielectric con.tant 18 taken to be real. Ttli. neglect of

conduction currenU i. valid when -;1K ( w «1. w here a 18 theo

conductivity of the ground.

1-163

-7-

DISCUSSION OF NUMERICAL RESULTS

A numerical evaluation of (6) was ca rried Oul for a wide ranlle

of the parameters tlo

' k b. K. and 6. The ranlle of intellration

was suitably .ubdivided and a C.Uulan quad.ature was applied to each

",ubinterval. The detailed tabulahons of the pattern factor W' and

the all-important parameter 1 + n are available from the authors.

Here. we will present some of the results in Ilraphical form and

discu" tile physical signiftcance of the radiahon patterns.

In Fig. Z, we illustrate tile magnitude of 1 + n aa a function of

the arazing angle '0' The ordinate Which ia ZO 10110 11 + n I ia

the decibel ratio of the radiation field with and without the ground

system. For this liaure. the radius b 01 the ground .creen is

(lOO/Z" p. 0 or about U. wavelenath•• and the relative dielectric

conatant of the (lo••le••) around i. 3. Varioua value. of 6. the

normali:&ed Icreen reactance. are .hown. Actually. 6" 0 correa­

ponda to a perfectly conducting Ie reen of radius b Ii. e •• the wire

spacing d is eflectively zero). The other two curvel (i. e •• for

6 " O. 1 and O. Z) correspond to typical valuel of the normalized

reactance. All the curvea in Fig. Z illustrate that 1+ n is approach­

ing a weU-elefined limit al ~o tendl to zero. The magnitude of the

orelinate for tbe.e lmall valuea of ~o is a mea.ure of the ability of

the ground Icreen to enhance both the low-angle Iky wave radiation

anel the ground wave from the dipole. Immediately, we lee tbat the

"be.t" around acreen i. not necellartly the one which i. perfectly

conductina. More will be laid about thi. below.

A particularly interuting feature of Fig. Z is the manner in

wMch 11 + n I variel with I/J for the different valuel of the reactanceo

parameter 6. In general, we lee that as "'0 is increased from zero.

11 + n I decreales to a minimum then rile. again to .ubsidiary

maximum and tben olciUates with a slnuloidal-1ike r\pple. As

1-164

ItIIIIIIII

-8-

indicated before [~1. for the case ~ = O. the pattern can be inter­

preted as an interference between the direct radiation of the dipole and

the diffraction from the edge of the ground screen. The situAtion is

simil~r for non-sero values of 6, but now the first null in the pattern

occurs at lower anllel, Thil il a consequence of the reduced phale

velocity of the'.round wave" excited by the dipole along the ground

The influence of the extent of the ground screen is illultrAted in

Fil. 3 for 6 = 0 and in Fig. 4 for 6 = 0.10. A s expected. at low

angl.s. the pattern modification factOr 11 + aI is proportionAl to

(kblt for 6 .. O. but thil il only approximately true for 6 = 0.1 •

Alao. we s.e from Fil. 3 thAt. for 6 =O. the factor 11 + aI hAs

a relatively weak dependence on 1P0

' at least for low angles. Again.

as expected,for 6 .. 0 the first null occurs at the lowest angles for

the larlest value of k b • A similar but more marked trend is seen

in Fia. 4. Here. aaain the relauve Slowness of the ground wave on

the screen is producina a more drastic mutilAtion of the pattern.

The dependence on the relative dielectric constant K of the

ground is illustrated in Fil. S for k b =100 and 6 = 0.10. While

only two values of K are shown (i. e .. 3 and 101, the results are sufficient

to demonstrate that 11 + al. at low angles, is greater for the lower

value of K [i.e., around acreena are most effective (or the poorer

ground conditions). Also, we note from Fig. ~ that the general (arm

of vertical radiation pattern Ii. e., the variation with 1P01 is not

critically dependent on K.

The dependence of the around screen modification factor 11 + alon the extent o( the ground screen 18 shown in Fig. b for K = 3 and

for the vanishingly small grasing angles. Various values of the re­

actance parameter 6 are shown. The dependence on k b, (or the

6 = 0 case, is fully consistent wltn the proportionality of a to (k bl'. as

dl~culaed in tne Appendix. A s.mllar trend is observed for other

1-165

-9-

values of 6 up to about 0.1. F 0" larger values of 6. there is a

marked degradation. particularly for the larger k b values. The

same e!fect is observed in Fig. 7 "'hen K = 10 .

The actual dependence of 11 + nI on 6 is illustrated in Fig. 8.

Here. we see that, for 6 values of the order ·~f O. 1 • the low-angle

radiation is maximized. This enhancement can be attributed to the

trapped surface wave character of the ground wave excited over an

impedance boundary that has an appreciable inductive component. In

fact. the magnitude of the Iround wave "attenuation function" WI

may exceed unity if the numerical distance parameter Ip I is in the

range from about 1 to 10, While at the same time, al'l p> 0 •

The.e coolditions are met over a portion of the distance range in the

present calculations.

The departures of the attenuation function Wg from unity over

the range of the distance p are illustrated in Tables 1& and Ib for a

selected set of parameters. The values of Wg based on (7) are shown

in the tables in complex polar form. It is evident. for the distance

ranges involved. 'N I exceeds unity by a substantial amount. How-I

ever. if the reactance 6 is increased beyond about O. 1, it is evident

that IW I decreases signi£lcantly below unity. Thus. the behaviorI

of WI indicated in Tables 1& and Ib is consistent ",ith the feature.

of the radiation patterns discussed above.

1-166

-10-

Table 1&

Amplitude and Phase o! W for K = 3g

6 = O.OZ O. I O.Z

kp 10 I. 055 (-3.4°) 1. ZZ4 (-19.7°) 1. HI (-41. 5°)

ZO I. 078 (-4.8°) I. 3Z3 (-Z8. 5°) 1. 306 (-60. Zo)

50 I.IZ7(-7.70) 1. 5Z3 (-47.1°) I. 335(-99. 3°)

100 I. 183 (-11. 1°) 1.744 (-69.8°) I. 179 (-146. 0°)

ZOO I.Z68(-15.90) Z. 004 (-105.3°) 0.696 (-146.8°)

300 I. 337 (-19.8°) Z.IZO(-135.Zo) 0.310(-99.8°)

Table lb

Amplitude and Pha8e o! Wg

for K .. 10

6 O.OZ O. I O.Z

kp 10 I. 053 (-3. 5°) 1.169 (-ZO. 1°) I. 075(_38.1°)

ZO I. 076 ( 4.9°) I. Z37(-Z9. 0°) I. 069 (- 54.4°)

50 I.IZ3(-7.90) 1. 36Z (-47. 4°) 0.990 (-86.9°)

100 1. 178( -II. 3°) 1. 477 (-69. 4°) O. 81Z (-IZ3. 0°)

ZOO 1. Z59 (-16. Zo) 1. 563 (-IOZ. 8°) 0.495(-169.4°)

300 t 3Z4 (-ZO. Zo) 1. 546(_130.1°) O. Z86 (-16Z. 7°)

1-167

-11-

CONCLUDING REMARKS

The results given in thia paper illustrate the rather significant

modification, for an inductive-type ground screen, of the radiation

from HF antenna.. It is believed the present calculations are the

£irst to sllow tile effect of a quasi-trapped suriace wave excited on

the llround screen. While the model consists of an azimuthal symmetric

system, the results can be applied with some confidence to sector­

shaped llround S••-':"'ns if the sector is directed toward the receivinll

point. Of course, the angular w;Jth of the sector should be sufficiently

1arlle to encompass at least one Fresnel zone, as indicated in some

deta,l elsewhere [S).

The major conclusion, from the present work, is that a perfectly

conducting llround screen may not be tbe best llround screen if low­

angle radiation is to be maximized. We empnasize, however, that

this conclusion must be tempered by the realization that the vertical

radiation patterns are less desirable if tbe low-angle radiation is

maximized.

1-168

APPENDIX:

-ll-

Analytical Simplifications

While the general equation (6) can be employed for numerical

work. it is desirable to investilate the analytical structure of the

intelral form and examine certain limitinI cases.

First of all. we decompose (6) in the following manner:

wbere

and

oa

ka

F S -ix (, 1:...."\- -- e W(x)~ +""iJ J t (xcos~) dxcos !Po I I 0

o

bk F S -ikp Q 1 "\- --.,- e W(kp) 1 + .-k )Jtlkpcos~) dp

cos.. I I p., 0o a

(9a)

(9b)

Now we usually choose the parameter ka so that, for 0 < x < ka.

W(x) can be approximated by 1 in the integral for 0a while, at the

same time, ka is sufficiently large that J t (kp cos !Po) can be ade­

quately represented by its asymptotic approximation in the intelral

for 0b'

With the judicious choice of ka indicated, we find that

where

oa

(10)

Ci (ka)

ka

e iff/ 4 Se -iX(l + 1....', Jt

('" cos I/l ) dxtJV 0

o

(11)

is an integral which has been tabu""ed and discussed rather

thoroulhly [BI. Thus, we can dispense with it.

1-169

-13 -

In the integral for nb

' we now utilize the fact that k p » 1.

Then. in the integrand,

I + _1_ ... Ii k P

and

. -i k p(l +coa "oIJ'- 1 e •

dp

bI [S W(kpl e -ik(l - coa !Polp

(ff kit p~a

(9bl ia approximated by

e iff/

4 k F

(Z coaa!p Ito

Then.

n ...b

(UI

Now, for small analel of 1/10

' we lee that the fint integral in aquare

bracket. above ia dominant becauae of the hiahly oaci1latlng term

exp [-i k (l + coa !Polp] in the aecond integral. In fact. thia aecond

intearal can be interpreted aa the contribution from the back edge of

the ground acreen. WhUe thia contribuuon could be retained without

difficulty, we will diacard it here.

Uaing the definitiona of

ia aiven by

F .''It/4 "'k);Sbnb ... ( a i L~.

Zcos !Pol a

b

i k; cz; Se -czp

a

W(kpl liven by (71. we now find that (61

-i k (1 - coal/lolpe d

? PP

(131

where a = -i k (7..'/"oll /Z. Now. the first integral in the aquare

bracket above can be converted to a Freanel intearal and tile aecond

1-170

lation. we lind thac

can be evaluaced by intearation by parn. Thull. alter lome manipu-

IIIIIII

-

n •b

+

'k(l .'1 (4kb/lll'lin(Cl/ZII -COli.. S 0

es+ik(l-eOIl~ I exp( -i(tr/l) ZZ] dz

o (4 kal Iflt lin (1/1 l'l)o

i (k eslt , ,es+ik(l-cOllfl) (~xp (- esb • ik(l'COI"o)b] erfc (i(Gbl ]

o

I(14)

U now "0 ~ O. we obt&in the limiting form

U. in addition. lince Ies a I hal already been auumed lmall.

Fei3ft/4 k t ."b tnb • zt (;) [e erte (i (esb) ) • 1]

II

n ..b

F i)"I. ~ t b t fe zi \,.-;) [e -e erfc (iles b) ] • e -ea erfc [i(ea) ] ]

(5)

(16)

In the extreme lbnitina cale where I es bl «1. we obt&in the aimple

form

which il W.,U known (sl. It correaponda Co the caae where the around

wave auenuation function WI iI effeetiyely unity over the wllole ex­

C.,nl of the Iround Ie reen. Thia can be aeen by returninl to 113) or

IIIIIIIII

(H) and aettinl "

F e i ,,/4n •b (ZCOI3 ,,)'

o

O. Wllinee ,(4 k bJ\r) lin (l/I IZ)

a,n I~ 721 r exp (-i hr/Z):Z] d zo 14 k aJ\r); ain 1$ IZ)

o

1-171

(l8)

·15-

which is also well known (5J. Then. it now we let ~ - 0o .

n • F(ZkI'll')t (bt _ at) e i ll'/4b

F(Zkb/ll')t e i ll'/4

which is consistent with (17) above.

REFERENCES

(19)

(1) J. R. Wait and W. A. Pope. "Input resistance o! LF unipole

aerials ... Wireless Enlineer. vol. 3Z. pp. 131-138. May 1955.

(ZJ J. R. Wait. ''The theory o! an antenna over an inhomoleneous

,round plane ... in Electromasnetic Theory and Antennas, E. C.

Jordan. Ed. Oxford. Enlland: Perlamon Pre... 1963.

pp. 1079-1098.

(3J W. E. Custabon. W. M. Cha.. and N. H. Balli. "Cround system

eHect on HF antenna propagation. "Report No. 1346. U.S. Navy

Electronics Laboratory (C.nter). San OielO, California. 191>1>.

(4) J. M. Horn. "A num.rical analysis a! HF .xt.nd.d-••ctor ground

syst.ms... R.port No. 1430. U. S. Navy El"ctronics Laboratory

(C.nt.r). San DieiO. California. Jan. 1967.

(5) J. R. Wait. "Characteristic. o! ant.nnas ov.r los.y earth...

Cn. Z3 in Ant.nna Theory. Part l. R. E. Collin and F. J. Zu;k.r.

Ed.. N.w York: McCraw-Hill. Inc•• 191>9

[I» J. R. Wait and C. A. Scalak, "N.w asymptotic .olution for the

electroma,netic £ielels o! a dipole over a .tratUi.d medium...

El.ctron. Letten. vol. 3. No.9. pp. 4Z1-4ZZ. S.pt. 191>7.

(7) R. J. KiDl' "EM wave propa,ation over a con.tant impedance

plan.... Radio Science, vol. 4. No.3, pp. ZS5-Z&8. April 19&9.

(8) J. R. Wait and L. C. Waltere. "1A£luence o! a ••ctor around

.creen on the field o! a vertical antenna... NBS Mono,rapta No.

&0. IS April 19&3.

1-172

IIIIIII

Fi.. I The e..ential ,eometry .howina a .ide view of the dipole on

the nat 'l'oWld plane.

16 - " kb =10014 " K= 3

--- \... ....\""10 " 0.1 \II) ,

~ \

£,

\ \8 \c::+ 02\

\. ,6

, /\ \(;\

\ I4

,I'- ....

t

t

tI

I,

2

Fi•• l The mo4ific:&tion. in d.cibel., of the radiation fi.ld of the

elipol.....ulliD' from the e.....nc. of Ut. ,round aC"e.n of

raeliua b. He...,.e.bow til. d.pend.nc. on til. ,raain,

anile .0 &Del til. nonnaUaed ...act&nc. 6 of Ut••ire m••h.

1-173

18

16

m 100'0c:

12CI kb=50+- 10

Fil. 1 The dependence on Irazinl anile for Yarioue wire-meeh

ecreen diameten for effectively zero wire epacinl.

m'0c:

CI+-

16

8

6

100

kbz50

8 z 0.10

Fl,." The dependence on ,razinl an,le for Yarioue wlre-me.h .creen

diameten for non-zero ware epac,nl.

1-174

t

16

14 kb =100

8=0.10

12

CD 10"0

.58c;

+- 6

4

2

Fil. S The dependence on Irazinl anile for two) dielectric COn8tanu

of the Irollnd.

CD"0

.:c::+-

FII. II The dependence on the wire-me.b .creen diameter for "'1'1011.

reactance. of the wire me.b for K A 3 •

1-175

16

14 K=IO

CD~ 12.=c::

10"'0=00

+-8 0.2

60 100 200 300

kbFil. 7 The dependence on the wire-me.h .creen dia.meter for variou.

reactance. of the wire me.h for K. 10 .

,J

.", =0°o

2

12 kb= 100

a:l 10"Cc:

c: 8+- 6

4

8Fia.' The dependence on die ellecti"e reactance of the wire mee"

lor two dUler.",c ·Ii.leeerie eOn.tanU 01 U,e aro_d.

1-176

1.20 Hill, D. A., and J. R. Wait, January 1973, "Calculated Pattem of a Vertical Antenna witha Finite Radial-Wire Ground System," Radio Science, Vol, 8, pp. 81-86

(Reprint No. 488)

Reprinted with the permission of the American Geophysical Union

1-177

ltIIdio .sc.,,", Volume 8, Number I. pqes 11-86, January 1973

Calculated pattern of a verticalanteDDa with afinite raclial-wire pound system

Daid A. Hill IIIfII Itllllu R. Wait

IIUIiI..t~ '0,. l'~IKOIrtm,,"i",'1D1 Sci~,,"$, Ol/ic~ 0' l'~l_mm ..llic,,'iOIl$US D~,."m,"' 0' C__rc" BOMld~,., CoIorllllo IOJ02

(Receiwed AulUst 7, 1972.)

Tbe apprapriate form for abe lUfface impeduc:e of a radial-wire pouDd lerecn is utilizedto obIaiD quantitative eIIec:ts of the impor1allt parametcn (number of radials, Icnath ofradia", wire radius, aDd around coastants) on the low-anile radiatioD of a wrtical antennl.An apprcWmltc correction term is derived to account for CUIRIIt. reftcc:ted from the edaeof abe screen. The dfed is ....u for Iarae acrectIS, but it may be impor1allt for IIIII1Icr SCreeDS.

. cap (-,.,)(1 + 1/lk,)J.(k, eoa ,y) dp

wberell is the fint-order Bessel function and W'(k,)is the unknown attenuation function.

The determiDation of W'(k,) is a considerabletask in itself, :lnd consequently, we choose to makethe simplifying assumption W'(kp ) = I. The validityof this assumption for an nponentiall)' tapered im-

"'

INTRODUCTION

Wire ground screens are commonly used to en­hance low-angle radiation from HF antennas. Thetheory of impedance ground screens has been treatedquite aenerally [Wait, 1963, 1967a; Collin tmdZuelen, 1969), and extensive numerical worlt hasbeen done IWait and Walle,s, 1963; Horn, 1967,1968). The complication of an exponentially varyingimpedance ground screen bas also been ell8mined[Wail, 1967b), and an integral-equation approachhas been used to determine the attenuation of theJrOund wave over the screen [Wait and Spies, 1969,1970).

Here we consider a radial-wire ground screenwhich bas a variable impedance as a result of itsvariable wire spacing. Through the use of such aground screen, the dependence of the low-angle radi­ation on the important parameters (number, length,and ndius of radials) can be directly examined..Also, the importance of the wave relected from theedge of the screen is evaluated by an approxmiatemethod. Such an evaluation is importat'lt since thepossibility of such waves is ipored wben using theulual impedance representation of the ground screen.

VAlUABLE IMPEDANCE FORMUlATION

The pometry of the transmitting antenna and cir­cular screen is shown in figure 1. The screen is ofradius a while the source is a Hertzian dipole of cur­rent moment Illacated at the ori!in. As indicated,tbe elevation an,le ." equals' - ",/2. In addition, Z

is the impedance of the ground, while Z'(,) is theparallel impedance combination of the screen aDd theground.

The far-zone radiated magnetic field witb noscreen present, H.O(",), is given by the geometrical­optical formula

H:Cit) - IlIkll exp (-iJu)J!4,,,"(J + R.) cos ,y (I)

where

R. - (siD ,y - (Z/lh)J/(sin ,y + (Z/"o»)

Z/". - t. -.'1(1 - (cos' ,y)/t.t"where c, is the relative complex dielectric constant ofthe ground, Ie is the wave number, and '10 is the in­trinsic impedance of free Ipace. In writing (1), andin what follows, exp(i.J) time dependence is as­IUmed.

With the screen present, the far-zone magneticfield H.(~) is liven by

H.(,y) - H:(,yXI + Q. + 0,) (2)

where 0,. k a correctinD for the wave reBected fromthe edge of the ICIftIl and wbere..Q. - (-Ic/eoa,y). W'(Ic,):lZ - Z'(p)JI",,1

"

1-178

12 HILL AND WAIT

V.,. I. Vertical electric dipOle located at the teDler of aradial-wire .rouDd ICrHn.

z ~---+---'--. p

far-zone pattern. Since a rigorous computation of thereflected current would necessitate the solution of anextremely complicated boundary-value problem, wechoose to make the following simplifying assump­tions: (I) the reflected current at the edge is thenegative of the primary screen current (equivalentto -I reflection coefficient), and (2) the reflectedcurrent propagates inward with the known propaga­tion constant for a wir;: located at the interface. Ifanything, assumption (]) above will be an overesti­mate of the magnitude of the reRected wave.

Now, the primary surface-current density carriedby the screen at ,. =a is given by

Z. - (iI.I".pl N) In (pI Nb)

where b is the ndius of the wire (assumed perfectlyconducting) aud N is the number of radials. By sub­stituting (3) into (2) and making a change of vari­able, we obtaia a form suitable for computation:

0. _ -(Z/,.)

cos'"[

(exp (-lx)J(I + I/IJt)J.(x cos "') (4)•• J + /1Jt/(Z/".)NJI (In (xl Nkob») dx

Evaluation of (4) and numerical results are dis­cussed Jater.

pedaDc:e bas been studied by Wail aM Spies (1970J,ad they point out that the assumption is essentiaUya pbysical-optics or Kirchhoff approximation. TheirDumerical results iDdieate that this approximationyields. slightly larJet' value for I + n. than themore riJOl')\lS c:aIculation ia some cases, but almostno difference in others.

The parallel impedance of the screen and groundfor grazing incidence is given by [Wait. 1959J:

(I - R.hin 1/1 ... (I + R.XZ/"..) flO)

J:(o) = [E,(o)]/(Z.(o») (5)

Furthermore, the radial electric field can be approxi­mated by

E,(o) ~ -Z,'(a)H; (0) (6)

where H.· is the mngnetic field which would eAist ifthe ground were perfectly conducting.

Using (3), (5). and (6), the reflected current,I/(a), at the edge can be written

J;(a) ~ [Z/(Z + Z.))H'-(a)

__ {kll(1 + I/ika) exp (-ika)= 211'011 + ika/(!\·(Z/".)) [In (ka/Nkb)lI (7)

Then the inward traveling current 1:(,.) is approxi­mated by

J;(p) ~ J;(a)(a/p) up [- ik ..(o - p») (8)

where k. =k« c. + ]) /2]'n is the propagation con­stant for the wire in the interface (Wait, 1972]. Thefactor (a/,.) results, of course, from the convergenceof the ingoing wave.

Now that the current density is known, the far­zone reradiated magnetic field H f1'( ",) can be calcu­lated from a straightforward surface integration.Thus

H;(I/I) ~ ll-ik(l - R.) sin 1/1 exp (-llcr»)/4..rl

.[ PJ;(P>[{W cos. exp (Ik,. cos t/I cos.) de#)] dp

(9)

where the factor (I - R,.) sin '" C3n be derived byreciprocity. The. integr3tion can be done exactly byusing the integral representation of the first orderBessel function. Also, we note that

(3)

at 1/1 ... 0

Z'(p) ~ ZZ./(Z + Z.)

wbere

CORREcnON FOR REFLECTED CURRENTS

Since the edge of the ground screen re-..ults in adilcoDtinuity in the surface impedance, a reftectedwave is to be expected. Equivalently, since the cur­rent of the ends of the radial wires must be zero tosatisfy the continuity equation, a reflected current isnecessary to produce zero wire current at the end.Since this current is normally neglected, it is desirableto determine whether it bas a significant effect on the

1-179

RADIAL-WIRE VERTICAL·ANTENNA PATTERN 83

We note that for large ka, n. becomes quite small.This is to be expected, since the individual wire cur­rents decrease, and the increased spacing results inan effectively small surface-current density. Equiva­lently, this can be thought of as the result of a largesurface impedance at the edge of the screen.

NUMERICAL RESULTS

In order to estimate the importance of reflectedcurnnts, the quantities: 1 + 0,,1 and 11 + 0" + n~l

were evaluated for various screen sizes, numbers ofradials, and complex dielectric constants. n. wasevaluated by a numerical integration of (4), and n.was evaluated from the asymptotic form in (17). Acomparison of the two quantities is given in Figure 2for a moderate-size screen (ka = 30). The value for

Finally, on substituting (16) into (14) and replac­ing k. by its definition in (8), we obtain

a.-- -'lZ(~)/f/.) up (-ikQ)

(2dG)'''(costl-)312«(1 +I i/cQ/lN(Z/ '10)11 [In(ka/Nkb»»

•{exp li(kQ cos tI- - 311'/4»)(I. + 1)/2)'12 + cos tI-

+ exp (-I(ko cos tI- - 3r/4»} (17)(e. + 1)/2]"5 - cos ft

where Z is the ",-dependent form as given by (I).Consequently, H.·(",) is given by

H:(ft) ~ (k(l + R,XZ/f/o) up (-ik,)]/z,

•[ pl"(p)/.(kp cos ft) dp (II)

The change in the far-zone pattern due to reflectedcurrent n. that was introduced in (2) can now be de­fined by:

0" ~ H."(ft)/ H.o(ft)

ClIi! (I + l/tkG)(Z(!f)/f/o) exp (-I(k + k.)o)- COl ft II + tko/(N(Z/..» In (kG/ Nkb)I

.r" 1.(% cos ft) exp (l(k./k)K) tbt (12)

where Z(",) indicates the oit-dependent form ratherthan the grazing incidence form that appears in thedenominator.

In general, the integral appearing in (12) requiresnumerical evaluation. However, for large screens(i.e., lea » I) and if k", has a significant imaginarypart [-1m (k,.4) » 1J, the main contribution tothe integral occurs for large 1l. Consequently, theBessel function is approximated asymptotically asfollows

J.~) -- 1I/(2n)"']

• lex&- (I(z - 3r/4») + exp (-I(z - 311'/4>11 (13)

Consequently, n. can be expressed in the form

I. -- f[-I(k/G)''']j(k. :± k cos ft)1

'exp IllG(k. :± k cos ft) =r (311'/4>11 (16)

(Z(ft)/f/o) exp (- I(k + k.)o)(14)

where

The leading term in the asymptotic ·~presentation of/. can be obtained either from the asymptotic expan­sion of the Fresnel integral for complex argument orfrom an integration by parts. Thus, in either case.

J. - ellp (=r 1311'/4)

·r" «Iexp (i(k./k:± cos JI.)KJI/%"'»d;c

By a change of variable, we find that

J. - /(exp (=r13r/4)]/(k./k :± cos ft)"'lI

le...... ".

. ° I(exp (a)]jz'''1 d.r (IS)

Icb corresponds lO #14 wire at a frequency of 30MHz. The effect of re8ected curreots is less than .1db for aU angles, and, fOr larger screens, the effect iseven smaller. For small screens in which the effectmay be of some importance, the asymptotic formin (17) will probably not be valid, and a numericalintegration of (12) is required. However, Cus shouldpose 110 problem since the integration interval (O­lea) will not be large.

The ellect of the number of radials for a largerground screen (i.e., lui = 60) is shown in Figures3 and 4 for two representative values of c•• It is seenthat significant improvement is obtained by increas­ing the number of radials. For both cases. the ellectof n. is insignificant.

1-180

14 HILL AND WAIT

O'-_..L-_......._--I.__J..--_...L-_-.J

o 25 30

Fi,. 4. The etfect of the Dumber of radial' for a larprdielectric CODlWlt: t. = 10 - i 2. All = 60. aDd kb =

0.512 X 10-1•

6N·'SO

5N='OO

- 4D~ NaSa

0 3d.- 2

5

-1'.410 1---1,.41 0. 41.1

I

1

4'--_-l-__L.-_-'-_--JL.-.-_-'-_---I

o 10 15 20 25 30

'" (deQrees)Pia. 1. TIle dect of eclae-reftected curreots: All = 30;

kb =0.512 X 10-1, •• =3 - I .3. aDd N = 150.

The effect of screen size is shown in Figures 5 and6. It is seen that little is to be gained by increasingu and that the pattern becomes more oscillatory atluau anlles for increased u. Again. the effect of0. is illlignificant.

CONCLUDING REMARKS

It was found that the effect of currents reOeetedfrom the edge of the screen is quite small when thescreen is reasonably large (i.e., kD > 30). Howevcr,the situation is not quite so clear-cut for smallerscreens, and a numerical integration is required toyield quantitative results.

For radial-wire screens, greater improvement canbe obtained by increasing the number of radialsthan by increasing the lengths of radials.

Since these calculations are based on a simplifiedmodel in which the unknown attenuation function ofthe ground wave is set equal to one, a worthwhile ex­tension would be to solve the integral equal for the

9N·/50,

§1

0c:: ••-

5

40

Fla. 3. TIle dect of the DUmber of radial, for a l1li8"diellclric COIIIWlt: •• =3 - I .3... = 60, ud kb =

0.512 X 10-1•

I

o •c::•

5

4L...._...L-_--L__.L-_-L-_......_---J

o 5 10 15 25 30'" (deQrees)

Fi,. 5. The effect of screen size for a ""In dielectric con·lIant: '0 = ] - , .3. N = 100. Ind Ab = 0.512 X 10-1.

1-181

RADIAL-WIRE VERTICAL·ANTENNA 'AlTERN 15

5r----r----r---,,.--.,---..,...---,

4

oc::•

3

Z~_....L.__......._--'__.l.___...L.._....J

o 10 15 25 30

'" (degrees)

FiJ. 6. The effect of ICreeD Iize for a Jaraer dielectric: COD­

ItaDt: •• =10 - I 2. N =100. ib =0.512 X 10"'.

attenuation function. Such a procedure would im­prove the accuracy of both n. and n•. However, wedo not feel that the conclusions would be appreci­ably modified.

AclcllOwled,,,,enu. We would lite to tbaDk PbilippBlac:f!smitb for iastiptial this study. aad Nra. Loys Gappafor ber belp ia PrepariDl tbe IIWIUlCript.

Edllor'. Note. Raden are mDiadecI tbat ia 1biWD1-

Irae:tion DOtalion. fI/be is to be interpreted to meanfI/(be).

REFERENCES

Collia, R. E., aad F. J. Zucker (Eds.) (1969), Anrtnnarlt~ory, vol. 2, pp. 316-432, McGraw·HiII. New York.

Hom, J. M. (1967). A numerical analysis of HF elltended­sector lrouad systems, Rrp. 14JO. 52 pp.. US NavyElectronics Laboratory. San Dielo. Calif.

Hom. J. M. (1961). HF vertical-pl.ne patteras of mono­poles aad elevated vertical dipoles witb aad witbout ell'

tended pound aystems, R~p. 1567. 142 pp.• US NavyElectroaic:s Laboratory. Saa Dielo. Calif.

Wait. J. R. (1959).011 tbe tbeory of reflectioa from a wirearid parallel to aa interface between bomoaeneous media.2. A"pl. Sd. Re••• Sect. B. 7. 35S-36O.

Wait. J. R. (1963). The tbeory of an antenna over an in·homoaeaeous lfOUad plane. in Elrctro"'fI,n~tic rheo,.,find Ant"'"".. part 2. edited by E. C. Jordaa. pp. 1079­1097. Perlamon, New York.

Wait. J. R. (l967a), 011 the theory of radiation from araised electric: dipole over an inhoDloaeacous aroundplane. Radio Sd., 2(9). 997-1004.

Wait. J. R. (l967b), Pattera of a line.r anteana erectedover a tapered arOUDd Kreen. Cfln. 1. Phys.• 41, 3091­3101.

Wait, J. R. (1972). Theory of wave propalalion alonl athia wire parallel to an iaterface, Radio Sci., 7(6). 675­679.

W.il, J. R.• and K. P. Spies (1969), Fields of electric diroleon radially iDhomoaeneous Iround surface, Electron. UII .•1(20). 478-479.

Wait, J. R.• and K. P. Spies (1970). latelnl equatioaappro.cb to tbe radiatioa from a vertical .nteaa. overaa iabomopneoul around plane. Rt«lio Sci., 1( I), 73-79.

W.it, J. R.• aad L. C. Walten (1963), Influence of •sector around KRea on the field of • vertical antenna,Nflt. B",. SttIIUI. (US) Mono,r.• 60, 1-23.

1·182

1.21 Hill, D. A., and J. R. Wait, March 1973, "Effect of Edge Reflections on the Performanceof Antenna Ground Screens,· IEEE Transactions on Antennas and Propagation, Vol. AP-21 ,pp.230-231

(Reprint No. 503)

Reprinted with the permission of the IEEE

1-183

(2)

(5)

Effect of Edce Reflections on the Perform&Dce ofADtuDa Ground ScreeDs

DAVID A. HILL AND JAMES R. WAIT

Abatr.cl-A lIMtJaod for _putiDc til••lfect of tile •••e reflected....reI f,.. tile ..,. of circ1IIu poud &Cr.... I. developed.JII_aricaI rwuJq ..... dlat til. n8ect.d •••• II Ullilllportut totil• ....-. 01 ftrJ Iaqe __ad oaIJ 01 ...u lmportuc.for lDOdarat•••_, Ro....r... til. acran size i. furtller~ til. nlati•• lmportuc. of tile r.Sected .... ilia......

I. INTftODt'CTJOS

"'ire pound llftftn••re eommonl~' U!!td to improve Ihf' pto.­fonnanre 01 HF antennu Opt'r.tinc in thf' prt!!'enre of • poorl~'

eoDdllctinC poIIIId. The improvement in the radiation Mid hasbeen d~ elltenllively (1}-[4], .nd thf' IIlf'lhnd 01 an.I~.,.i.~

• 11I bued on the compton.alion thMrem. In .ut'h tre.lmenlo, Iheeffeet of a ••vereflf'Cted in...rd from the ed.. of the l'<'reen hL~

been conaidend necli,ible. In thi. communication "'e deyelop •metllod lor ..a1u.ting the import.nre of the in..ard reSected ..ayefor circular cround SCreeIJS of coostant .urface impedance.

II. FOIl~ll'LATIOS

The &fOmetry i!l illustrated in Fill. I. A vertiul Hf'rUian dipolf'of current moment 11 i.~ located al Ihe cenlf'r of a cirt'ul.r rrollndscreen of radii'" a. The effecliye allrfart' impedance of Iht' Itrrllllld')'lItem ill a eon.~talll Z', .nd lhe around be)'ond radiu.~ a h.. a_tant .urfllCe impedant'e Z.

The lar·_mapetic field H.(II-.) CaD be writtea in the follo..inCf_ (I]:

H )ikll(l+R.)ens~.ellp(-ikr)

• (h - 4.... (l +0) (I)

• here R. - (ain 11-. - (ZI",)]i[.in 11-, + (Z;••l], k i!l thf' "a...­Dumber. '" .. tbe imlteda..rl' of free ap_••nd an up (iwl) de­pendenec j,. _umcd. If lhe ..-n....d j,. hOll\Oll!neolll, tbe ~iledlurfllCe iIIlpedanee ZI... is given by (4]

; -(!r[1 - (~~II-')r

• bare .. is the rel.liye t:omplell dieleclr~ conalant. Thf' lerm 0aecounta lor the~ of the ICreen and YaDishes if the ..reeaia abient.

As indicaled b)' (l) lhe lar-I.mf' p""ern ill modified II)' a lat'lnr1 + n. It can I~ .h"..·.. lhal II... It''''"''' ..·.ye • mudiMd by1 + n, .here 0 a1l1aled .. 11-. appruacllfll sero [I].

For CODftIIiIIIee. n is DOW aptil inlo t.o peru:

(3)

lIIa.lIIer1p, _"ed October :I. tll~:l: re..1Ied '"o,·....ber @. 19;~TIlM work •••up~ed 1»' the Air Forno C••_,. RMearrhLa_.lorIM.

Th. aulhors w..h the omee 0' T.I«OlnmunlcaUou. '.llIIulf' 'orfor T-r-.u "on Iiden.,.. I'. II. u......,.elu 0' Co••err?8ou1d•• Colo. 11I:I02.

FII l. \'Drllcal ele<1r1e dipole ., .,..,.,.. 0' dreular _ad __.

",hrft'll. i. thf' ILollal 'oml "'hirh nrj!'.... '. "."....tint,. f.lll. II... f'<11t"of Ihf' 'rrf't'n and 0, i.. a t'lIrrt'C't illn Irrm "'hit'h llC't'Ounl~ fll. Ihc,.lIf't"ioD. First of all "'I! kao,,' Irom previoWl .ork (1] thaI Q,

is &iyea b)'

Do. -F [ellP (-ik,,)lI',(k,,) (, + .~ ) J,(k"cna II-.)A:J" (4)COlI 11-. e ••"

..hf're F • (Z - Z') I",. J 1 i.. Ihe fi..I-order 8eAse1 funclion aadlI'.lk,,) • I - i(1r')'"l!lCp (-pI erft' 1;,'"), ..here, • - (ik,,/2)·(Z· ....)•. ID the preeedinc, II", i. a .......nd ..aYe aUenuati..n fllnrlinn..hit'h Ill'C'OlInla lor thf' mndifirlllinn ..f Ihl' nIAlt....lit' '"'1<1 h~' lhefinite ollrf_ impNMl'f' ?'. In emernJ, a nllmerical evaluation or(4) .. required, and the .ubatitllliun r • k" ill Wleful.

The modified turf_ imped.nce Z· is _umed to be the parallelt:ombinationof the pound impedanl'f'Z lalll-•• 0) andtbeeffeeliYI!impedance Z. oIlhe "'ire crid [4]. Th\lll

Z'::r~.(Z +Z.)

If 'hl' ,,·i,. «rid opat'in« i. murh I..... than" "'''Vl'll.'nj!lh And Ihe"'ir"" arf' highl~' t'ondul.'ling. Ihf'n Z, i. p...el~· indurliye (3], [4]and "'I! m.)· sel Z•• i"." ..here' is dimen.ionles.~ and independentof ".

An ""timate "r 0. ran he derivl'd b~' mo<lif~'inlt Ihl' mllllnl'liefit'ld over lhe I'('~n 10 int'ludt' a rt'ftf't'lt'd "'''Vf'. Malhemal",a1I~· •this i.< equiyaleat lo rt!1lTiting (-I) "'ilh the fullu"'ing'raasformatiun:

ellP (-ik,,)lI'.(k,,) (I + i:,,)

-exp (-ika)II",lka) (1 + i~) R("l (6)

..here

R ).FUp (-i1r/4lellp [-iklll-")]II·[ka_ )].(" 2(2..)'"(k(a _ ,,)]"' • ( "

The factor R(,,) ill • ,.fIetotion funclion ",hHoh ....... deriyed from .nearlier aDaI~"llis inyolying re8eclinn of a ..-ound ..ave Irom aland_ahound.,,· (5], ad we _ume here thai H("l i. nul affeeted b~' thecurvatlll'f of the bouDdllr)·. The oinllliar nalure "' R(,,) at" - •'Ullf"ta th.t the intecral be "plil inlo an inlecrable term inyolvinctbe .incularity ad a I'l!IIIIiDder. Th...

0.~ -"up [-i(ka + ../4)] tr (ka) (1 +..!..) I (7)2(2"COOhl'" • ika •

..bere

I. - 2(ka)'"JI (h_#.) +1- (loa - Z)-IIII-JI(kacoeh)o

+lI'.(ko - z) exp (-i(ka - z)]JI(z coa#.) I dz.

The intf'lrland no" is ...eU behaved, and (7) is a .uitable form lorDumerical evalualioa.

III. N"IIEItICU IlElllll.T!l

A numerical evalultion of (~) and (il "'M carried "ul for r,,"ll""01 intet'P."1 of #.. ka, ... and •. It ...... found thaI, lor la~ lC~n.

(kG ~ IIIUI, n. had an in.i,;nifit'Anl.ffecl «0.1 dUl ,",11,1' quantil)'I + o. In FiC. 2 "'e iIIu.<lrlle _nit". 01 ..-alill'; illridenet' f". apoorl~' ('OftduetiDC around. for I..·u !IOme"'hal smaller l'CreellS.E\".n here, the mwmum differenre in the curves ...·ith .nd "'ilh"nl1M ,.lIecled ..aye is onl)' abuut 0.2 dB aryl the vallie of • "'hit'h)'wld! the muimum yalne of II + 0 I alll-•• 0 j,. _ntillll)'unchaneed· M ka .,._. the ratio 10.11\ Q, I inCftllel. 110...•

1-184

....__ ON ANTI':NNU AND PIlUPAGA'ftON, .UL AP.2., NU. 2, IIAllnl ID73 2:11

,.-100_.. -S.-1.6

ff~--=5':--:!IO'~--:I5::'''''''''-:':!:'':--:::!Z5~'-~Wt.

t lO

cr'..:..

1\'. r.nNCLI'~I"N~

11le ..,.118 eonfirm tile .alidil~· fir earlier e.lC'lIl.t.i1l1ll1 'ur I••JI'OUIId _ ...here ....... relleeted 'rom tl~ edtte or tile IIeIftIl

ha" been MJleeted. In r.et, eYen ror moderate 1Ii_ Cia~ 25),the relIeetion" are rel.ti"')· unimpllrt~nt alld dll nllt ehanl" theJftIefai trendA. 11,...."" ... b dftre-... II~ relatiye importanee01 the refleeled 6eId iIIe~ .

_ . ...u __ are USIIalIy of minor interest~ beeauJoe they

do DOt fth8nee the radiation field lIIIfIieien'ly. n ....'.. 'or I......._ are Pown in F.... 3 and -I, with .1. hein. illllilnifieant Ionpapbinl .euney ill _h _. 11le optimll1n ..Iue of • dee...­qbtJy witb mere..in. t. and with inere...inll I., I .

The optimum ..Iue or , ror "'. - 0 )·ielcL. a ~. Mloirable patl.em'or hillier anp.. (4). Thi. ill ~hool-r. ....11 ill Fill:. 5 1\'hprp the pattl'm'or the optimum .alile of , 'or "'. "0 (a._ detprrnillt'd 'rom Fill. 3).. eompU't'd to the p.ttern or thp pPrieell~' C'lIl1duetinc diRk. For........, the DIIIla are ... to or.eur 'ur _aller .alllll of "'to

~·a.I­

~.a..A.I---

...0·.,., ..•

III'UII .. I.I•...... o..-'.ee .. ' fer ............._.

4!-~-,*-*,"--:':--tI 1.1 U U 1.4•.... 1. __ fIl ............ ......-........_.

II I' .. U 1.4 U•.... t. ID~I.'_~"."""::.''=.==e~- whII .....

1-185

1-186

1.22 Wait, J. R., September/October 1980, "Low Angle Radiation of an Antenna Over anIrregular Ground Plane," extracted from "Atti della Fondazione Giorgio Ronchi,"Vol. 35, pp. 576-583, (Florence, Italy)

(Reprint No. 689)

Reprinted with the permission of the Fondozione Giorgia Ronchi

1-187

extracted from fiAtt! della Fondazione Giorgio Ronchi"Anno XXXV, N.5 - Settembre-Ottobre 1980 -Pagg. 576-583

Firenze , Italia

Low-angle radiation of an antenna overan irregular ground plane

JAMES R. \VAIT(*j

Cunsuh.nr,lnsmUle lor Tdeeornmunic..ion Scicn<n. U. S. ~rrmrnt of Comm.-rct:,Boulder, G>lorO<lo IlOl02.

RIASStn.'TO.. U"",Jo ,I ItO,..",. Jt/lJ C_pt'Uol:Io.t. II OIlt.~ono It "pTtman,.J """,0 o,.J",~ J.<, ('limp. J, rIlliW%lo,,~ J, ",,'fl"It"". II ",OllOpoiU ptJflfl fOp'" "na

JIIPI",;t,t '''rca/llfe" "011 c'o"dUII"",;' Ii"".. Si cOI""in4 tl (IUD oJ '''''''''''''4 J, fO/",·

llu"t. r: 11 tlppll(flIIO I ''fI''''_" "I ,:(110 ,,, ('II, J'J1IU1IU ~ poJ/tJ lOp'" 'mil 1""1111/11''''11

Clreal." CO" pJrt't, mcl'lfllU, Si mOil'. thr II tllMiticno J'Il,,"""" ~ l'unl":/lJ,,,. Jdlcand... Ji 1('"" 10"0 "'''lJ('U'' dn ntl c/"o J. "".1 u,pt,/irlt ,.If,I",,,,t>mrn/C' P,,,,114

SUMMAltY.. U,,,,,, Ih~ ComPt"wllon T/~,r",. /ifJ/-ot'Je-, t'xpft'ssmns 11ft ubllllnc·J

/1)' Ihr: fdJIJJJun JidJs ,,} oJ ''''JlfOpul,· .ml""ntl lucoJuJ 0&'(" till "',"(111", emu"J JtI,·

IdCe ,,,,,,b ti",te ('o"Jue/U'Il\' Tbr Il:,mulh,dh' Jrm"'~/r,( eflJC II CtJnJ,Je,C"J JnJ fh..·,(,ulll 4J'~ tlpftl,cJ II) Ib~ n/rulllJn u·ht>,~ Ihr "'flffUp04.., If IfA"JIt'J on tln d ..·..·"h·J,,,,;,,1.,, pi4lll'tlU u:llh sin"",,, siJC'I. II If If,JKlllc,i Ihlll Ibc ,;nlt'''''' Roll", find ,b~

1."'''''4 U,.,,·C ,,"Xc:lltlllrJ,, J'~ r"/u,,,cJ 0,," 11:111 to, J II",If),,,,h lI.,t Jur,JCc

InrrocllK'ion.

Thcre ore a number of important applications where thc irrcl:ularIlround surfacc influcnccs thc dcsired performance of antennas. For e~am·

pic. in man\' H. F. communication "·srcms. the total transmission 1055 iscriticalh' dePendent on the nature of 'the radiation at (all' an!:lcs from theIransmittin!! antenna.

In this nOtc. we "ment a !leneral lirsr-order thea'" for calculatingthe radiation field uf a monopole antenna located on or over an irrelllJlarbounda,,·. The development is based on the elecrroma!!neric Compensa­tion Theorem in rhe form developed b,' Monteath f 1) over twen,,' yearsa,:c. The lull srope of the Compens:lri;'n Theorem has not a)\\,a,;s 'been

1-188

4

appreciated, since Monteith only illustrartd irs application to relarivelysimple problems. Also, ir has only been realjzed quire r«enrly rhar meformulation is sufliciendy general ro facilitate integral equarion solutionof rather complicattd problems (21. For the present, however, we shallinvoke physical approximalions wirh a view 10 obtaining relarively con·cise 6rst-order expressions for the radiation pattern of a vertical monopole.

Formulation.

As illustrared in Fig. I, we located a verlical electric dipole of effec­tive length /. al point A over a delerministic suriace ;: =f (x, )' I, where;: =0 is the reference surf"ce. Al some point B located in Ihe (x.:l plane.we have a similar electric dipole of etfecl;,'e lenglh Ib , The dislance orranlle between ;1 ;lnd B is R, and rhis srraillhr line sublends an anllie 9with Ihe posilive :·axis, The Ileneral problem is to calculare Ihe mUlualimpedancc belween rhe two dipoles for ;l prescribed surbce profile f (x, y)

;lnd irs electrical properties.Firsl of all. u'e assume Ihal the mutual impedancc ZAO belween the

dipoles is knou'n for the SImpler auxiliar\' problem uf havinll I (x. )" = 0ever,'where, and t;lkin~ rhe re~,on : < V /0 I"" :I h"mlll:t'nt"us half·spaee.The concspondinc mUlual impedance for the aC1\la( "r"tile ! (x, )'\ is denu·teJ as Z'AH, wliere the rmme "~OIlies the unknown llr sou~hl <luanUlI',Munteath's formulation III ot the Compcnsatlon Tht'urem tell. us imme·.lialdy that:

III

In [I), Ihe lermmal currents In Jlpoles t1 and B arc I. anJ I.. respecti.

vely, and E. anJ H. arc the kno",," field, of dipole B for the aUl(llialV

situation, whereas ['. and I j'. are the unknown fields for rhe acw;ll pro­file where : = I (x, ."1. Here, the intellrauon of rhese quantiries is overthe surlace Ihal comcld" '\'Ilh : =;1.<, '" and is closed bv am' surfacear some larlle nel!aul'e ,'alue of :. 'IX'" as~ume here Ihal il is al~'a\'S pos.sible 10 selecr rhls lower surtace so lh;lt us conmburion to the ',urlace

integral in [I) is nelllij!lble The unu ,'ector ~; is rhus rhe ourward normalro surbce : = I (::c. )'1, as indicated I.' Fill. 1. For relarively j!entle slo~s.

we can see rhar the Caneslan components at ~ are I!iven b\':

;::(-~,,1, -M,)"ll

The electrical characrerlStlc. "i the actual suriace arc to be descri·bed bl' Ihe Leonlo"lch or ,UrlJtc ,mpe<bnce bnundaf\' condition, This

'-189

surface impedance 1)' (x. y) may be a slowly varying function of the hori·zontal coordinates x and y. Strictly speaking. it should not vary signifi.cantil' in a distance equal to 'y-I:. where y is the propagation constantof the medium for :: < I lx, y). Then. for convenience, we dJaracterizc:the homogeneous hal£·space in the auxiliary problem by a constant surtaceimpedance 1).

ii ~Z =f(x,y)

YII

·ClS

A .....=-_+-_~_--:.-.,,;~ _

Flc. I

An .pproximate solution.

Adopting the various simplilications and physical approximations de·setibed abo\·e. we now can write down the required field expressions thatare to be inserted in [11. Here. we assume that the dipole B is etfectivelylocated at an infinite distance from the surface. Thus. the relevant unpri·med field components ovet S are:

(2)

III

E•• = .=.!.!~ ..n n.- i'R /'" <01. IinO (/'f eola -l- .-"feole ),

." RE " f}., I. I. 0 - j'R ,.4(> <0191;ne ( "f <ole - "f (010)

•• - ~COJ' I ,-, •

-'lHov

1·190

6

and:

(41

Jllm~J R. W,,;/

H•• _ -; j 1.1• •- ,jR /1i<J ce•••inO(.'j[ co.O +,- ,j/ co.O ).:'IR

These formul~s correspond to the radi~tion field of ~ Hertzian dipoie Orien­ted at right angles to the radius vector R and car~'ing a current equal tothe real part of 1. exp(iwtl, where w is the angular frequency. Here,k:: wk is the free-space \\·avenumber. and 11" a.. 120:: ohms. Also. asindicated in Fig. 1. qI is the azimuthal anglc with reierence to a cvlin­drica1 coordinate system (~. Q. ;). In 'Iddition. we should observe lhat inwriting [21. [3J. and [4], we arc. in elfect. ~ssuminj: that the rderencesurface at ; =0 is a relatively good conductor.

t

Now the primed or unknown field component' over S arc postulated10 have the form:

and:

('I

161

[71

• 1.. 1.. -,.t)H •• = !:'I t" (I + ,~)t cos•. F(·.',.) G(:.,:)

E' - ',.,' M E'•• --'1 ...-~ ••

where F (p, <;l) is an «attenuation function. that to a firsl approximationis replaced by unm·. In writing [5 J. [6 J. [i]. .md [8]. we assume thatthe field components J;e the same as Ihose of a \'ertica1 Hertzian dipolelocated at ; = O. but modified by a suitable height'j!ain function G (;•• ;)that depends onl\' on the heights of the dipole A and the observer overthe surface ; = O. It is important to note that these primed expressionsmelude near field etfects.

1-191

Lou"<I/fglt ,l1di<lIlO/f 0/ 11/f 11/f/t/f/f"

To proceed further, we observe that:

7

191 - -. " - - . of . or(1:111':' H.- E.·,,- H,}.,,::: £., Hn~- EhH IIIZD)"

+ E.~ H'e", - E'•• Hili. & - f'd HilII'M

Here. we [.'lte that the sum of the latter h"O terms can be replaced byTl' II'.. H" b\' virtue of [il:

Usinl! "ll the prC\'IC~l> equations. we lind without difficuhy that:

1101l'•• -Z.,,.

z.,. '"

{ (. "" <059 "nO [. .Jf:'/" Jl.', . , cosO 110 U rcosO) COSq:l. .

l)! ,)'-" 1 I -,It.~-5inO(05U/(0511)~_--,..-(059<05('f(05·')r2:;i:;·(I"".\')' '. Fh·.9)

where:

1111

a"'l:

(121 l. ~,-'''R2·' H

\X'c nOlC here that Z.. is the mUlual Imp"..lancc of the two .lipoic, if the\'were lueated on a flat l'Crtcctl\' conduclIn!: plane and 'cpar;1\ed b\' a di·stance R. ;\1,0, in wrllin:: [10 J. we ha"c' assumed that G 1:_,:) = 1.

An immediate simpli(,callon of [101 occurs ""hen Tl', I and aI/up ,treindependent of the aZimuth an!tle r:>. Then. we can use the followmgkno\\'n lormuias trom Gessel luncllon thcon':

h

zl:t .f :Zt:OSQ .It;> '= )" lZ)

and:

,.-;",' ( /Z COJ" (os. 4t;> = i, (Z)

1-192

8

Then. [101 becomes:

(I'I Z'••;:Z•• --Afl[rcoIQlin(AfcoIQ)- "',:" cO.(Afcola)]J,(.I,••;nQ).

where F (p) is azimulhallv svmmelric allenualion funclion.In Ihe case of a u~if~rmly flal surface I = iJl/iJp = 0 every",'here;

rhus [15 J reduces 10:

(16)

This expression or closely relaled forms have been used 10 predici Iheform of Ihe radialion pallem and Ihe enhancemenl of Ihe surface wavefic:lJ of antennas wilh larjle grounJ melallic jlrounJ >creens [3 J. [4 J.In such cases, 1)' is reJuced to be much less Ihan 1) and/or Ihe impedance1)' is made 10 be effeclively induclive so Ihal F (p I may be enhanced overIhe ground syslem.

Another spc::cial case is 10 ,,110"" Ihe surface (a be evcrywhcrc pcr­fectl" conducling. Tllen, 11' = 11 = U {or all \'"Iues of p. Cunse<juent/".( 15] reduces 10:

Z'... - z.."l.

.= - •.1 f 1,..0 COl (.If COlO) J. (.I~ ..nO). ~~

On sClllng F (p I = I, which is a rcasonablc approximalion. Ihis rcsullreduces 10 an cquivalcnl form dcm'ed by Pagc and Montc:llh (5).

Onc ralhcr rc"calinll c:<ample is to ICI 1) = O. corresponding 10 anassumed pc::rleclh' conducllng sea Ihal occupics Ihc rCllion P > b. WilhinIhis c.rcubr ,sland. wc havc a circubr plarcau of radius IJ wilh a flar lap.At Ihe same IImc, wc allow e_ 90·. corresponding 10 grazing incidcncc.For thIS case. we can \\'rilc:

1181 Z'•• ::: Z.II--'- n,.,.. n.1

1-193

9

where:

(191 O,-.j ~ (1+ j~ ),-i!l1JI(!Q)F(Q)JQ

and:

Thus n. is the modiJic~tion of the field resulting from the electrical pro­perty conrr~st of Ihe «island *. while n. is the corresponding modifi­calion due to the non-tlarness. As menrioned above. n, has been discus­sed on numerous ~rlier occasions. The lopographical effect representedby n. is usua!J)' ignored. although. in many applialions. lhe transmiuingmonopole is loc~led on a blutI or some olher elevated terrain feature. ToiIIustr~te its effect. we assume that the beach of the island has a uniformdow nward slope. so thai - al/ap is a positive conslant that we desillnaleby 1\. Also, 10 simplih Ihe discussion. we assume thai Ihe radius of theplaleau is much greater than a wavelenllth. Thus. k4 and kb> > I. and.consequently. J. (hpJ. can be replaced by the first lerm of its asymptoticexpansion over the intellration ranlle in [201. Thus. we find lhat:

With suitable assumptions aboul the form of f (p), Ihis inlellral an beevalualcd in lerms at error inrellrals of complex arllumenl (01). The essen­lial nature nf Ihe problem, howcver. appealS if we recollnize Ihal Ihecontribution from the r~pidly varyinll lerm i exp ( - 2ikp) can be nelllec·led. Also. over Ihe region iI < p < b. lhe slowly varying function F (p)can be replaced by F (il I. Thus. [21] reduces 10 Ihe remarkably simpleform:

[11\ 0. ~ (Z '~)t K ,"~'~ «U)t - (..)t\ 1'(_) _ lK ,'~/. (6 j,)t - (_:I.)t) 1'(_)

where A. is the frec-spxe wavelength. For example, if blA. =lOa.ilf). = 601, K = 0,1. and assuming f (iI) = 1. we sec thai:

0. ~ 0.• x /",.

This corresponds approximatelv to a 01 dB gain. For Ihis case. Ihe heillthof Ihe plateau above Ihe sea is 3.6>', and Ihe sloping beach eXlends overa radial dislance equal 10 ~pproximalely 36A..

AClualh·. the improvemenl of locating the monopole on an elevaledplateau applies also to the elfccti\'e ground wave radiated for a given

'-194

10 Jam~r R. W lUI

current moment (3 l. Thus. the main conclusion from this analysis is thatthe elevalion of the ground plane should be consideted in any design ofsurface wave launchers for high-frequency radio waves over the sea.

REFERE:-:crs

0) G. D. MosTEAnl. ApplzeD/IOIIS 01 th~ l'OmptlUtlIlO" th('()r,m to ...."'ttm,.wIOl,on and p,opa~a',on p'obl~mr. Proc. I.E.£. (London •. \'01. 98. pp. 2}·}0 119~11.

(2) J. R. \\/AIT .lnJ K. P. SPIES, InU1,rfl/ ~qulIOIt JPP"'Jd~h 'n ,!-t. 1.1,ilo1I10'1

/rom .J IItrlICill ""'t!",,d OVtr "n ",hotnolt"~OUJ grounJ pi"",. IUdlo xi., \'01. ,.No. I. pp ;}·;9. JJn. (1970).

(31 J R. \~AIT. Tht' thtor, 01 "" 11"""".1 ovt, In ",hom0r..tntoflr ~,.o""a riJIt("El<c"om.~nellc Theory and AnleM.'. E. L. Jordon. Ed. a.lurd. Ln~l.nd, Per,a·man Pres. (1963).

(~) J. R. \V"IT 3nd K. P. Srn:s, 0" th, '4JiJtlon I,om II L'UII(Ji JUIO/c' "'lIh." indue;'"t wlrt'-I."d "round svsum. l.El:.E. Trans. 'Antcnrus ;lnd Prot' .. \'01.AP·IH. Nu. ~. pp. }}.~.Jbl). luj,' 119701.

(5) II. PACE Jnd G. D. MONTEA",. Th, v"I,caI ,"';,."on {hIU<rn ..f .."J,um",av~ b'oadca"'n~ a.",ur. Pro<. I.E.E. (Lunolon). vol. 1028. pp. 279·297 119",

1-195

1-196

SECTION 2

DISCUSSION OF U.S. NAVY ELECTRONICS LABORATORY REPORTS BASED ONREPRINT 1.11

by

J. R. Wait5 March 1990

2·1

of H.F. extended-sector groundNEL Report 1430

Following the publication of the Monograph by Wait andWalters(1963), a series of unclassified reports were issued byinvestigators at the U.S. Navy Electronics Laboratory (now knownas the Naval Ocean System center) in San Oiego. We will identifythese as NEL 1346, 1359, 1567 and 1430 in the order we discussthem below.

W.E. Gustafson, W.M. Chase, and N.H. Balli, Ground system effecton high frequency antenna propagation, NEL Report 1346, 29 pgs.,4 Jan 1966G.D. Bernard, W.E. Gustafson, and W.M. Chase, HF extended groundsystems: results of a numerical analysis, NEL Report, 52 pgs.,24 Feb 1966 , NEL report 1359J.M. Horn, HF vertical-plane patterns of monopoles and elevatedvertical dipoles with and without extended ground systems,142 pgs.,25 Jun 1968, NEL Report 1567J.M. Horn, A numerical analysissystems, 52 pgs., 14 Jan ,1967,.................................................................

These four NEL reports were distributed widely includinglibraries and they were released to the Federal Clearinghouse nowknown as NTIS (National Technical Information Service) • Now such~esults are in the public domain. Thus we need not describe theircontent in great detail. But it does seem appropriate to identifysome of the significant results and conclusions reached by theauthors. In some cases, an alternative interpretation of the datais offered without meaning to detract from the noble efforts of theNEL'investigators .

In NEL 1346 it is pointed out that the ground based monopolewith extended ground screens may have a major advantage over elev­ated vertical antennas (with or without ground screens) if pattern}obing is to be minimized. With this motivation, the authors under­took physical. model r~ns for circular ground sreens employing bothsquare meshes of various sizes( t to 4 inches) and radial wireswith various angular separations ( 11 , 2t and 5 degrees). This exp­

erimental program became manageapl~ by using frequencies in theVHF range (130, 250 and 360 MHz) which were appropriate to modelfrequencies lower by a factor of 12 when a relatively well conduct-

2-2

ing ground site was selected e.g. 0.1 mhos/m). Some useful andmeaningful pattern plots were illustrated by the authors to confirmthe expectation of enhanced radiated fields at low take-off angles ..

Dr. Gary BeLnard, who was a consultant to NEl,presented a con­cise tabulation of the experimental data along with results gener­ated by computer calculations using the formulation in Wait(1963)and Wait and Walters(1963). This comparison of experiment and theoryis reproduced here from NEl Report 1346. The calculated quantitytabulated is equivalent to 20 In I 1 + 1\a I in dB . The source adoptedis a vertical (ground based) electric dipole at the center of thecircular screen. Actually the experimental data were obtained usinga quarter wave monopole with a small radius (impedance stabilizing)ground plane . The general agreement between experiment and theoryis reBsonably good. It is also worth noting that, for low take-offangles,the angular dependence of the field is small. This is part­icularly the case for 130 MHz (i.e ka = 16 ) which is the lowesttest frequency.

Further radiation pattern calculations are reported in NEL1359 which again are based on Wait and Walters(1963). The frequen­cies chosen were 4, 8, 16 and 32 MHz. The full circular groundscreen radius varied in octave steps from 2 to 128 wavelengths andthe grid spacings were 6, 12, 24 and 48 inches, The ground conduct­ivities chosen were 3,10, 30 and 100 milli-mhos/m with correspond­ing dielectric constants of 4, 10,20 and 30, respectively. In eachcase, computations were carried out for vertical (take-off) anglesof 2, 5, 10, 15, 20 and 25 degrees. For all computations, the groundsystem was circular (i.e. full azimuthal symmetry). The copper wiresize was No.l0 (American Wire Guage) . The source in all cases wasassumed to be a short vertical current element or (infinitesimal)electric dipole.

Some examples of the extensive calculations, from NEl 1359are shown in Table 2 where the quantity E in dB is again the quant-i ty 20 In I 1 + A a I • Here G represents the improvement for theantenna with extended ground screen compared with the same elementover the imperfect ground when the dipole moment is fixed • In thisexample,the ground conductivity is 10 milli-mhos/m and the dielec­tric constant is 10. In the case, where the radius is a = 16 wave­lengths or ka = 100, the results.fOr 32 MHz, for the smallest mesh-

2-3

spacing, are compatible with Fig.2a, in Wait and Walters(1963) car­ried out for a dielectric constant of 9, zero ground conductivity,and an assumed perfectly conducting ground screen.

In NEL 1567, many additional calculations are reported againfor the fully circular ground screen •• for the most part, the res­

ults are based on Wait(1967)'s paper for vertical electric dipoleof

finite length erected over the center of the ground screen. Manygraphical plots are presented in a format which allows the readerto see the relative enhancement of the low angle radiation when an

extended ground screen is employed. As J.M. Horn, the diligent

author,points out the advantages of such a ground screen are re­

duced somewhat for elevated antennas. Also, as indicated before,

the economic benefit,of large ground screens over well conducting

ground, is small.

In 1430; Horn deals with thE sector ground screen geometry,

using essentially the formulation of Wait(1963) and following the

methods outlined in Wait and Walters(1963). He adopts the ground

based quarter-wave monopole which would not be very different than

for the infinitesimal dipole model unless the ground screen wasless than several wavelengths in size. Horn includes a listing of

two computer programs. One is based on the mesh ground screenwhere the effective surface impedance Z' is essentially a constant

and the other is for various radial wire configurations where Z'~aries with radius.

In the four above mentioned NEL reports, it is stated that

the ground wave is neglected.I~eed,it is true that the results

are presented in the context of the modification of the radiation

fields - due to the presence of the ground screen. But the quaritity

I 1 + Jla I tabulated , for low angles (e.g. "'0 = 20), is applic­

able to the modification of the ground wave field as compared to

the case of no (or small) ground screen for a fixed current at

the terminals of the transmitting dipole. It is a pity that the

NEL investigators did not carry out their calcUlations for the

grazing condition where "'0 -+ 00 as done in Wait and Walters(1963).

While there is no reason to doubt the accuracy of the cal­

culations in the four NEL reports, one might question the validity

of the results in cases of very low ~round conductivity where sparse

ground screens are employed. In such cases some of the initial as­sumptions apPear to be violated.

2-4

TABLE 1. GAIN, IN DB, OF EXTENDED GROUND SYSTEM (20-FOOT­RADIUS) OVER IMPEDANCE GROUND (2-FOOT-RADIUS)

(repr~duced from NEL Report "1346)(20 FEET =2.5 WAVELENGTHS AT 130. 5 MC/S.)

Radial Wire Separation(degrees)

Elev. Angle(degrees)

Square Mesh Size(inches)

1/2 I 1 I 2 I 4

130.5 Mc/s

1. 25 I 2.5 I 5

5 4.9 4.9 5.1 5.1 4.7 4.3 3.8

10 5.2 5.2 5.5 5.4 4.9 4.4 3.9

15 4.9 4.9 5.1 4.9 4.9 4.3 3.6

20 5.0 5.0 5.2 4.8 4.8 4.1 3.4

25 4.8 4.8 4.9 4.2 4.5 3.9 2.8

250 Me/s

5 4.6 4.9 4.9 3.6 4.9 3.7 2.3

10 5.0 5.0 5.0 3.2 4.9 3.4 2.0

15 4.8 4.9 4.6 2.2 4.2 2.6 1.6

20 4.6 4.5 3.9 0.5 3.8 1.9 1.1

25 2.8 2.3 1.4 -2.8 1.4 -0.4 -0.3

360 Me/s

5 6.7 6.8 6.7 3.5 6.4 4.5 2.5

10 6.9 6.9 6.4 3.0 6.1 3.9 2.0

15 6.9 6.9 5.7 1.3 5.5 3.2 1.6

20 5.6 5.1 3.3 -1.2 3.4 1.7 1.0

25 2.4 1.6 -1.1 -1.5 1.3 1.1 0.6

18

2-5

TABLE 2. GAIN. IN DB. OF EXTENDED GROUND SYSTEM(20-FOOT-RADIUS) OVER NO GROUND SYSTEM FOR QUARTER­

WAVELENGTH ANTENNA. EXPERIMENTAL RESULTS. ANDSHORT DIPOLE. COMPUTED RESULTS.

(20 FEET = 2.5 WAVELENGTHS AT 130.5 MC/S.)(reproduced from NEL Report 1346) .

Quarter Wavelength

Measured and Modified. Short Dipole -- Computed

Elev. Angle(degrees)

Square Mesh Size(inches)

1/2 I 1 I 2 1/4 I 1/2 I 1 I 2 I 4

130.5 Mc/s

5 4.9 4.9 5. 1 4.2 4.2 4.2 4.2 I 3.9

10 5.1 5.1 5. 1 4.4 4.4 4.4 4.4 4. 1

15 5.5 5.4 5.4 4.9 4.9 4.9 4.8 4.4

20 5.0 5.1 5. 1 5.3 5.3 5.3 5.2 4.8-- --25 5.0 5.0 5.0 5.7 5.7 5.6 5.5 5.0

250 Mcfs

5 6.4 6.8 6.6 5.7 5.7 5.7 5.5 4.6

10 6.2 6.4 6.3 6.2 6.1 6. 1 5.8 4.9._--15 6.7 6.9 6.4 6.6 6.6 6.5 6.1 5.1

20 6.2 6.3 5.7 6.7 6.7 6.6 6. 1 5.0

25 5.2 4.9 4.1 6.2 6.2 6.0 5.5 4.3

360 Me/s

5 7.3 7.7 7. 1 6.7 6.7 6.6 6. 1 4.8....-._- f-._-_..10 7.1 6.5 6.7 7.2 7.2 7.0 6.4 5.0

15 7.3 7.6 6.3 7.5 7.5 7.7 6.6 5.0

20 6.2 6. 1 4.3 7.1 7.1 6.8 6.0 2.9

25 3.6 3.2 1.1 5.3 5.3 5.0 4.2 2.8

• Extrapolated from results with 2-foot-radius impedance ground.

20

2-6

Table 2,..Gain G in dB for large screen relative to small or noscreen for ground conductivity of 10 milli-mhos/m andrelative dielectric constant of 10 (reproduced fromNEL Report 13~9 by Gary Bernard et al.)

A. 4 Me/s0= 2 ,,= ~91.8 It O. ~". 983.5 It

flav.

~~dag!.-

2

5101520

2S

Mash Spacing (inch.. l

6 12 24 48--1.3 1.3 1.~ 1.3

13 1.4 1.4 1.4

IS IS 1 S 1.5

17 1.7 17 1620 2.0 2.0 18

2.3 23 22 20

0·,8"=19670"

flu. Mesh Sp.cing (inch..)-Angla (da,.) 6 12 24 ~8

2 1.9 1.9 2.0 1.9

5 2.0 2.0 2.0 1.910 2.3 :1.3 2.3 2.215 2.7 2.7 2.7 2.•20 3.1 3.1 3.0 2.7

25 3.3 3.3 3.1 2.7

00& 16 A. 393~.1 It

2 29 29 2.9 2.7

5 3 1 3.1 3.0 2.8

10 3.7 3.6 3.5 3.2

15 ~ 1 ~ 1 3.9 35

20 39 3.8 3.6 3.125 27 I 26 24 20

o = 32 A = 7868 It

2 6.2 6.1 59 5.3

5 68 6.7 6.~ 5.810 6.2 60 5.7 5015 33 3.2 31 27

20 2.6 26 2.5 2.1

2S 2 ~ 2.• 2.2 1.9

2 •. 3 4.3 4.2 3.85 4.7 46 4.5 4.1

10 5.4 5.3 5.1 4.515 4.5 4.4 4.2 3.620 2.5 2.~ 2.3 2.025 3 1 3.1 2.9 2.5

•• 64 A. 15.736 It

2 8.6 8.4 8.1 7.35 8.9 88 8.4 7.5

10 4.0 4.0 38 3.3

15 3.7 3.6 3.4 3.0

20 2.8 2.8 2.6 23

25 2.5 2.5 2.3 2.0

... 128 A. 31,472 It

flu. M..h Sp.cin, (inch••)

An,l" (d",.) 6 12 24 48

2 11.2 110 lOS 955 9.6 9.5 9.0 8.0

10 4.3 4.2 4.0 3.515 4 0 3 9 3.7 3.320 3.1 3 0 2.8 2.4

2S 2.6 :l5 24 2.1

2-7

19

0= 2 A.. 245.9 "

To"'e 2 CDllt.

8 8 Mc.•

Elov. M..h Spocing (inch..)Anglo (clog.) 6 12 2~ ~8

2 22 2.2 22 205 2.2 2.2 22 20

10 2.~ 2.~ 2.~ 2 I

IS 2.6 26 2.6 2.120 3.0 2.9 2.8 2 ~

25 3.3 3.3 3.1 2.6

o. 8 A.. 983.5 "

2 ~.5 A.A A.3 U5 A.7 ~.6 A.A 39

10 5.2 5.1 A9 A2

15 5.5 5.A 5.1 A.2

20 5.0 A.9 ~.5 3625 3.5 3.3 3.0 2.3

o .. 32 A. 393~.1 II

2 8.~ 8.3 7.9 685 8.8 8.7 8.2 7.1

10 7.7 7.5 7.0 5.815 ••• •.3 •.0 3.•

20 3.5 3.~ 3.2 2.6

25 3.3 3.2 2.9 2.3

Elev. M..h Spocing (inch..)A"gle (c1eg.) 6 12 ;Z~ ~8

2 3.1 3.1 31 2 75 3.0 3.2 31 18

10 3.5 3.5 3. 30IS •.0 3.9 37 3220 •.3 •. 1 ~O 3.32S ••• •.3 ..0 3.2

0" 16 A.. 1967.0 II

2 6.3 6.2 5.9 5 I5 6.6 6.5 6.2 5 3

10 7.0 6.9 6.5 5.515 5.8 5.6 52 ~.1

20 3.3 3.1 3.0 2.25 •. 1 •.0 3.6 2.9

o .. '" A.. 7868 "

2 11.0 10.7 10.2 8.85 11.0 10.8 10.2 8.8

10 5.3 5.2 •. 9 •.0IS ..9 •.8 •• 37

20 3.8 3.6 3.3 27

25 3.• 3.3 3.0 2.•

20

0" 128 A.. 15.736 "

Elo•. M..h Spoelng (inch..)

Ant'o (<I.g.) 6 11 2. 48

2 13.6 13.A 12.7 11.15 11.6 11.3 10.6 91

10 5.6 5.5 5.1 •. 2IS 5.2 S.I .7 3.920 .0 39 36 2.92S 3.• 3.3 3.1 2.5

2·8

o • 2 >. '" 122.9 "

EI.". Muh Spocing (inch••)

Angl. (eI.g.) 6 12 2~ ~8

2 3.2 3.2 3.1 2.~

5 3.3 3.3 3.1 2.~

10 3.~ 3.~ 3.2 2.515 3.7 3.7 3.~ 2.620 ~.o 3.9 3.6 2.725 ~.3 ~.2 3.8 1.5

2 6.1 6.0 5.6 ~.~

5 6.2 6.1 5.7 4.510 6.6 6.5 5.9 ~.6

15 6.8 6.6 5.9 ~.~

20 6.0 5.7 5.0 3.525 ~.1 3.9 3.2 2.1

o • 32 >.. 1966.~ ft

2 10.4 10.1 9.4 7.5

5 10.6 10.4 9.5 7.710 9.0 1.7 7.7 5.7IS 5.5 5.3 4.7 3.520 ~.~ ~.2 3.6 2.525 ~.O 3.1 3.3 2.3

EI.". Muh SplICing (inch••)Angl. (cI.g.) 6 12 2~ ~8

2 ~.4 ~.~ ~.2 3.35 ~.5 ~.5 ~.2 3.3

10 ~.8 ~.8 ~.~ 3.~

15 5.2 5.1 ~.6 3.620 5.~ 5.3 ~.7 3.525 5.~ 5.1 4.5 3.2

o .. 16 A. 983.2 fl

2 1.1 7.9 7.3 5.95 8.3 8.1 7;5 6.0

10 8.5 8.3 7.5 5.815 6.9 6.5 5.7 ~.O

20 ~.I 4.0 3.5 2.425 5.0 ~.7 4.0 2.7

... 6~ A. 3932.8 ft

2 12.9 12.6 11.7 9.65 12.1 12.4 11.4 9.2

10 6.5 6.3 5.6 ~.2

15 6.8 5.8 5.1 3.820 4.6 ~.4 3.1 2.625 ~.1 ~.o 3.~ 2.3

... 128 A. 7165.6 fl

EI.... M••h Speelng (Inch••)

Angl. (oI.i.) 6 12 2~ ~8

2 15.6 15.2 14.1 11.15 13.2 12.8 11.6 9.2

10 6.8 6.6 5.8 4.315 6.3 6.1 5.~ 3.920 ~.9 4.7 ~.O 2.725 ~.2 ~.o 3.5 2.~

2-9

21

o • 2 ).,. 61.5 It

Tolo'. 2 con'.

O. 32 Mel.

0,.4).. 122.9ft

EI••. Mesh Spocinll (inch.. )

Anlli. (d.,.) 6 12 24 48

2 4.0 3.9 3.3 2.15 4.0 4.0 3.3 2.1

10 4.2 4.1 3.4 2.115 4.4 4.3 3.5 2.120 4.7 4.4 3.6 2.1

25 4.9 4.6 3.6 2.1

0.8).. 245.9 ft

2 7.1 6.9 5.8 3.2

5 7.2 7.0 5.9 3.9

10 7.5 7.2 6.0 3.9

15 7.5 7.0 5.6 3.8

20 6.5 5.9 4.3 3.4

25 4.4 3.8 2.5 2.2

0" 32 ). =983.5 ft

2 11.5 11.0 9.5 6.7

5 11.6 11.1 9.5 6.510 9.6 8.9 7.1 4.1

15 6.1 5.7 .1.4 2S20 .18 4.3 3.2 16

25 .1.4 4.0 2.9 1.4

EI••. Mesh SpClcinll (inch.. ,

Anlll. (d·II·) 6 12 24 48

2 5.4 5.2 4 ..1 295 5.5 5.3 4.4 29

10 5.7 5.5 4.6 2.915 6.0 5.7 4.6 2.820 6.1 5.7 .1.5 2.625 5.9 5.4 4.0 2.1

o. \6 ).. 491.8 "

2 9.1 8.8 7.5 5.1

5 9.3 8.9 7.6 5.1

10 '.3 1.8 7.2 .1.6

15 7.3 6.7 5.0 2.5

20 4.6 4.2 3.1 1.625 5.3 4.8 3.4 1.6

0" 64). .. 1967.0 ft

2 140 13.4 11.7 84

5 137 13.0 11.1 7.7

\0 7.1 6.7 52 3.0

15 6.6 6.1 4.8 27

20 5.0 .1.5 3.2 I 6

25 4.5 4.1 3.0 1.5

22

o .. 128 ). .. 3934. I "

EI••. M..h Spocinll (inch.. )

Anlli. (d.II·) 6 12 24 48

2 166 16.0 140 1045 13.9 13.' 11.0 7.2

10 7.4 69 5.4 3.115 69 63 49 2720 5 3 48 34 17

25 46 4' 30 1.5

2-10

r· .... '. "

<

SEC'ftON 3

ANNCJ'fAiriiiUSTING OF SELECTEO'1tELATED PUBLICATIONS (1967-1989)

by ;

J.R.W.SMarch 1.190

'I

3·1r

." ..\

.. .. ..

3.1 Krause. L. 0., November 1967. "Enhancing H. F. received fields with large planar andcylindrical ground screens," IEEE Trans. on Antennas and Propagation, No.6, pp. 785-795

The author presents a series of interesting pattern calculations (using image theory)for tilted ground screens located over a conducting half-space. For such geometries, the lowangle radiation may be enhanced. Cylindrical shaped screens are also considered. Theauthor cautions on the use of the method for angles at grazing angles below 3°.

3.2 Collin. R. E. and F. J. Zucker. 1969, Antenna Theory, Pt. II, McGraw Hill, N.Y.

In Chap. 23 authored by J. R. Wait, eq.. 123 has a misprint; ejx should be e-jx. Eqs..23.122 is OK. Also the Fressel function J1 here was approximated by the first term of itsasymptotic expansion which can lead to 1 dB discrepancies in computed patterns but generalfeatures are correct. This point was confirmed by Lt. Joe Fortney. of RADC. Hanscom AFB,Bedford, Mass.., who recomputed the integral given by 23.123 (private communication Jan1990).

3.3 Waldman. A., January 1970. "Elevation steering of the pattern of vertically polarized elementsover a ground screen." IEEE Trans. on Antennas and Propagation. Vol. AP-18,pp. 105-107

Author presents calculation based on reprint 1.13 (Wait, , 967) for an elevated verticalantenna over the sector ground screen system. An approximation pattern multiplicationscheme is adopted. The objective is to develop a method to steer the principal maximum inthe low angle elevation pattern.

3.4 Balanis. C. A.• July 1970, "Pattern distortion due to edge diffractions," IEEE Trans. on Antennasand Propagation. Vol. AP-18, pp. 561-563

This is a systematic treatment for an arbitrary aperture source in a finite size ground planeof perfect conductivity. The ambient medium is free space. Excellent correlation withexperiment is shown.

3.5 Yu. J. S., K. J. Scott, and A. R. Spatuzzi, November 1970. "A modified elevation angle ofradiation from a monopole on ground screens," IEEE Trans. on Antennas and Propagation, Vol.AP-18. pp. 795-799

The authors employ an approximate diffraction technique to estimate the pattern of avertical monopole on a square wire mesh screen of finite extent. The experimental dataappears to justify the procedure for the pattern calculation for the pattern calculation for theparticular examples given.

3.6 Rafuse R. P., and J. Ruze. December 1975, "Low angle radiation from vertically polarizedantennas over radially heterogeneous flat ground." Radio Science. Vol. 10,pp.1011-1018

The authors construct an admittedly approximate analytical solution based on an assumedcurrent distribution on both the ground screen and the external ground surface. The resultsseem to be compatible with the more rigorous formulations based on the compensationtheorem. The latter can be used to form an integral equation to deduce the surface current

3-2

..... - ..

distribution over the ground screen rather than assuming it [e.g. see reprint 1.18 (Wait andSpies, 1970) and Hill and Wait, 1973»). Rafuse and Ruze introduce an "efll)iricalnormaUzation" of the near field surface impedance to bring their results into conformity withphysical expectations. The comment in the paper that the surface current at the edge of theground screen is discontinuous is a bit surprising. Of course, the radial electric field, indeed,is discontinuous. In an important observation, the authors confirm that the classical Fresnelzone description of the "active" area in front of the antenna does not hold. Indeed, it is muchsmaller as has been shown by J. Bach Anderson (1963).

3.7 Teng, C. J., and A. J. P. King, January-March 1981, "Surface Fields and Radiation Patterns ofa Vertical Electric Dipole Over a Radial-Wire Ground System," Electromagnetics, Vol. 2,pp. 129-146

The authors use the compensation theorem to deal with a vertical electric dipole locatedover a radial wire ground system at the interface of the flat earth and air. The radiation pattemswere found to be independent of the length of the radials beyond the point where theeffective impedance of the composite impedance was within 90 percent of the underlyingground.

3.8 Park, K. S., R. J. P. King, and C. J. Teng, April-June 1982, "Radiation Pattems of an HFVertical Dipole Near a Sloping Beach," Electromagentics, Vol. 2, pp. 129-146

The effective surface impedance of the beach is calculated from a twO-layer stratified earthmodel so the formulation proceeds as if the equivalent ground plane has a constant elevation.It is found that the low angle radiation is reduced somewhat from that of no beach. The usuallobing occurs in the elevation pattern when the dipole is raised above 0.3 wavelengths.

3.9 Burke, G. J., A. J. P. King, and E. K. Miller" September 1984, Surface Wave Excitation Study,Lawrence Livermore National Laboratory, Report 20214

In this report, results are shown for the terminal impedance of a vertical anteMa with aradial wire ground system for up to sixteen radials. The "Numerical Electromagnetics Code",known as NEC, is employed. An interesting comparison is made with corresponding resultsbased on the approximated compensation theorem approach. The agreement is good whenthe restrictions jr. the lalter method CJ.j'e adhered to. But, for a dielectric-like earth, agreementis poor because of reflections trolil the end of the radials. As Burke et al point out, resonanceeffects are most noticeable when the earth system is elevated and not grounded at the endpoints by stakes.

3.10 Weiner, M. M., S. P. Cruze, C. C. Li, and W. J. Wilson, 1987, Monopole Elements on CircularGround Planes, Artech House, Norwood, MA.

The analyses deal with the case where the monopole element and circular disk arelocated in free space. The models which are used and compared are the inducedelectromotive force (EMF) method, integral equation, method of moments, oblate spheroidalwave functions, scalar diffraction theory, geometric theory of diffraction, method of momentscombined with the geometric theory of diffraction, and the method of images.

3·3

3.11 Burke, G. J., January 1988, A Model for Insuluted Wires in the Method ofMoments Code (NEC),Lawrence Uvermore National laboratory, Report 21301

Among other things, Burke shows how the propagation constant of a buried insulatedwire depends on the depth of burial. Excellent agreement is obtained with Wait's paper(Canadian Journal of Physics, Vol. 50. pp. 2402-2409. 1972) who employed an analyticalmethod. The results are relevant to the radial wire ground system.

3.12 Nagy. lo, 1989. Input Impedance and Radiation Pattern ofa Top-Loaded Monopole Having a RadialWire Ground System. (Summary only), Proceedings of the International Radio SCientific Union(URSI) Symposium on Electromagnetic Theory. The Royal Institute of Technology.Stockholm

The author deals mainly with the input impedance calculation using the compensationtheorem along with first order perturbation. The radiation patterns. as presented. ignore thepresence of the ground system which is partly justified because of the short length of theradials.

3.13 James, G.l. and G. T. Poulton. August 14-17, 1989. EffectsofGroundScreenonHF Antennas(Summary only), Proceedings of the URSI Symposium on Electromagnetic Theory. The RoyalInstitute of Technology. Stockholm

Authors employ a GTD (geometrical theory of diffraction) technique to reduce radiationpatterns for a vertical antenna over a circular ground screen for both radial wire and squaremesh (bonded and unbonded) geometries. Approximate boundary conditions due toKontorovich et al are exploited.

3-4

APPENDIX A

BIOGRAPHY OF JAMES R. WAIT

Dr. James R. Wait was bom in Ottawa, Canada, on January 23, 1924. From 1942 to 1945 hewas a radar technician in the Canadian Army. He received his BASe. and MASc. degrees inengineering physics in 1948 and 1949, respectively, and his Ph.D degree in electromagnetic theoryin 1951, all from the University of Toronto, Toronto, Canada.

From 1948 to 1951, he was associated with Newmont Exploration ltd. of New York, N.Y., andJerome, Arizona, where his research led to several patents for electromagnetic and induced­polarization methods of geophysical prospecting. From 1952 to 1955, he was a section leader at theDefense Research Telecommunications Establishment, Ottawa, where he was primarily concernedwith electromagnetic problems. Since joining the National Bureau of Standards (NBS), Boulder,Colorado, in 1955, he was appointed adjunct professor of electrical engineering at the University ofColorado, Boulder. During 1960, he was a visiting research fellow at the Laboratory ofElectromagnetic Theory in the Technical University of Denmark, Copenhagen. For the academic year1966-1967. he was a visiting professor at Harvard University, Cambridge, Massachusetts. From 1963to 1967, he was a senior research fellow at Boulder, where he was also a consultant to the director ofthe Institute for Telecomrronication SCiences and Aeronomy. From 1967 to 1970. he was seniorscientist in the Office of the Direclor of the Environmental SCiences services Administration (ESSA)Research Laboratories. Boulder. In September of 1970, he became the senior scientist in the Officeof the Director of the Institute for Telecommunications Sciences (ITS) of the Office ofTelecommmications. In October. 1971, he retumed to his position in the Office of the Director of theEnvironmental Research Laboratories of the National Oceanic and Atmospheric Administration(NOAA ), formerly ESSA, but remained as a consultant to the director of ITS. In addition, he was afellow of the Cooperative Institute for Research in Environmental SCiences. In May, 1971. Dr. Waitwas a visiting professor at the Catholic University of Rio de Janeiro, Brazil, and in september, 1971, hewas a guest of the U.S.S.R Academy of SCiences in Moscow and Tbilisi.

Dr. Wait was awarded the Department of Commerce Gold Medal in 1959 for "highlydistinguished authorship in the field of radio propagation"; the Boulder SCientist Award sponsored bythe Scientific Research Society of America in 1960; the NBS Samuel Wesley Stratton Award in 1962;and the Arthur S. Flemming Award, Washington, D.C. Chamber of Commerce, and the Harry DiamondAward from the IEEE. both in 1964. He is also a fellow of the Institute of Electrical and ElectronicEngineers (IEEE) and the American Association for the Advancement of Science (AMS). Inseptember 1972, Dr. Wait receiVed an Outstanding Publication Award from the Office ofTelecommunications in Washington, D.C. Dr. Wait received NOAA's 1973 Scientific Research andAchievement Award "in recognition of his outstanding achievement as a scientist and research leaderin theoretical studies of electromagnetic wave propagation in the earth and its atmosphere". In 1975he received a Special Achievement Award from the Office of Telecommunications "in recognition ofoutstanding authorship of a great variety of publications and journal articles".

He was mainly responsible for the establishment of the journal Radio Science, which started in1959 as Section 0 of the NBS JOUTflQl ofResearch; he served three terms as editor. He has also servedthree terms as an associate editor of the Journal ofGeophysical Research. For four years, he was U.S.regional editor of Electronics Letters. Currently, he is a member of the Editorial Board ofGeoExplormion (Sweden) and he is co-editor of the Institution of Electrical Engineers (lEE) series,Monographs on Electromagnetic (EM) Waves (England).

A-1

Dr. Wait has published eight books and numerous papers on subjects ranging fromelectromagnetlcs to geophysics. He was secretary (1974 to 1978) of the U.S. National Comrnttee ofthe International Scientific Radio Union(URSI). He has been a U.S. delegate to URSI GeneralAssemblies In Boulder (1957), London (1960), Tokyo (1963), Ottawa (1969), Warsaw (1972), Uma(1975) and Helsinki (1978). In 19n, Dr. Wait was elected to the U.S. National Academy ofEngineering. Also, In June 1977, he was elected to be a fellow of the lEE. On July 31,1978, Dr. Waitreceived the Balth van der Pol Gold Medal at the General Assembly of URSI held In Helsinki, Finland.

In August, 1980, Dr. Walt became professor of electrical engineering and geosciences at theUniversity of Arizona in Tucson. Since then he received the IEEE Centennial Medal in 1984 and theIEEE Geoscience and Remota Sensing Society's Distinguished Achievement Award in 1985. He wasappointed regents professor in the University of Arizona in 1988. Jim Walt retired from the Universityin September 1989 to become a private consultant in his home office, 2210 East Waverly, Tucson AZ85719. He keeps in close touch with the campus (being a 10 minute bike ride away).

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APPENDIX B

COMPLETE LISTING (TO APRIL 1990) OF REFEREED PAPERSOF JAMES R. WAIT

by

J. R. WaitAugust 1990

Note: Reprints included in section 1 are denoted by an asterisk in front of the reprintnumber.

B·1

ReprintNLlllber

1 J.R. Leslie and J.R. Wait, Deteetion of overheated transmission 'Line jointsby m«Zns of a boZomettitl'~ Trans. AlEE, Vol. 68, 1-5, May 1949; also Elect.EDg., Vol. 68, November 1949.

2 Elee'f:%tanQ(Jnetia ztadiation in the eaz>th (A theo1'etiaaZ investigation) ~ Dept.of Electrical Engineering, University of Toronto, April 1950.

3 Tzransient eZeet1'omagnBtie p1'opagation in a eonducting medium~ Geophys.,Vol. XVI, 213-221, April 1951.

4 The magnetie dipole OIJer the ho1'i80ntaHy stratified earth~ Can. J. Phys.,Vol. 29, 577-592, November 1951.

5 A eonductint,J sphezee in a time va1'!fing magnetie fieZd~ Geophys., Vol. XVI,666-672, October 1951.

6 The oylindrieal ore body in the presence of a eab1.e ea1'1'ying an oseiZlatint,JOJa'2'ent~ Geophys., Vol. XVII, 378-386. April 1952.

7 Cu:t'1'ent-etrl"1"Ying tA1i1'e loops in a simple inhmlogeneous region:. J. Appl. Phys.,Vol. 23, 497-498, April 1952.

8 The basis of a method for measuring the compZe:r dieZee'Q'ie eonstant at miZ1.i­metre ~elengtha~ Radio Physics Laboratory, Project Report, 15 Sept. 1952.

9 Mutual inductance of ci1'eUits on a wo-layer ea:rth~ Can. J. Phys., Vol. 30,450-452, September 1952.

10 Etectromagnetie fields of eun-ent-earozoying tJi,re8 in a condueting mediuml

Can. J. Phys., Vol. 30, 512-523, September 1952.

11 The magnetie dipole antenna irrme1'sed in a eonducting medium:. Proc. IRE,Vol. 40, 1244-1245, October 1952.

12 A 71Dte of dipole radiation in a conductint,J medium Jl Geophys., Vol. XVII,978-979, October 1952.

13 Reflection of e'Lectromagnetic wGWes ob'LiqueZy from an inhmlogeneous medi1ll'l~

J. Appl. Phys., Vol. 23, 1403-1404, December 1952.

14 The B1.eet1'ic fields of a 'Long current-ca1"1'Yint,J lJi1'e on a stratified earthJl

J. Geophys. Res., Vol. 47, 481-485, December 1952.

15 Receiving properties of a lJire Zoop lJJith a sphez>oidaZ e07'e~ Can. J. Tech.,Vol. 31, 9-14, January 1953.

16 with J.E.T. Mousseau, Calcul.ated field patterns f01' ho1'i80ntaZ tmveZUng~e antennas~ Radio Physics Laboratory, Project Report No. 19-0-2,15 January 1933.

17 ~ient couplint,J in gzoaunded ci1'cuits~ Geophys., Vol. 18, 138-141,January 1953.

18 A t2-an8ient ~tie dil-'OZe sOW"ce in a dissipative medium~ J. Appl. Phys.,Vol. 24, 341-343, March 1953.

19 with L.L. Capbell, Effect of a 7"apge dieZeetZ'ie eonstant On f!1'ound-ltJavep1'opagation~ Can. J. Phys., Vol. 31,456-457, March 1953.

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20 On the feasibility of measuring g2'ound conductivity f1'orn an ai1'craft~

Defence Research TelecOllllllUnications Establishment, Ottawa, Canada(March 9, 1953).

21 with L.L. Campbell, The fie'tds of an e'tecmc dipo'te in a seni-infinitecont1.ucting medium~ J. Geophys. Res., Vol. 58, 21-28, March 1953.

22 Induction by a ho1'iaonta't'ty osci't'tating magnetic dipoZe oVe1' a conductinghomogeneous Ba2'th~ Trans. Amer. Geophys. Union, Vol. 34, 185-188, April 1953.

23 1'he potentiaZ of tlJo auP1'ent point SOUl'CeB in a homogeneous conductingpzooUzte sphezooid~ J. Appl. Phys., Vol. 24,496-497, April 1953.

24 Pzoopagation of zoadio waves over a stratified ground~ Geophys., Vol. 18,416-422, April 1953.

25 A conducting p~eab'te sphezoe in the presence of a coi't carrying anosci'tlating auP1'ent~ Can. J. Phys., Vol. 31, 670-678, May 1953.

26 with L.L. Campbell, T1'ansmission au1'VeB fop g1'ound L)atJe propagation at l"",zoadio fzoequencies~ Report R-l. D~f. Res. Tele. Est •• Radio Physics Lab.,April 1953.

27 1'he zoeceivi~ loop L)ith a hoZZOLJ p1'olate spheroidal core~ Can. J. Tech.,Vol. 31, 132-139, June 1953.

28 An appzeozimate method of cbtaining the ~ient 1'esponse f1'orn the f1'equency2'esponae~ Can. J. Tech., Vol. 31, 127-131. June 1953.

29 with L.L. Campbell, The fie'tde of an osciZ'tating magnetic dipo'te irrme1'sedin a seni-infinite conducting medium~ J. Geophys. Res•• Vol. 58. 167-178.June 1953.

30 Radiation from a vezotica't electric dipo'te over a st1'atified g1'ound~ Trans.IRE, Vol. AP-l, 9-12, Part I. July 1953.

31 Radiation resistance of a amaH ci1'auZar loop in the p1'esence of a conductingground~ J. Appl. Phys., Vol. 24. No.5, 646-649, May 1953.

32 The zoadiation fields of a horizontal dipole in a semi-infinite dissipativemedium~ J. Appl. Phys., Vol. 24. 958-959, July 1953.

33 Electzoomagnetic coupZing betL)een a ci1'aular loop and a conducting sphezoe~

Geophys., Vol. 18, 970-971, October 1953.

34 Radiation fzoom a 'tine SOUl'ce adjacent to a conducting haZf p'tane~ J. Appl.Phys., Vol. 24, No. 12, 1528-1529, December 1953.

35 Induction in a conducting sheet by a smaZZ current-carrying Zoop~ Appl.Sci. Res., Sec. B, Vol. 3, 230-235, May 1953.

36 Cowrplez TnagW4tic pezrmeabiZity of spherical partic'te8~ Proc. IRE, Vol. 41,No. 11, 1665-1667, November 1953.

37 The fieZds of a Zine souzoce of current OVe1' a st1'atified conductor~ Appl.Sci. Res., Sec. B, Vol. 3, 279-292, 1953. .

38 Radiation fzoom a g1'OUnd antenna~ Can. J. Tech., Vol. 32, 1-9, January 1954.

39 with W.C.G. Fraser, Radiation f1'Q1l a vezoticaZ dipo'te OVe1' a stzoatifiedgrownd~ Part II~ Trans. IRE, Vol. AP-3, No.4, October 1954.

B-3

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40 Note on the theoroy of roadio propagation over an ice-covered sea~ Def. Res.Tele. Est •• Radio Physics Lab•• Project Report 18-0-7. March 31. 1954.

* 41 with W.J. Surtees. Impedance of a top-l.oaded antenna of arbitrary l.engthove a ciraul.ar grounded screen~ J. Appl. Phys•• Vol. 25. No.5. 553-555.May 1954.

42 On anomal.ous propagation of radio waves in earth strata~ Geophys •• Vol. 19.342-343. April 1954.

* 43 with W.A. Pope. The characteristics of a vertica"l antenna Llith a radia"lconductor ground system~ Appl. Sci. Res •• Sec. B. Vol. 4. 177-195. 1954.

44 with K.F. Hill and W.A. Pope. Ref1,ection fran a mirror surface Llith anabsorbent coating~ J. Opt. Soc. Amer •• Vol. 44. No.6. 438-441. June 1954.

45 On the rel.ation between te"lZuric currents and the earth's magnetic fieU~

Geophys •• Vol. 19. 281-289. April 1954.

46 Mutual. coupHng of l.oops "lying on the ground~ Geophys •• Vol. 19. No.2.290-296. April 1954.

47 On the theoPy of an antenna Llith an infinite cornel' refZector~ Can. J.Phys•• Vol. 32. 365-371. May 1954.

48 Ref1ection fran a ~e grid paral.l.el. to a conducting pl.ane~ Can. J. Phys ••Vol. 32. 571-579. September 1954.

49 with S. Kahana. Radiation from a dot on a cyUndricaZZy tipped ~edge~ Can.J. Phys •• Vol. 32. 714-721. November 1954.

50 C. Froese and J .R. Wait. Ca"laul.ated diffNction patterns of die"lectric rodsat centimetric ~eZ~/ths~ Can. J. Phys•• Vol. 32. 775-781. December 1954.

51 with L.L. Campbell. FieLds of dipol.es in a semi-infinite conducting medium~

J. Geophys. Res•• Vol. 58. 21-28, March 1953; 167-168. June 1953; Def. Res.Tele. Est., Project Report 19-0-9, September 1954.

52 with W.A. Pope. EvaZuation of errors 'l",: an eight-e"lement adcock a'1tenna~

Trans. IRE. Vol. AP-3, No.4. 159-162. October 1954.

53 Theory of el.ectromagnetic surface waves over geol.ogical. conductors~

Geofisica pura e Appl., Vol. 28. 47-56. 1954•

. 54 Ref1ection at arbitrary incidence froom a para"lZel. ~re grid~ Appl. Sci.Res•• Vol. 14, No.6. 393-400, 1954.

55 On the scattering of spherical. waves by a cyUndricaZ object~ Apple Sci.Res •• Vol. 4. Sec. B. 464-468. 1955.

56 with C. Froese. Ref1ection of a trar.sient el.ectranagnetic wave at a conductingBUl'face~ J. Geophys. Res •• Vol. 60, No.1. 97-103, March 1955.

57 On the theory of the notch aerial.~ Def. Res. Tele. Est., Radio Physics Lab.,Project Report 19-0-14, May 17. 1955.

58 Mutual. eZectrcmagnetic coupZing of "loops ove a homogeneous ground~ Geophys.,Vol. 20, No.3, 630-637, July 1955.

* S9 with W.A. Pope, Input resistance of LF unipo"le aeia"ls~ Wireless Engr ••Vol. 32, 131-138. May 1955.

8-4

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60 Radiation characteztistics of azial slots on a c017liucting cyUnder~ WirelessEngr., Vol. 32, 316-332, December 1955.

61 Field produced by an arbitrary sl.ot on an el.Uptic cyUnder~ J. Appl. Pbys.,Vol. 26, No.4, 458-463, April 1955.

62 Radiation fl'Ol'fl an el.ectric dipol.e in the p:t'esence of a cor1"U{}ated cyUnder~

Appl. Sci. Res., Sec. B, Vol. 6, 117-123, 1955.

63 Scattering of a p1.ane lJal)e from a cil'(m1.aI' dielectrtic cyUnder at obl.iqueinci.dence~ Can. J, Pbys., Vol. 33, 189-195, February 1955.

64 Scattlftlling of el.ectremagnetic wal)eB frem a "l.ossy" strip on a conductingptane~ Can. J. Phys., Vol. 33, 383-390, April 1955.

65 with J. Kates, Radiation patterns of cil'cumferential sl.ots on moderatel.yLarge conducting cyl.indero8~ Proc. lEE (London), Pt. C, Vol. 103, 289-296,March 1956.

66 with H.H. Howe, Ampt-,.tude and phase curves for ground-r.Jat)e pl'opagation inthe band 200 ~~l-~ per second to 500 kil.ocycl.es, NBS Circular No. 574,May 1956. (available from NTIS No. PR272292/AS). '

67 Mi:J:ed path gl'ound tAmle p:t'opagation: 1. Short distances~ J. Res., NBS, Vol.57, No.1, 1-15, July 1956.

68 Radiation frcxn a verotical antenna over a auPtJed stratified ground~ J. Res.,NBS, Vol. 56, No.4, 237-244, April 1956.

69 Cu1t1'ents e:z:cited on a conducting surface of l.arge radius of CU1'Vature~ IRETrans., Vol. MTT-4, No.3, 144-145, July 1956.

* 70 Effect of the gl'ound screen or. the fiel.d radiated frem a monopol.e~ IRETrans., Vol. AP-4, No.2, 179-181, April 1956.

71 with M. O'Grady, Surface currents e:z:cited by an infinite sZot on half­p1.anes and rtibbons~ IRE Trans., Vol. AP-4, No.1. 47-50. January 1956.

'* 72 D.G. !!'rood and J .R. Wait. An i'Y(/)estigation of sl.ot radiators in rectangul.armetal. p1.ates~ Proc. lEE (London), Vol. 103, Part B, No.7, 103-109, Jan. 1956.

73 Radiation pattern of an antenna moun.ted on a SU1';"ace of 7.a:rge radius ofau1"l)ature~ Proc. IRE, Vol. 44. No.5, 1 pg., May 1956.

74 On the conductance of sl.ots~ IRE Trans., Vol. AP-4, No.2. 124-127, April1956.

75 Radiation resistance of dipoles in an interface between ~o diel.ect1'ics~

Can. J. Pbys., Vol. 34, 24-26. 1956.

76 with S. Kahana, CaZ~1.ated patteronB of ci1'CU1flferentiaZ sZots on a circularconducting oyl.inderl Can. J. Tech., Vol. 33, 77-97, 1955.

77 with R.E. Walpole, Cal.cu1.ated radiation characteristics of sZots cut inmetal. sheetB~ Can. J. Tech.• Vol. 33, 211-227, Part I, 1955.

78 Transient fields of a verticaZ dipole over a homogeneous curved groound~

Can. J. Pbys., Vol. 34. 27-35. January 1956.

79 with lC. Okaab1ao, Patterns of stub antenn:zs on oy1.indzoicaZ BtruCtures~

can. J. Pbya., Vol. 34, 190-202, February 1956.

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80 LotJJ f1'equency zoadiation f1'om a ho1'izontal antenna OVe1' a sphe1'ical ea1'th~

Can. J. Phys., Vol. 34, 586-595, June 1956.

81 ShieLdin,g of a transient elect1'omagnetic dipole field by a conductive sheet~

Can. J. Phys., Vol. 34, 890-893, August 1956.

82 with R.E. Walpole, Calculated 1'adiation chaztacte1'istic8 of slots cut inmetal sheets~ Can. J. Tech., Vol. 34, 60-70, Pt. II, 1956.

83 Mutual electzoomagnetic coupUng of loops ove1' a homogeneous ground - anadditional note~ Geophys., Vol. 21, No.2, 479-484, April 1956.

84 On the lJXn)ef()1'fTJ of a 1'adio atmospheztic at sho1't !'anges ~ Proe. IRE, Vol. 44,No.8, pg. 1052, August 1956.

'* 85 with D.G. Frood, The 1'adiation patte1'n8 and con4uctances of slots cut on1'ectangulal' metal plates~ Proe. IRE, Vol. 44, No. 10, pg. 1469, ~~t. 1956.

86 with A. Murphy, The geanet1'ical optics of VLF sky lJave p!'opagation~ Proe.IRE, Vol. 45, No. 6, 754-7~O, June 1957.

87 The mode thso1']J of VLF ionosphe1'ic p1'opagation fo!' finite g!'ound conductivity~

Proe. IRE, Vol. 45, No.6, 760-767, June 1957.

88 A.G. Jean, Jr., L.J. Lange and J.R. Wait, Ionospheric reflection coefficientsat VLF f1'01fl sfe1'ics measU1'gnents~ Geofisica Pura E Applieata, Vol. 38,147-153, 1957.

89 H.H. Howe and J .R. Wait, Mode calculations fo!' VLF ionosphe:roic pl'opagation(VLF Symposium Paper No. 36, Boulder, Colorado, January 1957).

90 On the meaBU1'ement of the conductivity of a fluid contained in a cyUntbticalbessel~ Can. J. Tech., Vol. 34, 410-412, 1957.

91 with A. Murphy, Influence of a 1'idge on the lOlJ-f1'equency ground lJave~

J. Research, NBS, Vol. 58, No. I, 1-5, January 1957.

92 with R.R. Rowe, The lJaVeguide mode theoroy of VLF ionospheric propagation~

Proe. IRE, Vol. 45, No. I, pg. 95, January 1957.

93 with A. Murphy, Multiple 1'eflections betlJeen the eal'th and the ionosphe1'ein VLF p1'opagation~ Geofisiea Pura E Appl., Vol. 35, 61-72, 1956.

94 The transient behavio1' of the elect1'anagnetic g!'ound lJXn)e on a sphe1'icaleal'th~ IRE Trans., Vol. AP-5, No.2, 198-202, April 1957.

95 with L.B. Perry, Calculations of ionosphmc 1'eflection coefficients atV81']J lOlJ zoadio f1'equentJies~ J. Geophys. Res., Vol. 62, No. I, 43-56,Karch 1957.

96 On the theo1']J of 1'eflection f1'om a lJiN g1'id paroZZel to an inte:rofoor.between homogeneous media~ Appl. Sci. Res., Sec. B, Vol. 6, 259-275, 1957.

97 AmpUtude and phase of the lOlJ-f1'equency g1'ound lJXn)e neal' a coastZine~

J. Research, NBS, Vol. 58, No.5, 237-242, Kay 1957.

98 The impedance of a lJi1'e groid paroZZel to a dieZectl'ic inte!'face~ IRE Trans.,Vol. KTT-5, No.2, 99-102, April 1957.

99 with W. Kientka, SZotted-cyUnde:ro anatenna lJith a dielectl'ic coating~

J. Re.earch, NBS. Vol. 58. No.6, 287-296, June 1957.

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100 Diffzeaction of a sphe:zoicaZ l.o1aVe pu"L8e by a haZf-p"Lane SC1'een~ Can. J. Phys.,Vol. 35, 693-696, 1957.

101 with J. Householder, Mized-path tp'ound-l.o1aVe pzoopagation: 2. Lazoge1' distances~J. Research, NBS, Vol. 59, No. I, 19-26, July 1957.

102 Insulated "Loop antenna irmIe1'sed in a conducting medium~ J. Research, NBS,Vol. 59, No.2, 133-137, August 1957.

103 Introduction to the VLF pape1'S~ Proe. IRE, Vol. 45, No.6, 739-740, June 1957.

104 The attenuation vs. f1'equency chaztacte1'istics of VLF 1'adio llXlVes~ Proe. IRE,Vol. 45, No.6, 768-771, June 1957.

105 On the mode theoPy of VLF ionosphe1'ic p1'opagation~ Rev. Geofisica Pura EApplicata (Milano), Vol. 37, 103-115, 1957.

106 A note on the pzoopagation of the t1'an8ient g1'ound l.o1aVe ~ Can. J. Phys., Vol.35, 1146-1151, 1957.

* 107 Pattezon of a f"Lush-mounted mi01'owatJe antenna~ J. Research, NBS, Vol. 59,No.4, 255-259, October 1957.

108 The effective e1.ectricaZ constants of soi"L at 1.(JIJ f1'equencies~ Proc. IRE,Vol. 45, No. 10, pg. 1411, October 1957.

109 A.G. Jean, L.J. Lange and J.R. Wait, Ionosphe1'ic 1'efZection coefficients atVLF /%'Om sfmes ",easU1'BtnentB~ Geofisica Pura E Applieata, Vol. 38, 147-153,1957.

110 Pztopagation of a puZse ac¥'Oss a coast 1.ine~ Proe. IRE, Vol. 45, No. 11,1550-1551, November 1957.

III with A.M. Conda, Radiation from sZots on die1.ect1'ic-cZad and c01"1'Ugatedcy1.inde1'S~ J. Research, NBS, Vol. 59, No.5, 307-316, November 1957.

112 Iw:J.uction by an osci"Llating magnetic dipoZee ave1' a '/;;b10-"LayB1' g1'OUnd~ Appl.Sci. Res., Sec. B, Vol. 7, 73-80, 1958.

113 On the caZcuZations of t1'Qnst'e1'se CU1'1'6nt 1.oss in bU1'ied z.n.1'e tp'ound systems~

Appl. Sci. Res., Sec. B, Vol. 7, 81-86, 1958.

114 E:t:citation of BU1'face l.o1aVes on conducting~ stzoatified dieZect1'ic-cZad3

and co1"1'U(Jated BU1'faces~ J. Research, NBS, Vol. 59, No.6, 365-377,Dec_ber 1957.

115 On the theo:oy of p1'opagation of e"Lect1'omagnetic t.)CWes aZong a CU1'Ved BU1'face~

Can. J. Phys., Vol. 36, 9-17, 1958~

116 A ZotJ-fzoequency anTQJ.Za:tt-s1.ot antenna~ J. Research, NBS, Vol. 60, No. I,59-64, January 1958.

117 and diacussions with R.B. Frische and H.V. Buttlar, Discussions on 'Atheo1'etioa1. study of induced e"Lect1'icaZ pozar.iaation'~ Geophysics, Vol. 23,No. I, 144-153, January 1958.

118 An eztension to the "'ode the01']J of VLF ionosphe1'ic p1'opagation~ J. Geophys.Res., Vol. 63, No. I, 125-135, Karch 1958.

119 with A.M. Conda, On the measUl'ement of tp'ound C01'IIiJ4ctivity at VLF~ IRETraua., Vol. AP-6, No.3, 273-277, July 1958.

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120

* 121

122

123

124

125

126

127

* 128

129

130

131

132

133

134

135

136

137

138

139

with A. Murphy, Further studies of the influence of a ridge on the 20',,)­

frequency ground wave, J. Research, ~BS, Vol. 61, No.1, 57-60, July 1958.

A study of earth currents near a VLF monopole antenna UJ'ith a radial UJi.reground system, Pro. IRE, Vol. 46, No.8, 1-2, August 1958.

Transmission and reflection of electromagnetic waves in the presence ofstratified media, J. Research, NBS, Vol. 61, ~o. 3, 205-232, Sept. 1958.

Propagation of ver>y-Zow-fr>equency pulses to gr>eat distances, J. Researcn,NBS, Vol. 61, No.3, 187-203, September 1958.

with A.M. Conda, Pattern of an antenna on a curved lossy suPface, IRETrans., Vol. AP-6, No.4, 348-359, October 1958.

A phenomenological theory of induced eZectPical pol~~zation, Can. J. Phys.,Vol. 36, 1634-1644, 1958.

A s1;u(J:,j of VLT:' fieZd streneth data: Both old and new, Geofisica Pura EApplicata (Milano), Vol. 41, 73-85, 1958/III.

On the eZectromagnetic response of an imperfectZy conducting thin dyke,Geophys., Vol. 24, No. I, 167-171, Februar.y 1959.

with A. Conda, The patterns of a sZot-aPPaY antenna on a f'~nite imper>fec:tgr>ound pl,ane, L'Onde Electrique, Special Supplement, pp. 21-30, 1958.

H.V. Cottony, R.S. Elliott, E.C. Jordan, V.H. Rumsey, K.X. Siegel, J.R. Wait,and D.C. Woodyard, National Committee Report, URSI Subcommission 6.3,Antennas and Waveguides, and annotated bibZiography, IRE Trans., Vol. AP-7,No.1, 87-98, January 1959.

Transmission Zoss curves for propagation at very l,ow radio fr>equencies,IRE Trans•• Vol. CS-6. 58-61, December 1958.

Downcoming radio ~aves, Electronic and Radio Engineer. Vol. 36, No.3,106-107, March 1959.

TPansmission of power in roadie 'F1'o;"::'Ja:r;ion, Electronic and Radio ::'ngineer,Vol. 36, No.4, 146-150, April 1959.

Radiation from a smaZl, Zoop ~.er8e2 in a ser.r:-infinite conduc:tir~ medium,Can. J. Phys., Vol. 37, 672-674, 1959.

Diur>nal, change of ionospheric ;.eights deduced from phase velocity meas;.aae­ments at VLF, Proc. IRE, Vol. 47. ~o. 5, pg. 998, May 1959.

On the theory of refl,ection from a wire grid paraZl.eZ to an in-;erfacebetween hanogeneous media (II), App::'. 5ci. Res •• Sec. B, Vol. 7, No.5,355-360, 1959.

The caZcuZation of the field ir. a homogeneous conductor UJ'ith a wavyintqrface~ Proc. IAE. Vol. 47, No.6, pg. 1155, June 1959.

EaPth currents near a monopo 7,e antenna lUith symrr:et~icaZ top l,oading, J.Research. NBS. Vol. 62, No.6, 247-255, June 1959.

with A.M. Conda, DiffPaction c: electromagnetic waves by smooth obs~acles

for grazing angl,es, J. Research, ~~S, Vol. 63D, No.2, 181-197, Sep~.­

Oct. 1959.

with W.E. Mientka, Cal,cuZated patterns of slotted 8ZZi~tic-cylinder

antennae, Appl. Sci. Res., Sec. B, Vol. 7. 449-462. i959.

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140 with J. Householder. Pattern synthesis fo~ sZotted-cyZinde~antennas~ J.Research. NBS. Vol. 630. No.3. 303-313. November-December 1959.

141 with A.M. Conda. On the diff~action of eZect1tunagnetic pul.ses by aurvedconducting surfaces~ Can. J. Phys •• Vol. 37, 1384-1396. 1959. (Addendaand corrigenda: Vol. 47. pg. 1050. May 1969).

142 Some soZutions fo~ eZect~omagrtetic p~obZems invoZving spheroidaZ~ spherical.and cyZindrical. bodies~ J. Research, NBS. Vol. 64B. 15-32. Jan.-Mar. 1960.

143 Terrest1tiaZ propagation of very-Zo1.J-frequency radio tJaVas~ J. Resaarch, NBS,Vol. 64D, No.2. 153-204. March-April 1960.

144 On the propagation of ELF radio 1.Jal)es and the infZuence of a non-homogeneousionosphere~ J. Geophys. Res., Vol. 65. No.2. 597-600. February 1960.

145 A.G. Jean. W.L. Taylor. and J.R. Wait. VLF phase c~cteristics deducedfrom atmospheric tJaVe fo~s~ J. Geophys. Res •• Vol. 65. No.3. 907-912.March 1960.

146 Preface to the surface tJave papers~ IRE Trans., Vol. AP-7. S132 (SpecialSupplement), December 1959.

147 Guiding of eZect1tomagnetic wves by unifo~Zy rough surfaces~ IRE Trans ••Vol. AP-7. Pts. 1. II. SI54-S168. December 1959.

148 Radio 1.Jal)e p~pagation in an inhomogeneous atmosphe~e~ NBS Tech. Note No. 24,September 10. 1959, (PB 151 383).

149 Dif.fzoactive cozorections to the geometrical. optics of Zen.> frequency p~opagation

in 'EZect1tamagnetic Wave Fropagation'~ edited by M. Desirant and J.L. Michiels.pp. 87-101, Academic Press. 1960.

150 Propagation of el.ectromagnetic puZses in a hunogeneous conducting earth~

Appl. Sci. Res., Sec. B, Vol. 8. 213-253. 1960.

151 with A.M. Conda. On the resonance excitation of a co~rugated cyZinde~

antenna~ Proc. lEE (London). Pt. C, Vol. 107, 362-366, June 1960.

152 with A.M. Conda. On the comt-utation of diffraction fieZds jor g~a2ing

angZes~ in 'EZect1tomagnetic Wave Propagation'~ edited by M. Desirantand J.L. Michiels. pp. 661-670, Academic Press. 1960.

153 with A.M. Conda, Radiation f~om a sZot on a Zarge co~gated cyZinder~

in 'El.ectromagnetic Wave Fropagatior.'~ edited by M. Desirant andJ.L. Michiels. pp. 103-109. Academic Press. 1960.

154 Mode theol"y and the propagation of extremeZy Zen.> frequency radio tJaves~

J. Research. NBS. Vol. 64D. No.4. 387-404. July-August 1960.

155 with N.F. Carter. FieZd strength caZcuZations for ELF radio tJaves~ NBSTech. Note.No. 52. March 1960 (PB 161 553).

156 On the el.ectrunagnetic response of a conducting sphe~e to a dipoZe fieZd~

Geophys •• Vol. 25. No.3. 649-658, July 1960.

157 On the theory of the sZ01.J-taiZ portion of atmospheric tJaVefo~s~ J. Geophys.Res •• Vol. 65. No.7. 1939-1946. July 1960.

158 On the ezcitation of eZectrunagnetic surface tJaves on a aurved 8Urface~

Trans. IRE. Vol. AP-B, No.4, 445-449, July 1960.

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159 A conference on the propagation of ELF electItCfTlagnetic r.1aI)es~ Proc. nEtVol. 48, No.9, pg. 1648, September 1960.

160 Inf1,usnce of sOUPce distance on the impedance chazoacteristics of ELF Ndiowaves~ Proc. IRE, Vol. 48, No.7, pg. 1338, July 1960.

161 with K. Spies, Influence of earth cwovatu1'e and the terzaestzoial magneticfie Zd on VLF pl'opagation~ J. Gee-phys. Res., Vol. 65, No.8, 2325-2331,August 1960.

162 Elect1'omagnetic Radiation fl'CfTI CyUndztical Stla'ctu1'es~ Cha. 1. Intzooduction~

2. S/,a>r)ey of the Uteroatuzoe on slots and l'elated topics~ 3. A:t:ial dipoleLlithin a coroner l'ef1,ectol' and slots on lVedges~ 4. The fields of a:z:iaZdipoles in the pztesence of a cil'cularo conducting cyUnder~ 5. Slots on acylindPicaZZy tipped l"edge~ 6. TzoansVe1'se haZf-lXlVe slots on isoZatedcylinde1's~ 7. Contiuctance of arial slots on infi?:ite cylinders and LJedges~

8. Residue series l'epl'esentation fUI' the fields of an a:cial sZot~ 9. FieUsof the a:z:iaZZy slotted cylinder in tenns of Fock functions~ 10. Residueseries l'eproesentation fol' ci1'cumferentiaZ slots on cylinde1's~ 11. FieUsof curzoent dist1'ibutions in rvedge l'egions~ 12. RadiaZ eZect1'ic dipoles incylintboical rvedge l'egions~ 13. Field of a slot on an elliptic cyZinde1'~

14. CaZculated Ndiation charoacte1'istics of sZots in metal sheets~ 15.Radiation fl'CfTI slot a:n'a7Js on a half-p'Lane~ 16. Slotted-cylinder antenmLlith a dielect1'ic coating. 17. Scatteroing of a plane lUaVe f1'CfTl a dieZect1'iccil'C1Uz.ar. cyZinde1' at oblique incidence~ 18. Diffzoaction by a convez cylin­dricaZ sU1'face~ Appendiz: Radiation f1'CfTl eurl'ent dist1'ibutions~ (PergamonPress, Inc., New York, NY, 1959).

163 A phenomenological theozry of ove1'Voltage fol' metalUc paroticles~ Ch. 3 of'Overvoltage Research and Geophysical Applications', edited by J .R. Wait(perg8Dlon Press, Inc .. , 1959).

164 The vazeiable-f1'equency method~ Ch. 4 of 'Overvoltage Research and Geo­physical Applications', edited by J .R. Wait (Perg8Dlon Press, Inc., 1959).

165 L.S. Collett, A.A. Brant, W.E. Bell, K.A. Ruddock, B.O. Siegel, andJ .R. Wait, Laboroatozry investigation of overvoltage~ Ch. 5 of 'OvervoltageResearch and Geophysical Applications', edited by J. R. Va!t (Pergamon Press,1959).

166 with L.S. Collett, C1'ite1'ia from the tItansient decay CU1'l)es~ Ch. 6a of'Overvoltage Research and Geophysical Applications', edited by J.R. Wait(Pergamon Press, 1959).

167 The eZectztomagnetic fieZds of a dipoZe in the pl'esence of a thin pZasmaBheet~ Appl. Sci. Res., Sec. B, Vol. 8, 397-417, 1961.

168 Pztopagation of electztomagnetic wves a1.ong a thin plasma sheet~ Can. J.Phys., Vol. 38, 1586-1595, 1960.

169 A nBlJ approach to the mode theozry of VLF pl'opagation~ J. Research, NBS,Vol. 65D (Radio Prop.), No. I, 37-46, January-February 1961.

170 On the calculation of APMAG anomaZieB~ Geophys., Vol. 25, 1290-1293,Dec.her 1960.

171 with K. Spies, A note on phase velocity of VLF roadio lXlVes~ J. Geophys.Res., Vol. 66, No.3, 992-993, March 1961.

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172 with A.M. Conda, Resonance cha.1oactePistics of a cozozougated cy'Lindep e:z:aitedby a magnetic dipo'Le~ IRE Trans., Vol. AP-9, No.4, 330-333, July 1961.

173 Some boundazoy va'Lue p1'Ob'Lems invo'Lving p'Lasma m8dia~ J. Research, NBS,Vol. 65D (Radio Prop.), No.2, 137-150, April-June 1961.

174 On the theozry of mized-path gpound-wave propagation on a spherica'L etZZ'th~

J. Research, NBS, 65D (Radio Prop.), No.4, 401-410, July-August 1961.

177 P1'Oceedings of the 1960 Conference on the Propagation of ELF Radio Waves~

NBS Technical Note No. 61 (PB-161 562), edited by J.R. Wait, June 6, 1960.

178 A compa:M,son betl.Jeen theopeticaZ and ezperimenta'L data on phase ve'Locityof vtF ~o waves~ Proc. IRE, Vol. 49. No.6, 1089-1090, June 1961.

179 rhe p1'Opagation of e1.ectpomagnetic waves along the eanh's sW'face~ in'Electromagnetic Waves'~ edited by R.E. Langer, pp. 243-290 (Universityof Wisconsin Press, Madison, WiS., 1962).

180 A.G. Jean, H.E. Taggart, and J .R. Wait, Ca'Libmtion of loop antenr.as atVLF~ J. Research, NBS, Vol. 65C, No.3, 189-193, July-September 1961.

181 A aiffmction theozry fop LF sky-wave propagation~ J. Geophys. Res., Vol. 66,No.6, 1713-1724, June 1961.

182 with A.M. Canda, A diffmction theozry fop LF sky-wave propagation - Anadditional note~ J. Geophys. Res., Vol. 66, No.6, 1725-1729, June 1961.

183 A.G. Jean, A.C. Murphy, J.R. Wait and D.F. Wasmundt, Obse'PVed attenuationmteofELFMdiowaves~J. Research, NBS, Vol. 65D (Radio Prop.), No.5,475-479, September-OCtober 1961.

184 The e'Lectzoomagnstic fields of a hoPiaontal dipo'Le in the presence of aconducting half-space~ Can. J. Phys., Vol. 39, 1017-1028, 1961.

185 K.P. Spies and J.R. Wait, Mode calculations fop VLF propagation in theeaPth-ionosphePe waveguide~ NBS Tech. Note No. 114, (PB 161 615),July 17, 1961.

186 A note concerning the e:z:citation of ELF e'Lectromagnetic waves~ J. ResearchNBS, Vol. 65D (Radio Prop.), No.5, 481-484, September-october 1961.

187 On the possibi'Lity of rejecting certain modes in VLF propagation~ Proc.IRE, Vol. 49, No.9, pg. 1429, September 1961.

188 On the impedance of long lJipe suspended over the gpound~ Proc. IRE, Vol. 49,No. 10, pg. 1576, October 1961.

189 A. Nennsberg (translator); J.R. Wait and J.B. Reubens (editors), ChapteP on'Long waves fpom book. 'Ionospheric lTopagation of Radio Waves' ~ by Ya. L.Al'pert, NBS Translation T5-60. August 5, 1961.

190 E:z:pected influence of a locaZi2ed change of ionosphere height on VLFppopagation~ J. Geophy8. Res., Vol. 66, No. 10, 3119-3123, October 1961.

191 COtmIents on K. Voaoff's paper 'CaZibration of pulsation detectop coils' ~J. Geophys. Res., Vol. 66, No. 10, pg. 3603, October 1961.

192 Average dscay Za1JB of VLF fieZds~ Proc. IRE, Vol. 50, No. I, 53-56, Jan. 1962.

193 C""",ents on paper by W.D. WestfaZZ 'Prediction of VLF diuzonal phase changescmd80Zarf1,are 8ff8ct'~ J. Geophys. Res., Vol. 67, No.2, 916-917, Feb. 1962.

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209

On t'h8 pzoopagation of VI;F and ELF radio 1I7al)es uhen the ionosp'h8re is notshazopZy bounded~ J. Research, NBS, Vol. 66D (Radio Prop.), No.1, 53-61,January-February 1962.

C.M. Jackson, J .R. Wait aad L.C. walters, NtlnezoicaZ resuZts for the suzofaceimpedance of a stratified conductor~ NBS Tech. Note No. 143 (PB 161 644),Karch 19, 1962.

An antlZysu of VI;F mode propagation jor a variabZe ionosphere height~ J.Research, NBS, Vol. 660 (Radio Prop.), No.4, 453-461, July-August 1962.

Mode convezosion in the eazoth-ionosphere fNtZVeguide~ NBS Tech. Note No. 151,June 8, 1962. (PB 191730)

rwzz,e p1'opagation azoound a CtD"Ved boundazry which contains an obstacZe~

Can. J. Phys., Vol. 40, 1010-1016, 1962.

PossibZe inf1,u.ence of the ionosphere on the impedance of a ground-basedantenna~ J. Res., NBS, Vol. 660 (Radio Prop.), No.5, 563-569, Sept.-OCt. 1962.

Thso~ of magneto-te1,1,uric fieZds~ J. Res., NBS, Vol. 66D, (Radio Prop.),No.5, 509-541, September-OC:tober 1962.

E:citation of modes at vezry tow frequency in the earth-ionosp'h8re tJal)eguide~

J. Geophys. Res., Vol. 67, No. 10, 3823-3828, September 1962.

A 7'IOte on the propagation of eZectromagnetic puZses ove the earth's BUZ'face~

Can. J. Phys., Vol. 40, 1264-1268, 1962.

In1:Jtoduction to the theoz?J of VI;F propagation~ Proc. IRE, Vol. SO, 1624-1647,July 1962.

Effective impedance of a we grid para1,1,eZ to the eazeth's suzeface~ IRETrans., Vol. AP-I0, No.5, 538-542, September 1962.

On t'h8 p1'opagation of ELF puZses in t'h8 earth-io7'IOsphere tJal)eguide~ Can..1. Phys., Vol. 40; 1360-1369, October 1962.

A 7'IOte on diurnaZ phase changes of vezoy-'LoI.>-frequency 1I7al)es for Zong paths~

J. Geophys. Res., Vol. 68, ~o. 1, 338-340, J,nuary 1, 1963.

E'Lectztcmagnetic Wavss in Stztatified Media (Perl_on Pres., Oxford, 1962):Chs. I. GtmIIl'aZ introduction~ II. RefZection of eZectrGmagnetic waves f1'fXflhoriaonta1,1,y stztatified media~ III. Ref1,Bction of eZsctzoomagnetic tJal)SSfztr:In iMDmogeJ'lSOU8 media with speciaZ pztOfiZss~ IV.A~ methodsfor treating refZections from inhcmogenllOUB media~ V. Pztopagation along asp'hsZ'icaZ IIW'/ace~ VI. Fuw1tzmenta1.s of mods theory of wave p1'opagation~

VII. ChaztactB1'i.stics of the modes fo:r .VI;F p1'opagation~ VIII. P:ropagationin etztatified magnBto-p'LaBma media~ IX. VI;F p1'opagation - J'hsoZ"Y and e:cpe:ri.­ment~ X. ELF (sz1::ftIlmBZy 'LOII1 frequency) p:ropagation - J'heorry and e:psriment~

U. ABJ1fIP1;otic dBoeZopnent for guided wave pzoopagation~ XII. Supen'ef:raotionand the the~ of tZ'opoephszoi.c dwJting.2nd. Ed. 1970) III1crofo1'lll (MIM ,fl4IIqell House,Falrvlew Park,Elmsford NY 10523 J.

!'he t'Mory of an antmzna OVI1l' an inha'flOgeJ'lSOU8 grouniJ. pZane~ in 'EZsctzoo­magntltic J'h«n-y and A~'~ edited by E.C. Jordan, PerlUlOn Prea.,Oxford, 1963.

A note on the eZflCtztomt:zgnetic rssponse of a stztatified «ZZ'th~ Geophya.,Vol. 27, Ro. 3, 382-385, June 1962.

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K. Tao aDd J.R. Wait. Ref7,ection mechanisms for sporoadic E~ in 'IonosphericSpol'at1:;,c E'~ edited by E.K. Smith and S. Matsushita. Pergamon Press. Oxford.1962.

Th8 mode theoroy of VLF l'adio propagation for a sphel'ical earoth and a concen­tric anisotl'opic ionosphel'e~ can. J. Phys •• Vol. 41, 299-315. 1963. (Errata:CJP !i, pg. 819 and pg. 2267, 1963).

with K.P. Spies. Height-gain foro VLF !'adio tuave8~ J. Research. NBS. Vol. 670(Radio Prop.). No.2, 183-187. March-April 1963.

with L.C. Walters. CuztVes for gl'ound ",ave pl'opagation ovel' mi:r:ed land andsea paths~ IEEE Trans., Vol. AP-ll. No.1. 38-45. January 1963.

A note on E-fietd and H-fietd Losses fol' groound-based antennas~ Proc. IEEE.Vol. 51, No.2, pg. 366, February 1963.

with L.C. Walters, Infl,uence of a sector ground sC!'een on the fietd of averotical antenna~ NBS Monograph No. 60, April 15. 1963. NTIS (COM-7S-l0053) ..

Conceroning sol,utions of the VLF mode probZem foro an anisot1'opic curvedionosphel'e~ J. Research, NBS, Vol. 67D (Radio l'rop.) , No.3. 297-302,May-June 1963.

with L.C. Walters, Refl,ection of VLF l'adio 1lXZt1es f1'Ol1l an inhomogeneousionosphe1'e. Pa:zot I. &:ponential.'Ly va:zoying isotl'opic modeZ~ J. Research,NBS, Vol. 670 (Radio Prop.). No.3, 361-367, May-Ju~e 1963.

with C.M. Jackson, Resonant charoacte1'istics of a c01TUgated Bphel'e~ J.Research, NBS, Vol. 670 (Radio Prop.), No.3, 347-353. May-June 1963.

Trans'Lated ezt1'acts froom the book 'Ei,ect1'Offlagnetic nelds AppZied toInduction Methods of El,ectricaL P1'ospecting'~ by A.B. Ve1i1dn and G.S.Fraatov (Translator: Alberta K. Garber), NBS Report, T7-60, May 6, 1963.

Vezty-ZOli) fzaequency proopagation in the earoth-ionocphe1'e UKlve-guide of non­unifo~ U){,dth~ Proc. Int. Conference on the Ionosphere, Imperial College.Inst. Pbys. and Pbys. Soc., London. 446--451. July 1962.

Influence of the 1,0U)B1' ionosphel'e on pZ"opagation of VLF U)al)eB to groeatdistances~ J. Res. NBS. Vol. 670 (Radio Prop.). No.4. 375-381, July­August 1963.

A note on antipodal focussing~ NBS Tech. Note No. 182. August 20, 1963.Available through NTIS (COM-75-l0283).with L.C. Walters, Reflection of VLF ztadio L7CZVes fl'an an inhomogeneousionosphezte~ Parot II. PB1'tu1'bed e:rponential model~ J. Res. NBS, Vol. 67D(Badio Prop.), No.5, 519-523. September-october 1963.

7'he possibiZity of guided eZect1'Ol1lagnetic '",aues in the earoth's C1'USt~

IEEE Trans., Vol. AP-l1, No.3. 330-335, May 1963.

The oblique proopagation of gl'ound waves acl'OSS a coast line - Pa1't I ~J. Research NBS, Vol. 67D (Radio Prop.), No.6, 617-624, Nov.-Dec. 1963.

with C.M. Jaclulon, The oblique pl'opagation of gl'ound tJal'es aol'OSS a coastline - Part II~ J. Research NBS. Vol. 67D (Radio Prop.), No.6, 625-630,Nov_ber-Dec:cber 1963.

L.C. Walters and J.R. Wait, Numez'icaZ caZcuZations fol' reflection ofelBO~netio L7CZVes fl'om a lOBsy ~etop~a~ NBS Tech. Note No. 205.Nov.-ber 21, 1963. (COM 73-10634)

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228 with C.H. Jackson, Calculations of the fieU near the ape:r: of a rvedge sW'face~

NBS Tech. Note No. 204, November 21, 1963 (PB 184 118).

229 RevielJ of mode t'h8ozoy of 1*adio proopagation in te1*1*estPial waveguides~ Revs.Geophys., Vol. I, No.4, 481-505, November 1963.

230 with L.C. Walters, RefZection of VLF 1*adio waves f1*om an inhcxnogeneousionosphe:re. ~ Pa1*t III. Exponential model bJith hyperboUc troansition~ J.Research NBS, Vol. 67D (Radio Prop.), No.6, 747-752, Nov.-Dec. 1963.

231 Calculated diffraction effects at VLF from a localized ionosphericdep1*ession~ NBS Tech. Note No. 208, January 16, 1964.

232 '!'heozoetical calcuZations of t'h8 impedance of a monopole antenm bJith a zeadial­bJi1*e ground systll'l on an impe1*fectly conducting ha'lf-space~ by S.W. Haley,R.J. King, and L.R. Branch (based on J.R. Wait's theory). AFeRL ScientificReport No. 26, December 13, 1963.

233 Tl.1o-dimensionaZ treatment of mode theozoy of the propagation of VLF l'adiowaves~ Radio Sci. J. Res. NBS, Vol. 68D, No. I, 81-93, January 1964.

234 Some 1*ema:rks on mode and NY theo1*ies "f VLF road';'o pl'opagation~ Radio Sci.J. Res. NBS, Vol. 68D, No. I, 79-80, January 1964.

235 with L.C. Walters, Reflection of electl'omagnetic waves fl'om a 'lossy magneto­pLasma~ Radio Sci. J. Res. NBS, Vol. 68D, No. I, 95-101, January 1964.

236 Oblique pztopagation of gzroundJ..Javes across a coastline~ Pal't III~ Radio Sci.J. Res. NBS, Vol. 68D, No.3, 291-296, March 1964.

237 '!'he 1*eZation between VLF propagation and D-Zaye1* chal'ac~e1*istics~ IEEETrans., Vol. AP~12, No.2, 239-240, Karch 1964.

238 On the theo1*y of Schumann 1*esonances in the eazoth-ionosphere Ca'lJity~ Can.J. Pbys., Vol. 42, 575-582, April 1964.

239 Electromagnetic scatte1*ing f1*om a 1*adiaZZy inhcxnogeneous sphere~ Appl. Sci.Res. Sec. B, Vol. 10, 441-450, 1964.

240 Electromagnetic fie'las in lossy media (The U.S. National Committee Reportfor Commission 6 of URSI, Subcomm. 6.3 E1ectromagnetics), Radio Scl. J. Res.NBS, Vol. 68D, No.4, 463-465, April 1964.

241 Some 1*flIfIal'ks concezoning non-1*eciprocity in 1*adio propagation~ IEEE Trans. ,Vol. AP-12, No.3, 372-373, Kay 1964.

242 On phase changes in vezoy-IObJ fl'equency p1*opagation induced by an ionosphe1*icdepression of finite e=tent~ J. Geophys. Res., Vol. 69, No.3, 441-445,February 1, 1964.

243 Electromagnetic surface uaves~ in Advances in Radio Resea1'ch~ edited byJ.A. &axton, Vol. 1, pp. 157-217, Academic Press. London, 1964 (See alsoRP-421, from EM Probing in Geophysics, 1970).

244 Influence of a ci1*cuZa1' ionosphenc dep1*ession or. VLF propagation~ RadioSci. J. Res. NBS, Vol. 68D, No.8, 907-914, August 1964.

245 A 7IOte on VHF 1*eftection f1*om a t1*oposphe1*ic laye1*~ Radio Sci. J. Res. NBS,Vol. 68D, No.7, 847-848, July 1964.

246 Pztofiu on CRPL~ Microwave J., Vol. 6, pg. 83, July 1963.

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247 with C.M. Jackson, Inf1,uence of the zoefractive inde:c pr'ofUe in VHF zoefZectionfzoom a tztoposphsric Zayer~ IEEE trans., Vol. 1P-12. No.4, 512-513, July 1964.

248 C01ICem:ng the theozwy of radiation fzo(X1l a s'totted conducting plane in a pZas1lr1.environment~ ~~S Tech. Note No. 223, September 28, 1964. (PB 168051)

249 with K.P. Spies, Pzoopagation of zoadio waves past a coast tine LJith a gztadualchange of suzoface impedance~ IEEE Trans., Vol. AP-12, No.5, 570-575,September 1964.

250 Cohel'ence the01'ies of tztopospheztic Mdio pr'opagation~ IEEE 'J:I:ans., Vol.AP-12, No.5, 649-651, September 1964.

251 Radiation fzoom sOUztces iTrrnezosed in comp1'essible pZasma medi.a~ Can. J. Ph .,Vol. 42, 1760-1780, September 1964.

252 L.C. Walters aDd J.R. Wait, Computation of a modified Fzoesnel integzoalarising in the theozoy of diffzoaction by a vareiable strl'een~ NBS Tech. NoteNo. 224, October 14, 1964 (PB 184 119).

253 TheoZ"JI of zoadiation f1'OTfl sOUztces iTrmezosed in anisotropic media~ J. Res.NBS, Vol. 68B, No.3, 119-136, July-September 1964.

254 TheoZ"JI of a s'totted-sphe1'e antenna mer.ed in a compzoessib'te pZasma~

Paret I. Radio Sci. J. Res. NBS, Vol. 68D, No. 10, 1127-1136, Oct. 1964.

255 1'heozoy of a sZotted-sphezoe antenna mersed in a compytessib'Lp pZasma~

~ II~ Radio Sci. J. Res. NBS, Vol. 68D, No. 10, 1137-1143, Oct. 1964.

256 Resonances of a spherical void in a compzoessibZe isotztopic pZasma~ J.Geophys. Res., Vol. 69, No. 19. 4189-4191, October I, 1964.

257 On the thsozoy of zoefZection of electzt(X1lagnetic 7.tXlVes fzoom the interfacebetween a compztessibZe magnetopZasma and a dielecmc~ Radio Sci. J. Res.NBS, Vol. 68D, No. II, 1187-1191. Noveaber 1964.

258 with K.P. Spies, A note on the insuZated loop antenna mersed in a con­du.cting medium~ Radio Sci. J. Res. NBS. Vol. 68D, No. 11, 1249-1240,November 1964.

259 with K.P. Spies, Cha1'acteztistics of the BCU'th-ionosphere waveguide fozo VLFztatIio waves~ NBS Tech. Note No. 300. December 30, 1964 (available frOiDthe Natioual Technical Information Service, Springfield, VA (PB 168 048»

260 E. Babar and J .R. Wait, NicztOLJaVe model techniques to study VLF ztatIiopzoopagation in the Bfl1'th-ionosphere waveguide~ Quasi-()ptics (ed. J. Fox),Polytechnic PI:eas of the Polytechnic I~titute of Brooklyn, pgs. 447-464,1964.

261 Scattezoi.ng of eZectztrnagnetic and e'tectroacoustic waves by a cy tindrica Zobject in a ~ssibZe p1.as1lr1.~ Radio Sci. J. Res., NBS. Vol. 690, No.2,247-256, February 1965.

262 with C.M. Jackaon, CaZcuZations of the bistatic soattezoing czooss sectionof a sphezoe with an impedance boundtrJwy condition~ Radio Sci. J. Res. NBS,Vol. 69D, No.2, 299-315, February 1965.

263 Cavity zoes01lClnDes fOl' a sphezticaZ. eazoth with a c01lCenmc anisot:roopic sheZZ~

J.A.T.P., Vol. 27, 81-89, 1965.

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264 A.G. Jean alld J .R. Wait, The use of e'Leetl'omagne:ie signa'Ls emitted fl'omnue'Lea1' e:cp1.osions to study 'Long-range VLF pl'Opagation~ J. Geophys. Res.,Vol. 70, No.5, 1258-1261, March 1, 1965.

265 Guided r..xwes in the earth-ionosphel'e eavity~ in 'Progl'ess in Radio Seienee~

1960-1963'~ edited by F. Horner, Vol. IV, pp. 98-112, Elsevier Pub. Co.,Amsterdam, 1965.

266 Waves eil'ou'Lating a1'ound a l'igid ey'LindrieaZ obstaeZe in a eompl'essib'Leptasma~ Radio Sci. J. Res. NBS, Vol. 69D. No.4. 567-577. April 1965.

267 Cone61'ning the meehanism of ref7,eetion of e'Leet1'omagnetie wves fl'om aninhomogeneous lossy p'Lasma~ Radio Sci. J. Res. NBS. Vol. 690, No.6.865-869, June 1965.

268 Justifieation fol' neg'Leeting the eom~essibi'Lityof the ionosphel'e in VZFMdio ~opagation~ IEEE Trar.s •• Vol. AP-13. No.3, 480-481, May 1965.

269 Nature of the eleet1'omagnetie fieU refZeeted fl'om a eoastUne~ ElectronicsLetters. Vol. 1. No.3, pg. 65, May 1965.

270 Inf7,uenee of an inhomogeneO'UB ~l'ound on the ~opagation of VLF zaadio tl1aVesin ths earth-ionosphl11'e bJaveguid8, Radio Sci. J. Res. NBS. Vol. 69D. No.7.969-975, July 1965.

271 On radiation of e'Lect1tomagnetic and e'Leet1'oaeoustie wves in a p'Lasma (I)~

Appl. Sci. Res., Sec. B. Vol. 11, 423-431. 1965.

272 Ea1'th-ionosphl!!1'e eavity resonances and the pr'opagation of ELF mdio wves~

Radio Sci. J. Res •• NBS. Vol. 69D. No.8. 1057-1070. August 1965.

273 Modes of proopagation in a bo'U11ded eompr'p.ssible p'Lasma~ Electronics Letters,Vol. 1. 193-194. Sept~ber 1965.

274 with K.P. Spies. In;fLuenee of finite ground conduetivity on the ~opagation

of VLF radio tl1aVes~ Radio Sci. J. Res. NBS. Vol. 69D, No. 10. 1359-1373,October 1965.

275 Propagation of pubes in d1:spel'sive media~ Radio Sci. J. Res. NBS. Vol. 69D.No. 11, 1387-1401, November 1965.

276 COffIfIents on H. Volland's 'Retrmt'ks on Austin's formu.'La' ~ Radio Sci. J. Res.NBS. Vol. 69D. No. 11. 1465-1467, November 1965.

277 On radiation of eleetztomagnetie and e'Leet1'OaCoustie tl1aVes in p'Lasma~ Appl.Sci. Res. Sec. B, Vol. 12, 130-138, 1965.

278 The long tl1aVelength limit in seatte1'ing fl'om a die'Leet1'ie eytinder atoblique incidenee~ can. J. Phys., Vol. 43, 2212-2215, December 1965.

279 with K.P. Spies, Theo.ry of a slotted-sphel'e antenna immel'sed in a eompl'essib'Leptasma, ~t III~ Radio Sci., Vol 1 (New Series), No.1, 21-26, Jan. 1966.

280 E. Bahar and J.R. Wait, Propagation in a modeZ tel'1'esmal wveguide ofnDnunifoZ'lfl height: TheoPy and cpel'iment, Radio Sci. J. Res., NBS. Vol. 69D,No. 11, 1445-1463, November 1965.

281 Some highlights of the URSI Symposium on Eleet1'omagnetie Wave TheoPy~ heldin Delft, The Nethel''LandB~ Sept. 6-11, 1965, Radio Sci. J. Res. NBS, Vol.690, No. 12. 1691-1693, December 1965.

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282 A note on su.rface ~es and (Jl'ound tJal)es~ IEEE Trans •• Vol. AP-l3. No.6.996-997. November 1965.

283 Pzeopagation of elect%'07rr:zgnetic pulses in tta"1'estriaZ tJal)eguides~ IEEE Trans••Vol. AP-13. No.6. 904-918. November 1965.

284 On the theo'JOY of tJal)e p1'Opagation in a bounded car1pzoessibZe pzasma~ Can. J.Phys •• Vol. 44. 293-302. 1966.

285 EZectzoomagnetic Ndiation f'¥'Om an acoustic point soUPce L1ithin a c01l1p1'essibZepZasma bubble~ Can. J. Phys •• Vol. 44. No.3. 467-475. March 1966.

286 2'heoPies of pzoolate spheroidal antenn:zs~ Radio Sci.. Vol. 1 (New Series).No.4. 475-512. April 1966.

287 Limiting behaviour of a thin plasma sheet for a transverse magnetic fieZd~

Electronics Letters. Vol. 2. No.4. pg. 156. April 1966.

288 Cha:ltaeteristics of a sZotted-sphezoe antenna mersed in a cOTTlpzoessibZe pZasma~

from Microwave Behaviour of Ferrimagnetics and PlaS1llB.s. Conference Pub. No.13. Institution of Electrical Engineers. London. September 1965.

289 R.J. ICing. S.W. Maley and J.R. Wait. Grou7dJave p'¥'Opagation aZong thzoee­section rrri:r:ed paths~ Proc. lEE (London). Vol. 113, No.5, 747-751. May 1966.

290 2'.ransverse pzoopagation of lIXnJBguide modes in cyZindzeicaZZy stNtifiedmag7'l8topla8ma~ Radio Sci.. Vol. 1 (New Series). No.6. 641-654, June 1966.

291 D.B. Large aDd J.R. Wait. Cavity 1'esonatozo modes in a cy7.indzoicaZZy stzoati­fied magnetop7.asma# Badio Sci.. Vol. 1 (New Series). No.6. 655-658.June 1965.

292 Radiation from a spherical aperture an,;enna mersed in a compressiblep7.asma~ IEEE Trans •• Vol. AP-14. No.3. 360-368, May 1966.

293 EZectzooJrrz.gnetic propagation in an idea7.iaed earth CPUBt tJal)3guide~ RadioSci•• Vol. 1 (New Series). No.8. 913-924. August 1966.

294 K.P. Spies and J .1.. Wait. On the caZcuZation of the ground wve attenuationfactor at ZOltJ frequencies~ IEEE Trans •• Vol. AP-14. No.4. 515-517. July 1966.

295 Ne1:tJol'k l'epresentation for spherical waves in a compressibZe pZasma~ IEEETrans •• Vol. AP-14, No.4. 517-519. July 1966.

296 Dipole l'esonances of a magnetopZasma coZumn~ Electronics Letters. Vol. 2,No.7. pg. 265, July 1966.

297 A possible metihanism fol' e:cessive mode convesion in the earth-ionosphere~BgUide# Radio Sci•• Vol. 1 (New Series). No.9, 1073-1076. Sept. 1966.

298 Smle highZights of a Symposium on Subsupface Propagation of E7.ectromagneticWaves~ Paris# ApzoiZ 2S-28~ 1966~ Badio Sci•• Vol. 1 (New Series). No.9,1115-1118. September 1966.

299 Influence of a comprBssible pZasma half-space on the impBdance of an eZectricdipoZe~ Proc. IEEE. Vol. 54. No.9. 1193-1194. September 1966.

300 Some factol's concezoning sZectzoQnag7'l8tic tJal)e propagation in the earth'st11'UBt~ Proc. IEEE, Vol. 54. No.8. 1020-1025. August 1966.

301 with E. Bahar. Simulation of curvatu.!'e in a straight model wavBguid8#Electronics Letters. v~l. 2. No. 10. pg. 358. October 1966.

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302 Radiation f1'OTfl a spherical antenna with an anisotropic plasma sheath~ Can.J. Phys., Vol. 44, No. 10,2303-2313, October 1966 (Errata: Can. J. Phys.,Vol. 45, pg. 2228, 1967.)

303 Electrcmagnetic fie'Lds of a dipoZe over an anisor;ropic haZf-space~ Can. J.Phys., Vol. 44, No. 10, 2387-2401, October 1966."

304 Theo:zoy of HF ground wave backscatter from sea ZJaVes~ J. Geophys. Res.,Vol. 71, No. 20. 4839-4842, October 15, 1966.

305 Quasi-static theo:zoy of a cylindrical impedance probe for magnetoplasma~

Radio Sci •• Vol. 1 (New Series), No. 11. 1297-1301, November 1966.

306 Influence of a sub-surface insulating layer on electromagnetic ground wavepropagation~ IEEE Trans •• Vol. AP-14. No.6, 755-759, November 1966.

307 Fields of a horisontal dipole over a stratified anisotropic half-space~

IEEE Trans., Vol. AP-14, No.6, 790-792, November 1966.

308 with K.P. Spies, On the theory of transient wave propagation in a dispersivemedium~ Appl. Sci. Res., Vol. 16. No.6, 455-465. 1966.

309 CcrmtentB on 'The variation of ground constants for the frequency range of30 to 70 MBs' (with author A. Rashid's reply)~ Proc. IEEE, Vol. 55, No.1.85-86, January 1967.

310 nlumination of an inhomogeneous spherical ear'th by an LF plane electro­magnetic wave~ Radio Sci., Vol. 2, (New Series), No. I, 111-118, Jan. 1967.

311 Theo:zoy of a plasma resonance probe in a magnetic fieZd~ J. Appl. Phys.,Vol. 37, No. 13, 4905-4907. December 1966.

312 Fundamental diffi.:mZty in the analysis of cylindrical 'WaVeguides withimpedance walls~ Electronics Letters, Vol. 3. No.2. 87-88, Feb. 1967.

313 Further note on the quasi-static theo:zoy of a cylindrical impedance probefor magnetoplasma~ Radio Sci.. Vol. 2 (New S~ries), No.2, 253-256.February 1967.

314 COtmlent on 'M~f)rowave propagation over mountain-diffraction paths' ~ IEEETrans •• Vol. AP-15, No.2, pg. 321. March 1967.

315 D.B. Large and J.R. Wait. Coupled modes in a spherical resonator with anonunifo~ impedance bou~~ Electronics Letters, Vol. 3, No.5, 221-223,May 1967.

316 Influence of a thin inhomogeneous surface "Layer on electromagnetic ground'WaVe propagation~ Proc. IEEE, Vol. 55. No.4, 568-569, April 1967.

317 Acoustic whispering-gaHery phenomena in circular' cyZinders~ Can. J. Phys.,Vol. 45, No.5, 1861-1869, May 1967.

318 with E.A. Brackett, Bibliography or. waves in plasmas~ ESSA Tech. Rept. No.IER-ITSA-39. April 1967, (PB 175 657).

319 D.B. Large and J .R. Wait, Resonances of the thin-sheU model of the earoth­ionosphere cavity with a dipolar magnetic fieZd~ Radio Sci., Vol. 2, No.7,695-702. July 1967.

320 Radiation from dipoles in an idealized jungle environment~ Radio Sci.,Vol. 2 (New Series), No.7, 747-750. July 1967.

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321

322

323

324

325

326

* 327

328

329

* 330

331

332

333

334

335

336

337

On assumed boundary conditions at an interface be~een a dielectric and acomppessible plasma6 Electronics Letters. Vol. 3. No.7. 317-318. July 1967.

The lJhispering gal lery nature of the earth-ionosphepe bJaVeguide at VLF ~ IEEETrans •• Vol. AP-15, No.4, 580-581. July 1967 (Correc~ion, IEEE Trans ••Vol. AP-16. No.1. 147, 1968).

TeP1'estzoial bJaVeguides~ in 'Electromagnetic Wave Theory'~ Ed. J. Brown,Proc. of Symposium at Delft. Netherlands, September 1965, pg. 195, PergamonPress, Oxford, New York, 1967.

On the theory of shielded suPface waves~ IEEE Trans., Vol. M'IT-15, No.7,410-414. July 1967.

R.J. King. S.W. Maley and J .R. Wait, E:z:perimental and theoretical studiesof propagation of ground lJaves across mi:ced paths~ in 'ElectrCN7lagnetic WaveTheory'~ Ed. J. Brown, Proe. Symposium at Delft, Netherlands, September 1965,pp. 217-224, Pergamon Press, Oxford, New York, 1967.

E. Bahar and J .R. Wait, Propagation in a model terrestrial lJave-guide ofnon-uniform height - Theory and ~periment~ in 'EZectzoomagne#c Wave Theory' ~Ed. J. Brown, Proc. Symposium at Delft, Netherlands, September 1965, pp.211-216, Pergamon Press, Oxford, New York, 1967.

Patteron of a Unear antenna erected over a tapel'ed gl'ound screen~ Can. J.Phys., Vol. 45, No.9, 3091-3101, 1967.

Launching a surface lJave over the earth6 Electronics Letters, Vol. 3, No.9,396-397, September 1967.

with G.A. Schlak, NelJ asymptotic solution fol' the elec~omagnetic fieldsof a dipole over a st1'atified medium6 Electronics Letters. Vol. 3, No.9,421-422, September 1967.

On the theory of l'adiation fl'om a l'aised electric dipole over ar. inhomo­geneous g1'ound pZane~ Radio Sci., Vol. 2 (New Series), No.9. 997-1004.September 1967 ..

Electzoomagnetic lJhisperoing gallery ~odes in a dieZectl'ic l'od~ Radio Sci.,Vol. 2 (New Series). No.9. 1005-1017. September 1967.

Theory of diffl'action by a curved inhomogeneous body~ J. Math. Phys., Vol. 8.No.4, 920-925, 1967.

G.A. Schlak and J.R. Wait, ElectzoCN7lagnetic bJaVe propagation over nonpax'aZZelstPatified conducting medium6 Can. J. Phys., Vol. 45. No. 11. 3697-3720, 1967.

Comment on 'Radiation from linear l'esonant antenna in lJeakly ionized plasma'~

Int. J. Electronics. Vol. 22, No.4, 389. 1967.

CCfmlents on a paper '1. numerical irwestigation of classical appr'oximationsused in VLF pl'opagation' by R.I.. Pappert~ E.E. Gossard~ and I.J. Rot17nuZZel'~

Radio Sci •• Vol. 2 (New Series). No. 11. 1393. November 1967.

David B. Large and J .R. Wait. Resonances of a cyZintboical earth-ionosphel'ecavity model with a dipolar magnetic fieZd~ J. Geophys. Res •• Vol. 72, No. 21.5395-5400, November I, 1967.

K.P. Spies and J .R. Wait, On the calculation of antenna patterns fol' aninhomogeneous spheroicaZ ea1'th~ aadio Sci•• Vol. 2 (New Series), No. II,1361-1378. November 1967.

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338 Planar analogue of the earth-ionosphere waveguide~ Electronics Letters,Vol. 3, No. II, 518-519, November 1967.

339 Applications and limitations of the mode theory of long wave radio propa­gation~ Reprinted from MF, LF, and VLF Radio Propagation Conference, Pub.No. 36, November 1967, pp. 57-62 (Institution of Electrical Engineers,Savoy Place, London).

340 Asymptotic thoery for dipoZe radiation in the presence of ~ lossy sZablying on a conducting half-space~ IEEE Trans., Vol. AP-15, No.3, 645-648,1967.

341 K.P. Spies and J.R. Wait, Phase of the electromagnetic fieUi on the surfaceof an inhonogeneous earth for sky-wave iZ1,umination~ IEEE Trans., Vol. AP-15,No.5, 708-709, September 1967.

342 Y.S. Yeh and J.R. Wait, On the theory of radiation from an annular slot ina compressible plasma~ Radio Sci., Vol. 3 (New Series), No.2, 171-180,February 1968.

343 EZectrcxnagnetics and PZasmas~ Holt, Rinehart and Winston, Inc., New York,NY, 1968.

344 CalcuZated mode conversion at the sunrise boundary in the earth-ionospherewaveguide~ Electronics Letters, Vol. 4, No.7, 131-132, April 5, 1968.

345 G.A. Schlak and J.R. Wait, Attenuation function for propagation over a non­paralZeZ st.r-atified ground~ Can. J. Phys., Vol. 46, No.9, 1135-1136,May 1, 1968.

346 Mode conversion and refraction effects in the earth-ionosphere waveguidefor VZF ~!o waves~ J. Geophys. Res., Space Phys., Vol. 73, No. 11,3537­3548, June 1, 1968.

347 D.B. Large and J .R. Wait, Influence of a radial magnetic fieUi on theresonances of a spherical plasma cavity~ Radio Sci., Vol. 3 (New Series),No.6, 633-637, June 1968.

348 COffITIents on S. W. Lee~ 'Wave propagation in earth-ionosphere waveguide' ~Appl. Sci. Res., Vol. 18, 458-459, March 1968.

349 K.P. Spies and J.R. Wait, Calculated mode conversion coefficients for' agraded height change in the earth-ionosphere waveguide at VLF~ ESSA Res.Labs., Tech. Rept. ElL 77-0Dl, May 1968.

350 CortITIents on 'Propagation of e.1,.f. radio waves to great distances below theanisotropic ionosphere'~ by M. Yamashita~ J.A.T.P., Vol. 29, p. 937,1967,J.A.T.P., Vol. 30, No.6, 1247-1248, 1968.

351 Cc:KmIents on 'An asymptotic field calculation in the penunbra region' ~ IEEETrans., Vol. AP-16, No.2, pg. 260, Karch 1968.

352 with K.P. Spies, On the calculation of mode conversion at a graded heightchange in the earth-ionosphere waveguide at VLF~ Radio Sci., Vol. 3 (NewSeries), No.8, 787-791, August 1968.

353 with D.B. Large, EZectroqnetic fields in a spherical cavity with a con­centric anisotropic shell syste'n~ Can. J. Phys., Vol. 46, No. 13, 1505­1509, July 1, 1968.

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354 D.B. Large and J.R. Wait. Theory of e"Lectromagnetic coup"Ling phenonena inthe earth-ionosphere cavity, J. Geophys. Res., Vol. 73, No. 13, 4335-4362,July 1, 1968.

355 Modal, am"Lysis of i1'1'Bgu"Lazt terrain effects in radio propagation and theray interpretation, Electronics Letters, Vol. 4, No. 17, 367-368,August 23rd, 1968.

356 Note on mode conversion at VLF in the earth-ionosphera lJa1)Bguide, J. Geophys.Res., Space Physics, Vol. 73, No. 17, 5801-5804, September 1, 1968.

357 ~isperi~-gat"Lery modes in a tropospheric "Layer, Electronics Letters, Vol. 4.No. 18, 377-378, September 1968.

358 On the theory of VLF propagation for' a step model, of the nonuniform earth­ionosphere uaveguide, Can. J. Phys., Vol. 46, No. 17, 1979-1983. Sept. 1. 1968.

359 Diffraction and saatterine of the e"Lectranagnetic {J1'ountiJuave by te1'1'ainfeatures, Radio Sci., Vol. 3 (New Serip.s), No. 10, 995-1003, October 1968.

360 EZ.ect1'orragnetic induction in a smaU conducting sphere above a resistiveha"Lf space, Radio Sci., Vol. 3 (New Series), No. 10, 1030-1034, Oct. 1968.

361 with K.P. Spies. VLF mode ca"Lcu"Lations for propagation across a "Land/seaboU7'Jtialty in the earth-ionosphere waveguide, ESSA Tech. Rept. ERL 87-oD-2,September 1968 (40 cents).

362 Scattezoing fran a thin sheet obstacle on a curved surface, Can. J. Phys.,Vol. 46, No. 23, 2601-2605, 1968.

363 with K.P. Spies. On the diffraction by a knife-edge obstac"Le on a conductingearth, Radio Sci •• Vol. 3 (New Series), No. 12, 1179-1181, December 1968.

364 Recent theoretical, advances in the terrestrial, propagation of VLF e"Lectro­magnetic waves, in 'Advances in Electronics and EZ.ectron Physics' Ed. L.Marton, Vol. 25. pp. 145-209, Academic Press, New York and London, 1968.

365 Optimisation of the ground wave over a stratified earth, Electronics Letters,Vol. 4, No. 26. 575-576, 27 December 1968.

366 EZ.ectromagnetic transient response of a sphericaZ conductine shell, ElectronicsLetters, Vol. 4. No. 26, 576-577. 27 December 1968.

367 with K.P. Spies, Radio pzeopagation over a ayZindriaa"L hill inatuding theeffect of a s~ounted obstaatp, IEEE Trans., Vol. AP-16, No.6, 700-705,November 1968.

368 ReJ!ection from subrefractive layers, Electronics Letters, Vol. 5, No.4,64-65. 20 February 1969.

369 with K.P. Spies, Phase shift for totaZ re:tection from a bounded dieZectric~er, Can. J. Phys., Vol. 47, No.7, 826-830, 1969.

370 E"Leot1'c:mlagnetio radiation fran conical, structures, in 'Antenna Theory',Ed. by R.E. Collin and F.J. Zucker, Chapter 12. Part 1, pp. 483-522(McGraw-Hill Book Co., New York, 1969).

371 Eteotromagnetic radiation from spheroidal, structures, in 'Antenna Theory'Edited by R.E. Collin and F.J. Zucker, Part I, Chapter 13, pp. 523-559(McGraw-Hill Book Co., New York. 1969).

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372

373

374

375

376

377

378

379

380

381

382

383

384

385

386

387

* 38~

* 389

390

391

Chazoacteroistios of antennas over ~OS8Y ea:rth, in 'Ant:m;1fl Theory', Ed. byR.E. Collin and F.J. Zucker, Part 2, Chapter 23, pp. 386-437 (McGraw-HillBook Co., New York, 1969).

EZeotromasnetic fieZds of sou:t'ces in ~ossy media, in 'Antenna Theory', Ed.by R.E. Collin and F.J. Zucker, Part 2, Chapter 24, pp. 438-513 (McGraw-HillBook Co., New York, 1969).

Antennas in p~asma, in 'Antenna Theory', Ed. by R.E. Collin and F.J. Zucker,Part 2, Chapter 25, pp. 514-555 CMcGraw-Hil1 Book Co., New York, 1969).

An approach to the mode conversion p'l'ob~em in nol-aA.nifornJ crooustic waveguides,J. Math. Phys., Vol. 10, No.3, 447-451, March 1969.

with K.P. Spies, IntePrial guiding of miozoCNaVes by an elevated tI'oposphezoicZayer, Radio Sci., Vol. 4, No.4, 319-326, April 1969.

RefZection of a pZane tI'ansient etectI'anagnetic wave fran a coZd tosstessPZasma stab, Radio Sci., Vol. 4, No.4, 401-405, April 1969.

On the propagation of a puZse through an antipode, Can. J. Phys., Vol. 47,No. 12, 1327-1330, 1969.

Tztansient zoesponse of the penumbrat C!U1'7'ents for ptane wave diffraction bya cyZinder, Can. J. Phys., Vol. 47, No. 12, 1307-1312, 1969.

On the propagation of acoustic gravity waves over a spherical earth, Pureand Applied Geophys., Vol. 74, 35-44, 1969/111.

with K.P. Spies, EZectI'omagnetic induction in a conducting sphere with aconoentric sheZZ, Radio Sci., Vol. 4, No.6, 557-560, June 1969.

On mode convel'sion of VLF radio waves at a Zand-sea boundtzroy, IEEE Trans.,Vol. AP-17, No.2, 216-220, March 1969.

Image theOl'y of a quasistatic magnetic dipoZe ovezo a dissipative half-space,Electronics Letters, Vol. 5, No 13, 281-282, 26 June 1969.

ObZique zoef1,ection of a plane impulsive electI'anagnetic wave fran a p~asma

half-spcroe, Phys. Fluids, Vol. 12, No.7, pg. 1521, July 1969.

Distortion of elect.romagnetic pulses at totaZZy ref1,ecting layers, IEEETrans., Vol. AP-17, No.3, 385-388, May 1969.

Critical zoef1,eation of pZane waves in the time danain, Proc. IEEE, Vol. 57,No.8, 1453-1454., August 1969.

Distol'tion of ELF puZse after pl'opagation throough an antipode, J. Geophys.Res., Space Phys., Vol. 74, No. 11, 2982-2986, June I, 1969.

Impedance chazoacte1'istias of elect.ric dipoles over a conduating hal!-spaae,Radio Sci., Vol. 4, No. 10, 971-975, October 1969.

Su:t'faae-wave effeats r.ri.th Za:rge antenna ea:rth sC2'eens, Electronics Letters,Vol. 5, No. 22, 552-553, 30th October 1969.

On the optimum l'eceived band:J.1idth for pl'opagated puZsed signats, Proc. IEEE,Vol. 57, No. 10, 1784-1785, October 1969.

with K.P. Spies, On the imase repzoesentation of the quasi-static fieZds ofa line Cur1'ent BOU1'Ce above the ground, Can. J. Phys., Vol. 47, No. 23,2731-2733, 1969.

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392 E2ec~gnetic induction in a soZid conducting Bp~~e encZosed by a thinconducting spheztio:z.Z sheU~ Geophys., Vol. 34, No.5, J53-/59, Oc~. 1969.

393 with K.P. Spies, Quasi-static ~ansient response of a conducting ~eabZesphere~ Geophys., Vol. 34, No.5, 789-792, October 1969.

* 394 with K.P. Spies, FieZds of eZectztic dipoZe on rodiany inhcmogeneous ~ound

s~faoe~ Electronics Letters, Vol. 5, No. 20, 478-479, 2nd October 1969.

395 Concezomng the theory of scatter of HE' radio ~ound waves from peztiodicsea ~es~ ESSA Tech. Rept. ERL 145-oD 3 (Supt. of Documents, U.S. Gov'tPrtg. Office, Wash. DC 20402 ), December 1969. (also see Rp-423)

*396 Transient response of a dipoZe OVer a circular ~ound screen~ IEEE Trans.,Vol. AP-17, No.6, 806-S09, November 1969.

*397 with K.P. Spies, IntegraZ equation ap~oach to the radiation from a verticalantenna 0IJe:t> an inhomogeneous gJ~ound pZane~ Radio Sc:1., Vol. 5, No. I,73-79, January 1970... ,

398 D.C. Chang and J .R. Wait, AppraisaZ of near-fieU soZutions for a He:t>taiandipoZe otoiU' a conducting haZf-space~ Can. J. Phys., Vol. 48, No.5, 737-743,1970.

399 Transient analysis for an electric dipole on a disk ~ound sc~een~ IEE-IERE(India Division) Proc., Vol. 8, No.1, 10-15, January-March 1970.

400 On the input impedance of a Hertzian dipote over a ftat surface~ IEEE Trans.,Vol. AP-18, No.1, 119-121, January 1970.

401 Critical refZection of a p"Lane-lJave carrie?' puZse UJith a Gaussian enveZope~

IEEE Traus., Vol. AP-18, No.1, 145-146. January 1970.

402 R.Gabillard, P. Degauque, and J. R. Wait, Rayonnement des sources eZect1'o­magnetiques pZacees dans des mil-ieux absorbants (Paper presented at AGARD(NATO) Meeting, Paris, June 22-26), 'Electromagnetics of the Sea' (a)-Revue;(b)-Bibliography.

403 Comments on 'On the ?'ej!ection coefficient of a plasma ~ofiZe of exponentiallytapered elec~on density and fued coll-ision f?'equency' ~ IEEE Trans., Vol.AP-lS, No.2, 297-299, March 1970.

404 C7'OUnd-fIXlI)e attenuation function fo?' a Gaussian modutated can'ie?' signaZ~

Electronics Letters, Vol. 6, No.4, 106-107, 19 February 1970.

405 Oblique reflection of an elect7'omagnetic plane wave from a st iated imped­ance surface~ J. Hath. Phys., Vol. 11, No.4, 1437-1440. April 1970.

406 EZectromagnetic fieZdS of a puZsed dipoZe in dissipative and dispe:t>sivemedia~ Radio Sci., Vol. 5, No.4, 733-735, April 1970.

407 On Zauwhing an aaimuthaZ 8U"r'face lJave on a cyl-intiJ'icaZ inrpedawe bountJmoy~

AFCRL Scientific Report No. 49. (Contract No. PRO-Y-70-8~8), Harch 1970(Revised, July 1970). Also Acta Physica Austr., Vol. 32, 122-130. Springer­Verlag, 1970.

408 Factoztiaation method applied to eZectromagnetic lJave propagation in a curved~eguid.e lJith wm-unifoMl tt1az.7.s~ Radio Sci. t Vol. 5, No.7, 1059-1068,July 1970.

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409 with K.P. Spies~ TPansient field cal.oulations fol' an elect1>ic dipole inhomogeneous conducting media~ ESSA Tech. Hemo. ERLTM-OD5. August 1970.(PB 194 024).

410 David C. Chang and James R. Wait. Theory of a ver'tical tubuZa'l' antenna llocated above a conducting half-space~ IEEE Trans •• Vol. AP-1S. No.2.lS2-1SS. March 1970.

* 411 with K.P. Spies~ On the radiation fl'CR1 a vertical dipole 1I1ith an inductivefJJire-grid ground system~ IEEE Trans •• Vol. AP-1S. No.4. 55S-560. July 1970.

412 Ezact fields of a pulsed dipole in a hCRIogeneous cold pl.asma~ J. Appl. Phys ••Vol. 41. No. 7. 2SS0-2SS4~ June 1970.

413 Transient fields far an electric dipole in a dissipative medi~~ Can. J.Phys •• Vol. 4S~ No. 16. lS5S-1S62. 1970.

414 A pitfall in th2 scal.a'l' e'Lect1'CRIagnetic f01'l7'lU'Lation of Kirchhoff theory~

IEEE Trans.~ Vol. E-13. No.2. 9S-99. August 1970.

415 Ana'Lysis of VLF propagation in the earth-ionosphere 1JXZVeguide over a mi:J:ed?,ana/sea path (Part I)~ lEE-IERE (India Division) Proc. ~ Vol. S. No.4.144-151~ September-october 1970; Part II. Vol. S. No. 5. 192-200~ Novem.ber­December 1970.

416 Disto.rtion of pulsed signaZs when the group de'Lay is a non-Zinea:r functionof frequency~ Proc. IEEE. Vol. 5S. No. 8~ 1292-1294. August 1970.

417 with K.P. Spies. TPansient magnetic fieU of a pulsed e'Lecrnc dipole in adissiPative medium~ IEEE Trans •• Vol. AP-18~ No. 5~ 1l4-716~ Sept. 1970.

418 Pzsopagation of e'LeotrCRIagnetic lAUZVes OVer' a smooth multisection CUMJedearth - an e=act theory~ J. Math. Phys •• Vol. 11, No. 9~ 2S51-2860.September 1970.

419 Electrornagnetics of the sea~ Science. Vol. 170 (Meetings Section). 1124-1130,4 December 1970. Full Proceedings of the conference. I Electromagnetics ofthe Sea'. available from the National Technical Information Service (NTIS).Springfield. VA 22151 (AD 716 305)

420 On the reflection of elect1>omagnetic pulses f1'CR1 the ionosphezte~ Radio Sci. ~

Vol. 5, No. 12. 1461-1467. December 1970.

421 Theory of Ground Wave Propagation, Chapter 5 in 'E'Lect1'CRlagnetic Pzsobingin Geoph,ysics'~ Ed. J.R. Wait. pp. 163-207. Golem Press, Boulder, Colo ••1971 (from Rp-243).

422 Ground Wave Pulses, Appendi:c A in 'Electromagnetic Probing in Geophysics' ~Ed. J.R. Wait. pp. 349-360, Golem Press. Boulder. Colo •• 1971.

423 Conce'l'ning the Theory of Scatter of BF Radio Ground Waves f1'CRl PeriodicSea Waves, Appendi:c B in 'Elect1'CRlagnetic Pzoobing in Geophysics' ~ Ed.J.R. Wait. pp. 361-369. Golem Press. Boulder. Colo •• 1971 (See also Rp-395).

424 On the Theory of Radio Propagation Over a Slightly Roughened Curved Earth~

Appendi:c C in 'E2ectroTfD.gnetic Probing in Geophysics'~ Ed. J. R. Wait,pp. 370-381. Golem Press. Boulder. Colo •• 1971.

425 Rayner K. Hasich and James R. Wait. On the calcuZation of an average SUI'faceimpedance 101' a periodic BU1'lace~ OT Research Report. OT/rrSRR1, Oct. 1970,(Also AFCRL Scientific Rept. No. SO. Contract No. PRO-Y-7o-818).Available from NTIS AD 714 090

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426 On BW'face rJaVe pl'Opagation around a bend~ Acta Phys. Austr., Vol. 32,113-121, Springer-Verlag, 1970.

427 PenU1'bation anal.ysis fop pefl.ection fporn tluo-dimensionaZ pezoiodic searuaveB~ Radio Sci., Vol. 6, No.3, 387-391, Karch 1971.

428 Propagation of VLF eZectpomagnetic waves aaross Zand/sea boundaries~

'Electromagnetics of the Sea', AGARD Conference Proceedings No. 77, NATOAdv. Group for Aero. Res. and Dev., November 1970.

429 with K.P. Spies, Ez1al.uation of the sU1'face el.ect1'omagnetic fieZds fo1' abU1'ied magnetic dipol.e SOU1'ce~ AFCRL Sci. Rept. No. 52, Contract No.PRo-Y-71-872, February 1971.

430 Kenneth P. Spies and James R. Wait, On cal.cuZations of the modaZ paroamete1'sof an ideaUaed earth-Ol'Ust wveguide~ ONR Technical Report No.1, OrderNo. NA-QNR-15-71, March 17, 1971 (AD 721 371 - NTIS).

431 with K.P. Spies, VLF mode p1'opagation ac1'OSS an el.evated coastUne~ RadioSci., Vol. 5, Nos. 8-9, 1169-1173, August-September 197J.

432 F. Einaudi and J.R. Wait, AnaZysis of the e:r:citation of the earth-ionosphe1'eruaveguide by a satel.Ute-bo1'ne antenna~ Can. J. Phys., Vol. 49, No.4,447-457, February 1, 1971 (See Rp-436, Pt. II).

433 Editop's Preface and Tabl.e of Contents~ from 'Electromagnetic Probing inGeophysics', the Golem Press, Boulder, Colo., pp. 5-12, 1971.

434 On the theozty of guided el.ect1'omagnetic ruaves On a nonideal. cyUnd:t>icaZstructuzte~ Can. J. Phys., Vol. 49, No.7, 864-869, April 1, 1971.

435 AJo2tay technique fop el.ectromagnetic positionaZ detezmination of a buztiedl'eceiving point~ Electronics Letters, Vol. 7, No.8, 186-187, 22 Apr. 1971.

436 F. Einaudi and J .R. Wait, AnaZysis of the e:rcitation of the earth-ionosphe1'eruaveguide by a sateZUte-bopne antenna - II~ Can. J. Phys., Vol. 49, No. 11,1452-1460, June I, 1971 (See Rp-432, Pt. I).

437 C1'ite1'ia fo1' l.ocating an osciZl.ating magnetic dipole bu1'ied in the earth~

Proc. IEEE (Letters), Vol. 59, No.6, 1033-1035, June 1971.

438 Kenneth P. Spies and James R. Wait, Attenuation cal.cuZations fo1' a 2- l.aye1'ea1'th-C2"U8t ruavBguide fo1' sub-sU1'face el.ectromagnetic p1'opagation~ ONRTechnical Report No.2, NA-ONR-15-71, June 1971 (AD 726 754 - NTIS,Springfield, VA 22151).

439 El.ectramagnetic-pulse propagat{on il1 a sirrrpZe dispe1'sive medium~ ElectronicsLetters, Vol. 7, No. II, 285-286, 3 June 1971.

440 Tzoansient e:r:citation of the ea1'th by a line SOU1'ce of CW'1'ent~ Proc. IEEE(Letters), Vol. 59, No.8, 1287-1288, August 1971.

441 El.ectromagnetic induction technique for l.ocating a burtied sOU1'ce~ IEEE Trans.Geasci. Elect., Vol GE-9, No.2, 95-98, April 1971.

442 with D.A. Hill, Tztansient signaZs [1'am a bU1'ied magnetic dipoZe~ J. Appl.Phys., Vol. 42, No. 10, 3866-3869, September 1971.

443 EZec~etic ruave p1'Opagation in t1w ea1'th's C1'USt~ P~oc. of the Inter­national Symposium on Electromagnetic Wave Theory, Tbilisi, U.S.S.R.,September 9-15, 1971;, Plenary Session 6P, pp. 830-835.

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444 R.N. Grubb and J .R. Wait, In situ mealna'anents of the compl(J.% p1"opagationconstant in rocks for frequencies fl'(XT/ 1 MHa to 10 MHa~ Electronics Letters,Vol. 7, No. 17, 506-507, 26 August 1971.

445 with K.P. Spies, Subsurface electr(XT/agnetic fieUs of a line sOUX'ce on aconducti~ half-space~ Radio Sci., Vol. 6, Nos. 8-9. 781-786, Aug.-Sept. 1971.

446 with K.P. Spies, Electr(XT/agnetic fiel-ds of a smaU loop buried in a strati­fied earth~ IEEE Trans., Vol. AP-19, No.5, 717-718, September 1971.

447 with James A. Fuller, On radio propagation through the earth~ IEEE Trans.,Vol. AP-19, No.6, 796-798,-November 1971.

448 A report on electromagnetics in T.bil-isi~ IEEE Spectrum, 99-100, December 1971.

449 Analytical investigations of eZectromagnetic wave p1"opagation in the earth'scrust~ in 'The Structure and Physical, Properties of the Earth's Crust'~ Ed.J.G. Heacock, pp. 315-324, Geophys. Mono Series, Vol. 14, AGU, Washington,DC, 1971.

450 with K.P. Spies, Note on calcul-ations of p1"opagation parameters for anidealized earth-crust UJaJ)eguide~ in 'The Structure and Physical Propertiesof the Earth's C1'Ust'~ Ed. J.G. Heacack, pp. 325-331, Geophys. Mono Series,Vol. 14, AGU, Washington, DC, 1971.

451 On diffraction by a rippled conve:r surface~ IEEE Trans., Vol. AP-19, No.4,562-565, July 1971.

452 Inf1,uence of earth curvature on the 8ublna'face electromagnetic fieUs of atine source~ Electronic Letters, Vol. 7, No. 23, 697-699, 18 November 1971.

453 R. Gabillard, P. Degauque, and J.R. Wait, Sublna'face el,ectzoomagnetictel,ecommunication - a review~ IEEE Trans., Vol. COK-19, No.6, 1217-1228,December 1971.

454 D.A. Hill and J.R. Wait, Reftection of pulses from a linearl-y varyingionosphere model UJith a vertical magnetic fiel,d~ Radio Sci., Vol. 6, No. 11,933-937, November 1971.

455 D.A. Hill and J .R. Wait, Scatterine of transient plane UJaVe by a periodicgrating~ Radio Sci., Vol. 6, No. 11, 1003-1009, November 1971.

456 D.A. Hill and J .R. Wait, Ref1.ection of el,ectr(XT/agnetic pulses from apeztturbed linear ionosphere model-~ Radio Sci., Vol. 6, No. 12, 1039-1043,December 1971.

457 David A. Hill and James R. Wait, Theory of the el,ectr(XT/agnetic transientresponse of the ionosphere f01' pl-ane wave e:r:citation~ PAGEOPH, Vol. 90,No. VII, 169-186, 1971.

458 David A. Hill and James R. Wait, El-ectr(XT/agnetic scattering of a smaZZspherical, obstacl,e near the ground~ Can. J. Phys., Vol. 50, No.3, 237-243,February 1, 1972.

459 Ground UJaVe pulse p1"opagation~ Revista BrasUeira de Tecnologia, Vol. 2,No.4, 193-199, December 1971.

460 with K.P. Spies, Wave p1"opaga~ion aZong a thin diel,ectric coated UJireburied in the earth~ AFCRL Sci. &ept. No. 56, Contract No. PRo-Y-71-872,January 26, 1972.

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461 ,.nth K.P. Spies. Note on detel'mining eleotrical gzoound constants from themutuaZ impedance of BmaZZ oopZantl1" ZOOPB~ J. Appl. Phys •• Vol. 43. No.3,890-891. Karch 1972.

462 ~ient dipole ~iation in a oonducting medium~ Revista Brasileira deTecnologia, Vol. 3. No.1. 29-37, Karch 1972.

463 The effect of a buried conductor on the subsurface fie lds for Une sourceercitation~ Radio Sci •• Vol. 7, No.5, 587-591. Kay 1972.

464 Theory of lI)(We propagation a7,ong a thin "";'re paraZZel to an interiace~

Radio Sci•• Vol. 7. No.6. 675-679. June 1972.

465 Kenneth P. Spies and James R. Wait, Propagation of electromagnetic lJavesin the earth crust lI)(Weguide from ELF to v.LF~ ONR Technical Report No.3.NA-oNR-15-71. 1 May 1972 (AD 748 331 - NTIS. Springfield. VA 22151).

466 On the theory of transient electztomagnetic sounding over a stratified earth~

Can. J. Phys., Vol. 50. No. II, 1055-1061. 1 June 1972.

467 with R.E. Wilkerson, The sub-BU:l'face magnetic fields produced by a Uneczatl'ent SfAa'oe on a non-f1.at ea:rth~ PAGEOPH. Vol. 95, No. Ill, 150-156.1972.

468 with R.H. Ott. On caZauZating transient electztomagnetic fieZds of a smaZZC'UZ"!'ent-OCU"l'Ying loop over a haI'Iogeneous earth~ PAGEOPH. Vol. 95. No. III.157-162. 1972.

469 Absozoption mode for E.L.F. electztomagnetic propagation in the earth-crustlJaveguide~ Electronics Letters, Vol. 8, No. 11. 292-294. 1 June 1972.

470 with David A. Hill, ~ient magnetic fields produced by a step-function­e:cited loop buried in the Ba:t'th~ Electronics Letters. Vol. 8, No. 11.294-295, 1 June 1~;~.

471 Kenneth P. Spies ~d Jamaa R. Wait. Detel'mining electrical ground constantsfrom the mutuaZ impet:1tmce of sma7.Z copZana:r Zoops~ IEEE Trans •• Vol. AP-20,No.4. 501-502. July 1972.

472 with Kenneth P. Spies, Subsurface electromagnetic fields of a cirauZar loopof azatztent Zocatsd above g1'OUnd~ IEEE Trans •• Vol. AP-20. No.4. 520-522.July 1972.

473 James A. Fuller and James R. Wait. EZectztomagnetic pulse transmission inhomogeneous displ11'sive rock~ IEEE Trans •• Vol. AP-20. No.4. 530-533.July 1972.

474 The Sanguine Concept from the Proc. of the 1972 IEEE Conf. on "Engineeringin the OCean Environment", Newport. RI, Sept. 13-15. 1972. pp. 84-87 andProject Sanguine~ Science, Vol. 178. 272-275, 20 October 1972.

475 with Kenneth P. Spies. Attenuation of eZsqtromagnetic lJaPes in the Ba:t'th­crust lI)(Wsguide fZ'Om ELF to VLF~ Radio Sci. (Letters). Vol. 7. No.6.689-690. June 1972.

476 Electztomagnetic soattl11'ing from a lJire grid pa2'aZZeZ to a plana:r stratifiedmedium~ IEEE Trans. Antennas and Prop •• Vol. AP-20, No.5, 672-675.Sept_ber 1972.

477 James A. Fuller and James R. Wait, EM coupling of coa:riaZ and copZana:r Zoopsin unifo~ dissipative media~ Proc. IEEE (Letters). Vol. 60. No.8. 993­994. Ausust 1972.

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418

419

480

481

482

483

484

485

486

487

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489

490

491

492

493

494

Locating an oscillating magnetic dipole in the earth~ Electronics Letters,Vol. 8, No. 16, 404-406, 10 August 1972.

Theory of electromagnetic reflection fram a parallel grid of dielectriccoatee ~ires buried in the earth~ Can. J. Phys., Vol. 50, No. 18, 2149­2157, September 15, 1972.Erratum and Addenda: can. J. Phy~., Vol. 52, No. 22, ~302-2303, 1974.

with David A. Hill, EZectr(Tflagnetic surface fields produced by a pulse­~cited loop buried in the earth~ J. Appl. Phys., Vol. 43, No. 10, 3988­3991, October 1972.

David A. Hill and James R. Wait, The transient electromagnetic response ofa spherical sheZZ of arbit;rary thickness~ Radio Sci., Vol. 7, No. 10, 931­935, October 1972.

with Kenneth P. Spies, On mode conversion in the ea1'th-cl"UE't lAJaveguide~ ONRTechnical Report No.4, NA-ONR-15-71. 27 October 1972 (AD 756 693 - NTIS,Springfield, VA 22151).

N01"TfIQ,l mode model for electromagnetic ppopagation in the earth Cl"Ust l.I)(ZVe­guide~ IEEE Trans., Vol. AP-20, No.6. 811-813. November 1972.

Electromagnetic uave propagation aZong a buried insuZated 7A1ire~ Can. J.Phys., Vol. SO, No. 20, 2402-2409, 15 October 1972.

with J .A. Fuller, APgand representations of the mutual eZectr(Tflagneticcoupling of loops on a two-Zayer ea1'th~ Geoexplor •• Vol. 10, 221-227, 1972.

with David A. Hill. Trans;~nt electromagnetic fields of a finite cirouZarloop in the presence of a conducting half-space~ J. Appl. Phys., Vol. 43.No. II, 4532-4534, November 1972.

with Kenneth P. Spies, Dipole ~citation of ultral~-frequencyeZec~gnetic

~es in the earth crust uave guide~ JGR. Vol. 77. No. 35, 7118-7120,10 December 1972.

David A. Hill and James R. Wait. CalcuZated pattePn of a vertical antenna7A1ith a finite radial-7A1ire ground system~ Radio Sci., Vol. 8. No. I, 81-86January 1973. (AFCRL Sci. Rept. No. 59, August I, 1972, Contract No. PRO­y-71-872).

James A. Fuller and James R. Wait, High-frequency electromagnetic couplingbetween smaZZ cop"Lanar Zoops over an inhomogeneous ground~ Geophys., Vol.37, No.6, 997-1004, December 1972.

Lorv-angle radiation of an antenna over an irreguZar ground pZane~ AFCRLSci. Rept. No. 60, Contract No. PRo-Y-71-872, November 7, 1972.

with James A. Fuller, Characteristics of an insulated linear antenna in af1,uid-or aize-fiZled borehole~ ONR Technical Report No.5, NA-ONR-lS-71,28 February 1973 (AD 757 977 - NTIS, Springfield, VA 22151).

with Kenneth P. Spies, E1,ect1'(Tflagnetic pl'Opagation in an ideaUaed earthcrust waveguide~ Pt. I~ PAGEOPH, Vol. 101, No. IX, 174-187, 1972.

with Kenneth P. Spies, Electromagnetic propagation in an idea1,ised earthcrust waveguide. ~. II~ PAGEOPH, Vol. 101, No. IX, 188-193, 1972.

On ths theory of r.N%Ve pJ'opagation along a thi.n di.electric coated 7A1ire ina stratified medium~ Int. J. Electronics, Vol. 34, No.2, 265-272, 1973.

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496

497

498

499

500

501

502

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504

505

506

507

508

509

E~eat~netic indUction in conductive 8he~~8~ Can. J. Phys •• Vol.51.No.2. 209-218. 1973.

Resistance of earth e~ectrodesl Electronics Letters. Vol. 9. No.4. 90-91.22 February 1973.

Randolph H. Ott and James R. Wait. Firost o1'dero effects of te1'1'Qin on theroadiation patte1'n of a non-di1'ectiona~ LF beacon~ Archiv fur ElecktronikuDd Ubertragungstechnik (AEU) - Electronics and Communication. Band 27.Heft 3. 106-110. 1973.

James A. Fuller aDd James R. Wait, Mutua~ e~ect1'anat,Jneticcoupling ofco~l loops in a bo1'eholel Radio Sci•• Vol. 8. No.5. 453-457. Hay 1973.

Remote sensing of obse1'Vable8 in geophysics l extracted from Proceedings atNSF Workshop on Future Directions of Electromagnetics of Continuous Media.20-25. Williamsburg.VA. December 1972.

COf/fT/ents on a Zette1' by .T.Ga~ejs~ 'On the Fie~d Reproesentations in the ELFRange' and on the zoepZy by .T.R• .TohleI' and R.L. LMs~ JGR. Vol. 78. No. 18.3627. June 20. 1973.

David A. Hill and James R. Wait. Radiation patteron of a ZOLJ-froequency beaconantenna in the pI'e8ence of a semi-elliptic teM"ain i1'1'egu7.a:rity~ Archiv furElektronik und Ubertragungstecbnik (AEU) - Electronics and Communication.Band 27. 239-243, 1973.

with James A. Fuller. Tmnsmission line theoPy fo1' an insuZated ZineaPantenna in a fluid-o:r ail'-fiZ ~d boI'ehole~ Appl. Phys. 1. 311-316.June 1973.

David A. Hill and James R. Wait. Effect of edge reflections on the pBf'­fOZ'fflance of antenna groound 8C1'eens I IEEE Trans.. AP-21. No.2. 230-231.March 1973.

David A. Hill and James R. Wait. E~ect1'c:mtagnetic t1'ansient 1'e8ponse of asphe1'ical conducting 8hell overo a conducting haZf-8Pace~ Int. J. Electronics.Vol. 34. No.6. 795-805. 1973.

David A. Hill and JRes R. Wait. Subsu:rface elect1'arragnetic fields of agJ'OUnded cabZe of finite Zength~ Can. J. Phys •• VoL 51. No. 14. 1534-1540.1973.

with K.P. Spies. L<N-f1'equency intpedance of a ci:reularo Zoop overo a conductingg~nd~ Electronics Letters. Vol. 9. No. 15. 346-348. 26th July 1973.Errata. ElectroD:-:Lett •• Vol. 10. pg. 248. 1974.

David A. Hill and James R. Wait. Radiation patte1'n of a Zow-fI'equency beaconantenna Zocated on a semi-elliptic t~in i1'1'egularoitYI AEU. Archiv furEkektronik und Ubertragungstechn1k - Electronics and Communication. Vol. 27.No. 7-8. 293-296. July-August 1973.

David A. Hill aDd Juaes R. Wait. E~ect1'a'fIagnetic t1'ansient 1'esponse of asmall IJizoe loop buzoied in a ha'flogeneous conducting earoth~ PAGEOPH. Vol.105. 869-878. 1973/IV.

with Kenneth P. Spies. SOU1'ce-to-ob8e~el'distance dependence of the sU1'faceiJrtpt4dance and IJave ti tt fo1' a line-sou:rce e:rci ted wo- ~e1' ea1'thl ONiTechnical Report No.6. NA-oNR-15-71. 4 September 1973. (AD 767 089 - NTIS.Spr1nlfield. VA 22151).

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510 David A. Hill and James R. Wait, The eZectl'omagnetic l'esponse of a bzatiedsp'hsl'e for bwoied-dipoZe e=citation~ Radio Science, Vol. 8, Nos. 8-9,813-818, August-September 1973.

511 with Kenneth P. Spies, Subsurface eZectl'omagnetic fieZds of a Zine SOU1'Ceon a ~o-Zaye1' ea1'th~ Radio Science, Vol. 8, Nos. 8-9, 805-810, August­September 1973.

512 R.H. Ott and J.R. Wait, A fil'st appl'oach to the pl'opagation of ZatemZtJaVes in an inhomogeneous jungZe~ OT Technical Memorandum 73-154, Nov. 1973.

513 David A. Hill and James R. Wait, EZect1'omagnetic scatte1'ing of an a1'bitl'al'ypZane ",ave by ~o non-intel'secting perpendicuZa1' "'il'e gl'ids~ Can. J. Phys.,Vol. 52, No.3, 227-237, 1974.

514 R.B. Ott and J.R. Wait, EXcitation mechanisms fol' tl'a~1i8Sion th1'oughf01'est-cove1'ed and vegetated media~ Technical Report No. ACC-ACQ-8-73,November 1973. NTIS Accession No. AD 771-915,

515 with David A. Hill, Ezcitation of a homogeneous conductive cyUndel' offinite Zength by a pl'esc1'ibed axiaZ CU1'1'ent dist1'ibution~ Radio Science,Vol. 8, No. 12, 1169-1176, Dec~ber 1973.

516 David A. Hill and James R. Wait, Perturbation of magnetic dipoZe fieZdsby a pel'fectZy conducting pl'oZate sphel'oid~ Radio Science, Vol. 9, No. I,71-73, January 1974.

517 State of k~Zeage of anaZyticaZ techniques fol' th1'u-th~-ea1'th eZectl'omag­netic t,,)CZVe pl'obZems l'eZevant to mine l'escu.~ Reprinted from Proceedingsof "Tbru-the-Earth Electromagnetics Workshop", pp. 9-14, Final Report onU.S. Bureau of Mine Contract No. G 133023, 31 December 1973. (PB 231 154)

518 with Kenneth P. Spies, Range dependence :;f the surface impedance and t,,)CZVetiZt f01' a Zine-sOUJIce e:cited ~o-Zayer earth~ IEEE Trans. on Antennasand Propagation, Vol. AP-21, No.6, pp. 905-907, November 1973.

519 Ray J. King and James R. Wait, EZectl'omagnetic gl'oundztJave pl'opagation theoroyand e:r:p~ent~ Proceedings of "Teoria Mathematica de11'electromagnetismo",Roma, 19/22, Febbraro, 1974, Published in the series 'Symposium Mathematica',No. U, Academic Press.

520 David A. Bill and James R. Wai t, SubBW'face eZect1'ic fieZds of a gl'oundedcabZe of finite Zength f01' both frequency and time domain~ Pure and AppliedGeophysics (pAGEOPB), Vol. Ill, 2324-2332, 1973/X.

521 em t'hs eZect1'omagnetic puZse l'esponse of a dipoZe ovel' a pZane BW'face~ Can.J. Phys., Vol. 52, No.2, 193-196, 1974.

522 ReviBlJ of gl'ound l.l)tZve pl'Opagation ovel' non-uniform surface~ Paper reprintedfrom Conference Pre-Print No. 144, AGARD Heeting on Electromagnetic WavePropagation Involving Irregular Surfaces and Inhomogeneous media, 22-1-22-20,Held in the Bague, Netherlands, 25-29 March 1974.

523 David A. Bill and James R. Wait, Diffusion of eZect1'omagnetic puZses intothe ea1'th fl'Om a Zine sOU1'ce~ IEEE Trans. AP, Vol. 22, No. I, 145-146, 1974.

524 D. C. Chang and J. R. Wai t, E%tl'e",eZy ZOLJ fl'equency (ELF) pl'Opagation aZong ah01'iaontaZ we located above 01' bu1'ied in the ea1'th~ IEEE Trans. on COIIIIlUni­cations, Special Issue (ed. J.R. Wait), Vol. COM-22, No.4, 421-426, April1974.

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525 Excerpts from IEEE Special Issue edited by J.R. Wait: Historical Back­gJ'ownd CD1d Intl'oduction to the Special Issue on E:r:tl'emeZy Low Froequency(ELF) Conrrrunication; Dedication to Janis Galejs; Guide to Papezos in ThisIssue; The Guest Editol"s l?ast Intel'est -!on E:r:tl'emely Low F1'equency (ELF)EZectzoomagnetics, IEEE Trans. on Communications (Special Issue). Vol.COM-22. No.4. April 1974.

526 J.T. deBettencourt. D. Davidson and J.R. Wait. LEEE Guide fol' radiomethods ofmeas'Ul'ing earth conductivity, IEEE Std 356-1974. pp. 1-30.Feb. 20. 1974. (available from the Institute of Electrical & ElectronicsEngineers, Inc •• 345 East 47 Street. New York. NY 10017), also publishedin IEEE Trans •• AP-22. No.2. 373-400. March 1974.

527 S.F. Mahmoud and J.R. Wait. Theozoy oflJave propagation along a thin lJireinside a rectangular lJaveguide, Radio Science. Vol. 9. No.3. 417-420.March 1974.

528 COtmIents on "The Use of the LOl'entz Pecip1'ocity Theol'em to Pzoove Equalityof the Open Cil'cuit Voltages of a Receiving DipoZe and a Monopole", IEEETrans •• EHC-16. No.1. p. 51. February 1974.

529 Canments on "Shielding Pel'f01'l1lance of Metallic CyZindel's" and COtmIents byC. W. Harrison, Jr., and Reply to D. Schiebel', IEEE Trans •• EHC-16. No.1.p. 52. February 1974.

530 The01'Y of the terzoestrial propagation of VLF electromagnetic lJaves (Thisis an up-dated and revised version of the author's chapter in Vol. 25 ofAdvances in Electronics and Electronic Physics. ed. by L. Marton. AcademicPress, Inc •• 1968). Also used as Lecture Notes prepared for NATO AdvancedStudy Institute, Spatind, Norway. April 17-27, 1974.

531 with R.H. Ott and T. Telfer (editors). Pzooceedings of Workshop on RadioSystems in Forested ana/ol' Vegetated Environments, Advanced ConceptsOffice, U.S. Army Communications Command. Fort Huachuca. AZ 85613.February 1974. Available from NTIS Accession No. AD 780-716.

532 Applied Mized-Path Theories, Presented at the Workshop on Radio Systemsin Forested and/or Vegetated Environments. U.S. Army CommunicationsCommand, Fort Huachuca. AZ, 6-9 November 1973.

533 COl'l'ection to lettezo "On Electl'omagnetic Induction in Elongated areBodies'~ Geophysics, v. 38, no. 5, p. S8t-S85, Geophysics. p. 235.April 1974.

534 D.A. Hill and J.R. Wait. Pezoturbation of magnetic dipole field by afinitely conducting circular cylindezo, Rivista Italiana di Geof1sica.Vol. XXII. No. 5/6, pp. 421-424, 1973.

535 S.F. Mahmoud and J .R. Wait. Guided electl'(71lagnetic lJaVes in a curvedl'ectangular mine tunnel, Radio Science, Vol. 9, No.5, 567-572, May 1974.

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536 D.A. Hill and J .R. Wait, EZectl'omagl;etic l'esponse of a conducting cyUndel'of finite Length~ Geofisica Internacional, Vol. 12, No.4, 245-266,October 1972.

537 h wave-optical field solution fol' a line source ovel' nonpaPallelstl'atified ea1'th~ Radio Science, Vol. 9, No.6, 593-597, June 1974.

538 Propagation undel' the eaPtr.'s surface~ Conference Publication No. 114,URSI Symposium on Electromagnetic Wave Theory, pp. 80-85, The Institutionof Electrical Engineers, London, July 1974.

539 S.F. Mahmoud and J.R. Wait, EM waVe pl'opagation in l'ectanguLaP minetunneLs containing an axial wil'e~ Conference Publication No. 114,URSI Symposium on Electromagnetic Wave Theory, pp. 89-91, TheInstitution of Electrical Engineers, London, July 1974.

540 with D.A. Hill, Guided eLec~l'omagnetic waves aLong an axial. conductOl'in a cil'culaP tunneZ~ Succinct paper, IEEE Transactions on Antennasand Propagation, Vol. AP-22, No.4, 627-630, July 1974.

541 Recent anaLytical investigations of eL~ctromagnetic ground W~Je propa­gation over inhomogeneous eaPth modeLs~ Proceedings IEEE, Vol. 62,No.8, 1061-1072, August 1974.

542 with D.A. Hill, Coa=ial and bifilar modes on a transmission Zir~ in acircuZar tunnel~ Applied Physics, Vol. 4, No.4, 307-312, September 1974.

543 Theory of transmission of eLa~tromagneticwaVes aLong muLti-conductorLines in the proximity of walZs of mine tunnels~ Proceedings of theInternational Colloquium on Leaky Feeder Communication Systems, 8-10April 1974, Guildford Surrey, England (pp. 97-107). also published inThe Radio and Electronic Engineer, Vol. 45, No.5, pp. 229-232, May 1975.

544 Theory of the teroztestriaL propagat{:Jn of VLF electromagnetic lJaves~

J.A. Holtet (ed.), ELF/VLF Radio Wave Propagation, pp. 129-147, D.Reidel Publishing Co., Dordracht-Ho:ladd, 1974.

545 D.A. Hill and J .R. Wait, E:t:cit,::r;ior: of monofilaP and bifilaP modes ona transmission line in a circuLaP tunnel~ J. Appl. Phys., Vol. 45, No.8,3402-3406, August 1974.

546 Theory for the excitation ar~ propagation of electromagnetic waves guidedby an elevated refractive ir.iex d~ 'ontinuity in the troposphere~ Can. J.Phys., Vol. 52, No. 19,1852-1861, :974.

547 The enigma of the continuous epectrwn in acoustic gravity wave propagationin an isothermal atmosphere~ J. Geophys. Res., Vol. 79, No. 33, 5055-5056,November 20, 1974.

548 D.A. Hill and J.R. Wait, Gap exc~tation of an a-~aZ conductor in acircuZar tunnel~ Journal of Applied Physics, Vol. 45, No. II, 4774-4777,November 1974.

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549 with D.A. Hill, EZect~omagnetic fieZds of a dieZect~c coated coaxiaZ'cabZe ~th an interTUpted shieZd, OT Technical Memorandum 75-192,January 1975.

550 s.r. Mahmoud and J.R. Wait, Geomet~icaZ optical app~oach foreZect~omagnetic~ave p~opagation in ~ectanguZ~ mine tunneZs, RadioScience, Vol. 9, No. 12, 1147-1158, December 1974.

551 Comments or. "The ho~zontaZ ~i~e antenna ove~ a conducting o~ dieZect~c

haZf-space: C~ent and admittance", Radio Science, Vol. 9, No. 12,p. 1165, December 1974.

552 with L. Thrane and R.J. King, The t~ansient eZectric fieZd ~esponse of an~ay of p~aZZeZ ~i~es on the e~th's s~face, IEEE Trans. on Antennasand Propagation, Vol. AP-23, No.2, 261-264, March 1975.

553 with D.A. Hill and S.r. Mahmoud, AnaZyticaZ studies of eZect~~agnetic

~aves on mono-fU~ and bi-fil~ stl'uctu~es in ci~cula~ and ~ectanguZ~

mine tunneZs, Proceedings of International Conference "Radio: Roads,Tunnels and Mines", Liege (Belgium), 1-5 April 1974.

554 with K.P. Spies, MaiJneto-TeZl~ic fields fo~ a segmented ove~b~deY;,

ONR Tech. Rept. No.7, 31 May 1974 and J. Geomag. Geoelect., 26, 449-458, 1974. (Tech. Rept. available from h~IS, AD 781740) --

555 Comments or. 'Trar~missior. 0: ~i~c~lar poLzrized ~aves be~een eZe~ated

antenna', IEEE Trans. on Broadcastin~, Vol. BC-21, No.2, pg. 39-40,June 1975.

556 with D.A. Hill, Proopagation along a b~aided coaziaZ cabZe ir. a ci~~l(II'

tunneZ, IEEE Trans. on MTT, Vol. MTT-23, No.5, 401-405, May 1975.

557 No~e on excitation of the eZe~t~o~~3netic e~th-c~st waveguide, IEEETrans. AP, Succinct Papers, 425-428, May 1975.

558 with D.A. Hill, On the eZect~vmagr.ctic fieZd of a dieZect~ic coatedcoa:r:iaZ cabZe ~th an inte~u!'ted shieZd, IEEE Trans. on Ant. andProp., Vol. AP-23 , No.4, 470-479, July 1975.

559 with D.A. Hill, Theo~y of the transmission of eZect~omagneti~~e8

down a mine hoist, Radio Science, Vol. 10, No.6, 625-632, June 1975.

560 Radio-~e p~opagation - G~OUI~ st~ct~e and topog~aphy, Contribution<;0 the "Review of Radio Science, 1972-74" prepared for 18th GeneralAssembly of the International Union of Radio Science (URSI), held inLima, Peru, 11-19 August 1975.

561 Radio Wave Proopagation, by t.. Piauenard (Ne~ Yo~k: John WHey & Sons,1974, 340 pages), Book ReView, IEEE Circuits & Systems, Vol. 7, No.7,June 1975. Also published in the IEEE Proceedings, Vol. 63, No. 10,p. 1534, October 1975.

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562 Theory of EM wve propagation through tunnels, Radio Science, Vol. 10,No.7, 753-759, July 1975.

563 with D.A. Hill, EleC1:rcxnagnetic fieZds of a dielec1:ric coated coarialcable ~th an interrupted shield - Quasi-static approach, IEEE Trans.on Antennas and Prop., 6S0-682, September 1975.

564 D.A. Hill and J.R. Wait, Elec1:rcxnagnetic fields of a coa:cial cable withan inte~ted shield Zoca~ed in a circular tunnel, Journal of AppliedPhysics, Vol. 46, No. 10; 4352-4356, October 1975.

565 Some corrections to published formulas on biconical st;ructures irmIersedin sphericalZy s1:ratified media, IEEE Trans. on Antennas and Prop.,Vol. AP-23, No.6, p. 864, November 1975.

566 D.A. Hill and J.R. Wait, Coupling betlJeen a radiating coaxial cable anda dipole antenna, IEEE Trans. on Communications, Vol. 23, 1354-1357,November 1975 (Correction, Vol. 24, p. 479, April 1976).

567 W.J. Hughes and J.R. Wait, Effective wave tilt and surface impedanceover a laterally inhcxnogeneous tlJo-layer earth, Radio Science, Vol. 10,No. II, 1001-100S, November 1975.

568 Scattering frCTf1 a break in the shield of a braided coaxial cable - Theory,Archiv fur Elektronik und Ubertragungstechnik, Electronics and Communi­cation (AEU), Band 29, 467-473, 1975.

569 W.J. Hughes and J.R. Wait, Elec1:ramagnetic induction ~ver a tlJo-layerearth ~th a sinusoidal overburden, Pure and Applied Geophysics (PAGEOPH),Vol. 113, 591-599, 1975.

570 J.A. Fuller and J.R. Wait, A Pulsed Dipole in the Earth, 'Topics inApplied Physics' (Editor L.B. ~elsen), Vol. 10, C~lpter 5, 238-270,(Springer-Verlag), 1976.

571 Scxne basic elec1:rcxnagnetic aspects of ULP field variations in theatmosphere, Pure and Applied Geophysics (PAGEOPH), Vol. 114, 15-28,1976.

572 C.H. Stoyer and J.R. Wait, Analysis of source location errore for amagnetic dipole buried in a laterally inhcxnogeneous oonducting earth,Pure and Applied Geophysics (PAGEOPH), Vol. 114, 39-51, 1976.

573 S.F. Mahmoud and J .R. Wait, Calculated channel characteristics of abraided coaxial oalle in a mine tunnel, IEEE Trans. on Communications,Vol. COM-24, No.1, 82-87, January 1976.

574 Note on the theory of transmission of electrcxnagnetic lJatIes in a coalseam, Vol. II, No.4, 263-265, April 1976.

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575 D.A. Hill and J.R. Wait, Calculated transmission loss f~ a leaky feedercOTmlUnication system in a circ:ulcrr tunnel~ Radio Science. Vol. 11. No.4,315-321. April 1976.

576 D.A. Hill and J.R. Wait. Scattering from a break in the shield of abraided coa:r:iaZ cable - Nwnerica.l results" AEU (Archiv fur Elektronikund Ubertragungstechnik). Vol. 30. No.3, 117-121. 1976.

577 J.R. Wait and D.C. Chang, Theory of electromagnetic scattering from alayered mediwn ~th a laterally varying substrate~ Radio Science, Vol.II, No.3. 221-229. March 1976.

578 D.A. Hill and J .R. Wait. Electromagnetic scattering of an arbitraryplane wave by a uire mesh ~ith bonded junctions~ Canadian Journal ofPhysics. Vol. 54. No.4. 353-361. 1976.

579 Electromagnetic field analysis fo1' a coa:cia.l cable ~ith periodic slots~

IEEE Trans. on Electromagnetic Compatibility, Vol. EMC-I9. No. I, 7-13,February 1977.

580 Electromagnetic the01'y of the loosely braided coa:r:ial cable~ Part I~

IEEE Trans. Microwave Theory and Techniques, Vol. MTT-24. No.9,547-553, September 1976.

581 D.A. Rill and J.R. Wait, Propagation along a braided coa:r:ial cablelocated close to a tunnel wall~ IEEE Trans. on Microwave Theory andTech., Vol. MTT-24, No.7. 476-480. July 1976.

582 J .R. Wait and D.A. Hill, Low-frE.quency radio transmission in a circuZa:rotunneZ containing a ~re conducto1' near the wall~ Electronics Letters,Vol. 12, No. 13, 346-347, 24th June 1976.

583 J .R. Wait and D.A. Hill, Electromacnetic scattering by ~o perpendic:uZa:ro~re grids over a conducting haZf-sPaCe~ Radio Science. Vol. 11. Nos. 8,9,725-730. August-September 1976.

584 Long-wave behaviour of the Beverage wave aerial~ Electronics Letters.Vol. 12, No. 14, 358-359. 8th July 1976.

585 The~ies of scattering from ~e grid and mesh st.ructU1'e8~ Summary ofinvited review paper for Nat'l Conf. on EM Scattering, June 15-18. 1976.University of Illinois at Chicago Circle, Chicago, IL.

586 D.A. Hill and J .R. Wait. AnaZysis of electromagnetic scattering from~re~esh Btructures~ Electronics Letters, Vol. 12. No. 17. 427-428,19th August 1976.

587 S.H. Cho and J.R. Wait, Transmission of creeping-~avemodes around aperiodically modulated impedance cyUnder~ Canadian Journal of Physics,Vol. 54. No. 18. 1839-1849, 1976.

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588 Lette1' to the Edito1' 1'ega1'ding "SanguineISeafCl.1'e1''' by G1'ove1' EHis(Sie1'1'a CZub BuZZetin~ ApPil 1976), Sierra Club Bulletin. Vol. 61.No. 10. November/December 1976.

589 D.C. Chang and J.R. Wait, An analysis of a 1'esonant loop as anelectromagnetic senso.r of cval seam thickness~ From Pr~ceedings ofURSI Conference on Propagation in Non-Ionized Media. LaBaule. France.28 April-6 May 1977. pp. 141-146 (Radio Sci., Vol. 13. p. 399. 1978).

590 with D.A. Hill, Sub-supface EM t1'ansmission in tunneZ 1.IXZVeguides~ FromProceedings of DRSI Conference on Propagation in Non-Ionized Media,LaBaule, France. 28 April-6 May 1977. pp. 125-128.

591 S.H. Cho and J.R. Wait, AnaZytical study of whispe1'ing gaZZe1'y t1'ans­mission in a non-unifo~ t1'oposphe1'e~ From Proceedings of DRSI Conferenceon Propagation in Non-Ionized Media. LaBaule. France. 28 April-6 May 1977.pp. 37-41.

592 The eccent1'icaZZy located wi1'e in a cyZind1'ical cavity in a conductingmedium and the limit of a planar' bou:ndt:Jzty~ Radio Science. Vol. 11,No. II, 897-899, November 1976.

593 with D.A. Hill, Impedance of an elect1'ic dipole located in a cylindricalcavity in a dissipative medium, Applied Physics, Vol. 11. No.4,351-356, December 1976.

594 R. J. King and J .R. Wait, Elect1'omagn~tic {J1'0undJJave p7'opagation-theo1'yand ezp61'iment, Ist1tuto Nazionale di Alta Mathematica. SymposiaKathematica, Vol. XVIII, Academic Press, 1976.

595 D.A. Hill and J.R. Wait. Analysis of Mdio fl'equency t1'ansmission alonga t1'oZZey wi1'e in a mine tunnel, IEEE Trans. on Electromagnetic Com­patibility, Vol. EMC-18, No.4, 170-174, November 1976.

596 D.A. Hill and J.R. Wait. Electl'omagnetic sUl'f~e ~e pl'opagation eve1'a bonded wi1'e mesh, IEEE Trans. on Electromagnetic Compatibility. Vol.EMC-19, No. I, 2-7, February 1977.

597 with D.A. Hill, Attenuation on a sUl'face 1.IXZVe G-line suspended withina ci1'culCl.1' tu.nneZ, Journal of Applied Physics, Vol. 47, No. 12, pp.5472-5473, December 1976.

598 Lette1' to the Edito1' conceroni1"ofJ the pape1' "Pot.Je1' dissipated in dipoleezcitation of an eZectromagnetic field above a conducting half-space"by L. V. Kochmanova and V.P. Pe1'ov, Radiotechnika i Electronika, Vol.21, No. II, p. 2451, 1976.

599 C.H. Stoyer and J.R. Wait, Resistivity p7'obing of an "e:r:ponential"Ba1'th with a homogeneous ovel'buzoder., Geoexploration. Vol. 15, 11-18.1977 •

600 Letttn' to the Edito1' on the F01't1'an "Sphel'e" P1'ogroam fol' G1'ound WaveCaZcuzations~ Microwave Journal. Vol. 19. No.8. p. 68. August 1976.

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601 Analysis for the electromagnetic fieZds 0; a dipole located within ametal-cased borehole, Journal of Applied Physics, Vol. 48, No.3,1009-1012, March 1977.

602 with D.A. Hill, Inj1uence of spatial dispersion of the shield transferimpedance of a braided coaxic! cable, IEEE Trans. on Microwave Theoryand Techniques, Vol. 25, 72-74, January 1977.

603 with D.A. Hill, Radio freque~cy ~ansmission via a trolley wire in atunnel with a rail return, IEEE Trans. on Antenn~s and Propagat.,Vol. AP-2S, No.2, 248-253, March 1977.

604 EXcitation of a coaxial cable or wire conductor located over the groundby a dipole radiator, Archiv fur Elektronik und Ubertragungstechnik (AEU)Vol. 31, 121-127, 1977. (Errata, AEU, Vol. 31, p. 230,1977).

605 D.A. Hill and J.R. Wait, Cou~Zing hetween a dipole antenna and ar. infinitecable over an ideal ground plane, Radio Science, Vol. 12, No.2, 231-238,March/April 1977.

606 with D.A. Hill, Slectromag1;et::c s;lielding of sources within a metal­cased bore hole, IEtE Transactions on Geoscience Electronics, Vol.GE-lS, No.2, 108-112, April 1977.

607 Propagation of ELF electromagnetic waves and Project Sanguine/Seafarer,IEEE Journal of Oceanic Engineering, Vol. OE-2, No.2, 161-172, April 1977.

608 On the theory of scattering fY'OlTI a periodicaUy loaded wire grid, IEEETrans. Antennas and Propagation, Vol. AP-2S, No.3, 409-413, May 1977.

609 with D.A. Hill, Electromagnetic fields of a dipole source in a ciY'cular>tunnel containing a surface wave line, Int. J. Electronics, Vol. 42,No.4, 377-391, 1977.

610 Low-frequency behaviour of th~ Be~erage aeriaZ, Electronics Letters,Vol. 13, No.9, p. 272, 28th April 1977.

611 Quasi-static limit foY' the pY'opagating mode along a thin wire in acircular tunnel, IEEE Transactions on Antennas and Propagation,Vol. 25, No.3, 441-443, May 1977.

612 D.A. Hill and J.R. Wait, Ef~CCt 0: a lossy jacket or. the external fieldsof a coaxial cable with an inteY'rupted shield, IEEE Trans. on Antennasand Propagation, Vol. AP-25, No.5, p. 726, September 1977.

613 D.A. Hill and J.R. Wait, Analysis of radio frequency transmission in asemicircular mine tunnel coni-aining two axial conductors, IEEE Trans.on Communications, Vol. CQM-25 , No.9, 1046-1050, September 1977.

614 Reply to Professor R.W.P. Kin; his comments on the exponential curY'entmodes on an infinitely long wir~ o~er the earth, Proceedings of the IEEE,Vol. 65, No.7, p. 1062, July 1977.

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615 R.K. Rosich and J.R. Wait, A generQZ perturbatio~ eoZut~on fer re;le~tior.

frem two-dimensional periodic surfaces, Radio Science, Vol. 12, ~o. 5,719-729, September/October 1977.

616 MUltiple scattering between a buried line conductor and the earth'ssurface, Geophysics, Vol. 42, ~o. 7, 1470-1472, December 1977.

617 S.H. Cho and J.R. Wait, EZect"pema;J'Yi.etic l.oXZve pl'opa?ation in a lateralZynonunifo~ troposphel'e, Radio Science, Vol. 13, Xo. 2, March/April 1978.(The complete text of this article is available on microfiche. Orderfrom the American Geophysical ~nion, 1909 K Street K.~., Washington, DC20006. Document 785-001; $1.00).

618 Seme earth l'esistiJity problens involving buried cables, Colorado Schoolof Mines Quarterly, Vol. 73, Xo. 1, 1-21, January 1978.

619 Excitation of ar. enserr.bZe 0,.1 naraZi.eZ cables by ar; extel'na: ii:;;:Jlc overa layered groune, AEU (Arcniv' fur Elektronik unc Ubertragungst~chnik),Vol. 31, No. 12, 489-493, 1977.

620 D.A. Hill and J.R. Wait, Ele~t1'emarnetic scattering ;l'Dm ar. ur~or3ed

l'ecta~uZar wire mesh located neQ1' the air-ground inte~face, IEEETrans. on Electromagnetic Compatibility, Vol. EMC-19, ~o. 4, 402-406,November 1977.

621 Analysis of alternating current excitation 0.<:' a l...'il'e rope bI- Q tOl'oidalcoil, D.A. Hill and J.R. Wait, J. Appl. Phys., Vol. 48, No. 12, 4893­4897, December 1977.

622 EXcitation of CU1'1'ents on a bvyied in~~lated cable, J. Appl. Phys.,Vol. 49, No.2, 876-880, February 1978.

623 R.K. Arora and J.R. Wait, Re.f'raction theories of'radiowave propagationthrough the troposphere - A l'evia~J Summary published in Radio Science,Vol. 13, No.3, 599-600, May-June :978. Full paper available as Micro­fiche Document 785-002 available from American r.eophysical Union, 1909K St. N.W., Washington, DC 20006.

624 R.G. Geyer and J.R. Wait, Electrar:a;:metic fields of a vertical magneticdipole above a laterally inhemogeneous thin layer of conductive overburden,Pure and Applied Geophysics, Vol. 116, 181-197, 1978.

625 Comments on "Measured Field 0.<:' a Directional' Antenna SUUnerged in a Lake"IEEE Trans. on Antennas and Propagation, Vol. AP-26, No.2, p. 366,March 1978.

626 D.A. Hill and J.R. Wait, The01'etical and numerical studies of un.re meshstructures, Sensor and Simulation Note No. 231, pp. 1-80, 10 June 1977,Published by Kirtland Air Force Base, Albuquerque, NM 87117, U.S.A.

627 with D.A. Hill, Analysis of the dedicated communication line in a minetunne l for a shunt- loaded tro Hey un.re, IEEE Trans. on COlllll1unica tions,Vol. COM-26, No.3, 355-361, March 1978.

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628

629

630

631

632

633

634

635

636

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640

D.C. Chang and J.R. Wait, An anaZycis of resonant Zoop as an eZectronag­netic sensor of coaZ seam thickness, Radio Science, Vol. 13, No.2, p.399, March/April 1978 (The full text of this article in the Proceedingsof the La Baule Symposium. Copies of the Symposium proceedings may beobtained from P. Misme, C.N.F.R.S., 3 Avenue de la Republique, 92131Issy-les-Moulineaux, France.

D.A. Hill and J.R. Wait, The impedance of dipoZes in a circuZar tunneZwith ar. aria! conductor, IEEE Transactions on Geoscience Electronics,Vol. GE-16, No.2, 118-126, April 1978.

Guided eZectromagnetic waves in a periodicaZZy nonunif~ tunneZ, IEEETransactions on Antennas and Propagation, Vol. AP-26, No.4, 623-625,July 1978.

D.B. Seidel and J.R. Wait, Transmission. modes in a braided cClariaZ cabZeand coup Zing to a tunneZ e~~i~onment, IEEE Transactions on MicrowaveTheory and Techniques, Vol. MTT-26, No.7, 494-499, July 1978.

K.R. Umashankar and J.R. Wait, Electromagnetic coupling to an infini~e

cabZe placed behind a sZot-perforated screen, IEEE Transactions onElectromagnetic Compatibility, Vol. EMC-20, No.3, August 1978, pp. 406-411.

D.A. Hill and J.R. Wait, Scattering by a slender void in a homogeneousconducting ~ire rope, Applied Physics, Vol. 16, 391-398, 1978.

Excitation of an eZevated cabZe over a stratified earth by ar. o~erhead

current system, Int. J. Electronics, Vol. 44, No.6, 609-616, 1978.

D.A. Hill and J.R. Wait, EZectromagnetic surface-~ve propagation overa rectangular-bonded ~ire mesh, IEEE Transactions on ElectromagneticCompatibility, Vol. EMC-20, No.4, 488-494, November 1978.Errata, EKe-2l, No.3, 276, Aug. 1979.

Concise theory of radio transmiseiQ~ in the earth-ionosphere ~ave guide,Reviews of Geophysics and Space Physics, Vol. 16, ~o. 3, 320-326,August 1978.

D.A. Hill and J.R. Wait, Electromagnetic basis of drill-rod telemetry,Electronics Letters, Vol. 14, No. 17, 532-533, 17th August 1978.

Theories of' scatterir.g from ~ire grid and mesh structures, Reprintecifrom book "Electromagnetic Scattering", Edited by P.L.E. Uslenghi,pp. 253-287, Academic Press, 1978.

D.A. Hil~ and J.R. Wait, Surface ~ve propagation on a rectangu~

bonded l.nre mesh located over the ground, Radio Science, Vol. 13,No.5, 793-799, September/October 1978.

EZectromagnetic response of an anisotropic conducting cyUnder to anexternal source, Radio Science, Vol. 13, No.5, 789-792, September/October 1978.

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641 COImIents on "Inf1,uence of the Earth in the HF Transmission of LPDAntennasl~' tEEE Trans. on Antennas and Propagation, Vol. AP-26.No.5, p. 756, September 1978.

642 D.B. Seidel and J .R. Wait, Role of controZZed mode conversion in 'Leakyfeeder mine communication 8ystems~ IEEE Trans. on Antennas and Propa­gation, Vol AP-26, No.5, 690-694, September 1978.

643 S.H. Cho and J.R. Wait, Analysis of microwave ducti1'l{J in an inhomo­geneous troposphere~ Pure and Applied Geophysics, Vol. 116. No.6,1118-1142, 1978.

644 D.A. Hill and J.R. Wait, Theory of electromagnetic methods for nondes­tructive testi1'l{J of ~re ropes~ From Proceedings of the Fourth WestVirginia University Conference on Coal Mine E1ectrotechnology, August2-4, 1978.

645 The electromagnetic basis for nondestructive testi1'l{J of cyZindricalconductors~ IEEE Trans. on Instrumentation and Measurement. Vol. IM-27 ,No.3, 235-238, September 1978.

646 Corrments on 'Light lJaVeS guided by a si1'l{Jle cupved surface r ~ AppliedOptics, Vol. 17, No. II, p. 6978. 1978.

647 D.A. Hill and J .R. Wait, Bcznd1JJidth of a leaky coa:rial cable in a cir­cutar tunne'L~ IEEE Transactions on C01lllDUnications, Vol. COM-26, No. 11pp. 1765-1771, November 1978.

648 D.A. Bill and J .R. Wait, Ezaitation of the Zenneck surface la1CZve by avertical aPe1'ture~ Radio Science, Vol. 13. No.6, pp. 969-977, Nov/Dec1978.

649 Electromagnetic response of an anisotropic cyZindPical sheZZ~ Elec­tronic Letters, Vol. 14, No. 24, pp. 750-752, 23 Nov. 1978.

650 D.A. Hill and J.R. Wait, Electromagnetic fieU perturbation by an in­ternal void in a conducti1'l{J cyZinder e%cited by a ~re loop~ AppliedPhysics, Vol. 18, pp. 141-147, 1979.

651 Theory of Comm.cnication in Tunne'Ls Using Open Transmission and LeakyCabZes~ ORSI Meeting, XIXth General Assembly, Helsinki. Finland, 31July-S August 1978.

652 D.B. Seidel and J.R. Wait. Radio transmission in an elZiptical tunneZIJith a contained a:r:ial conductor~ J. Appl. Phys., Vol. 50(2) p. 602­60S, Feb. 1979.

653 Comments on Papers DiscUBsi1'l{J the ~inciptes Underlying the Computationof I.P. Parameters in a Heterogeneous Medium~ Geophysical Prospecting,Vol. 27, No.1, 292, March 1979.

654 with K.R. U1IUl8hankar, Analysis of the Earth resistivity response ofbJatied CabUJB~ Pure and Applied Geophysics, Vol. 117, PP' 711-7421978/79.

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655 with D.A. Hill. Theory of t1'ansmission of eZeetr01lagnetic waves al,ong adril,l rod in eondueting roek~ IEEE Trans. on Geosc. Elec •• Vol. GE-17.No.2, 21-24, April 1979.

656 On defining transmission toss in radio-wave propagation~ Teleco1lllll.Journal, Vol. 46-V!1979, p. 297.

657 James R. Wait (Editor). EM Guided Waves in Mine Environments~ Proeeed­ings of a Workshop~ u.s. Bureau of Mines. Open File Report. 134-78.31 May, 1978, available from N.T.I.S., PB-289742.

658 with Robert L. Gardner, Eleetr01lagnetie nondestructive testing ofcyZindricaZZy 1.ayered eonduetors~ IEEE Trans. on Instru. and Measmnt ••Vol. IM-28, No.2, pp. 159-161, June 1979.

659 Applications of eleetromagnetie theory to geophysieal ezploration~

Proc. IEEE, Vol. 67, No.7, pp. 979-980, July 1979.

660 Samir F. Mahmound. Adel Z. Botros and James R. Wait, Transient el,eetro­magnetic fieZds of a vertical, magnetic dipoZe on a ttJo-layer earth~

Proc. IEEE, Vol. 67. No.7. pp. 1022-1029, July 1979.

661 Reviet.J of e"Leetrarzagnetic methods in nondestruetive testing of LJireropes~ Proc. IEEE. Vol. 67. No.6. pp. 892-903. June 1979.

662 with D.A. Hill and D.B. Seidel, Proineipl,es of R.F. eorrrnunication intunnels using open transmission l,ines and l,eaky eabl,es~ Proc. of AGARD!NATO Confer. "Special Topics in H.F. Propagation" Lisbon. pp. 36-1 ­36-3, Hay 1979.

663 with D.A. Hill. E:citation of the R.F. surfaee LJave by vertical, andhorizontal apertures. Radio Science. Vol. 14, No.5. pp. 767·780,S.pt!Oct. 1979.

664 D.A. Hill and J.R. Wait, Comparison of loop and dipoZe antp.nnas in leakyfeeder CCf'ff'fflmication system8~ Int. :J. Ueettonics, \101. 47, No.2. 155­166. 1979.

665 D.B. Seidel and J.R. Wait. Mode cCMversion by tunnel, non-unifozwmities inleaky feeder eCf'ff'fflmication systems~ IEEE Trans •• Vol. AP-27, No.4. pp.560-564, July. 1979.

666 Book review of NikoZa TesZa Col,orado Springs Notes 1899-1900~ (ed. byA1ekaandar Marincic, No1ite. Belgrade. Yugoslavia, 1978, IEEE Spectrum.Vol. 16, No.8. Aug. 1979.

667 with D.A. Hill, E1.ectrO'Tlagnetic interaetion bettJeen a eondueting eyZinderand a 80Z,snoid in relative motion~ Jour. Appl. Phys •• Vol. 50. No.8.pp. 5115-5119, Aug. 1979.

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668 David A. Hill and James R. Wait, CaZcuLated admittance of ar. idealizeddrill ~od antenna in a lossy medium~ IEEE Trans. on Antennas andPropagation, Vol. AP-27, No.5, September 1979.

669 E:z:act 8ta'face impedance for a cylindrical conductor~ Elect. Ltrs.,Vol. 15, No. 20, 659-660, September 1979.

670 with D.A. Hill, Dynamic electromagnetic response of a hanogeneousconducting cylinder f01' syrrrnebtic e:ccitation~ Appl. Phys., Vol. 20,89-96, 1979.

671 On naturaZ slot antennas~ Radio Science, Vol. 14, No.5, pp. 765-766,Sept/Oct. 1979.

672 Vignette of guided ~Ves research~ (2 page summary), Combined Session,National Radio Science Meeting, Boulder, Colorado, 5-8 November 1979.

673 Ezact sta'face impedance for a spherical conduct01'~ Proc. IEEE, Vol. 68,No.2, February 1980.

674 Elecbtanagnetic sta'face impedance for a "Layered earth for generale:ccitation~ Radio Science, Vol. 15, No.1, pgs. 129-134, Jan/Feb. 1980.

675 D.A. Hill and James R. Wait, Propagation aZong a coa:::iaZ cabZe tJith ahelical shieZd~ IEEE Trans. on Microw. Theory and Tech., Vol. MTT-28No.2, Feb. 1980.

676 K. Tsubota and J .R. Wait, The frequency and the time danain l'esponseof a buried azial conductor~ Geophysics, Vol. 45, No.5, 941-951, May1980.

671 D.A. 8111 ad J.es It. Wait, Ground wve att..-rtion frmction fCJlO a.,,~z. .,,-tla with arbitrary 811Z'fac" imp«lQlltJfl. lladio Science. Vol.15. No.3. PIS. 637-643. May/June 1980.

678 COfIl1Z«1l11Oi:le tlIttZZgri. fCJlO a l'f01fWfifDlWl tl'optJephftoic .... gui.d.~ bdioScience. Vol. 15. 10. 3. PIS. 667-673. May/June 1980.

679 D.A. Bill and J.It. Wait, n"ctJ"omtIg7Uttic thllOrY of the Z008eZy bl'aidedcoa:riaZ cabu: pazot II - IhIttmccZ ZO"8IlZte~ IEEE Trans. on Micro. TheoryMTT-28. Mo. 4. AprU 1980.

680 David A. B11l and J... R. Wait. Ckl the excitation of the Zennec:ksurface wave CJY8r the grolDl at 10 tIIz. tExcitat1cD de I' onde deZenDeck G1 eo1 • 10 tItz.), Annales des Teleccm,,11catians, t:aIa 3S,.... 5/6, Ha1-JuiD. 1980.

681 with D. A.' B11l, Fielda of • bDrizantal 1aap of ubltrm:y 8bIIpeburied in • tIIO-laJu urth. Rad10 Sdece, Vol. 15, Ro. S,903-912, Sept. - OCt. 1980.

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'lbeorles for nd10 grOIDi ¥aYe~ ewer a 11111d.8ect1cnpath, Badia Sc:ieDce, Vol. lS, No. S, 971-976, Sept. - Oct. 1980.

Beply to M. G. Margan' S CCXlii&lt on "A new in~tation to the VLFiqMIdance tneasUreIIIents of island slot antemas. ' Radio Science.Vol. lS, tb. 5, 969, Sept. - Oct. 1980.

Coupled t:rapospher.i.c mde analysis for a range dependent UDdel.Archiv for Electranick wid tJebe~steehnik(El~tronics andCCIIiIIiilCadCi, &id 34, Heft 10;m:4%6, ociOber, 1980.

Propagat1on in rectangular tunnel with iJq)erfectly conductuJgwalls, Electtcaics Letters, Vol. 16, No. 13, 521-522, 19 June 1980.

D. A. Hill and J. R. Wait, Elect:rallBgnetic characteristics of acoaxial cable with periodic slots, IEEE Trans. a-te, Vol. 22, No.4,303-307, Hay 1980. (eo-Elec~ac Qiiiiidbility)

M. A. Ralston and J. R. Wait, '1beory of law-frequency c:cnductivityprabq of mot etruetu:res in coal 1IIlDes, Jad10 Science, Vol. IS,No.6, 110S-1108, Nov. - Dec., 1980. (S\IImiiY of HLiOfichearticle) .. Paper h a".Ua.ble on microfiche fromAmerican Geophysical Union. ZOOO Florida Ave NW, WashingtonDC, Z0009,

E1.ec:tmDIgnetic .bttering frc::m an Isolated Irregularity in a'l'xopaspheric Duct, Acta Physic:a Austriaca, Vol. 52, 193-202, 1980.

tow-qle radiatian of an antenna over an irregular~ plane."cti Della Fondaziane Gio~iO Ronchi, norence, Vol. 35, No.5,..ietfOiitite:oEtCi6ii 1980, 5 -583.

Coupled IIIDde malysia for ground wave pmpagation averj~ paths, Electranaptics,· Vol. 1; 91-100, (Hemi.spb£reT\i)l'l..ltWlg <mporatiaii) 1981.

la: "A gramded vertical long-wire source system for plalle waveIJ8II'I8totelluric: analog 1IDde1ing" Geophysics, Vol. 46, No.6. 934­935, Jwle 1981.

D. A. lU.U mel J. R. Wait, HF Grtuld wave pxopsgat1on aver Sea Icefor • apberical earth IIlOdel, IEEE Trans., Vol. AP-29. tb. 3,S2S-527, Hay 1981.

D. A. HUla J. I. Wait, HF Iad10 wave tranIIIIlsalon ewer sea iceIDd z.Dt.e ...q pau1bil1t1a, IEEE Tnns., (GeoecieDce _t.Dte s..ma>, Vol. <iE-19, No.4, 204-209, Oct. 1981.

D. A. HUl mel J. Il. Wait, !IF GrolInd ww prop8pt1on over mbredla, sea and ..-tee paths, IEEE Trans., Vol. GE-19, No.4, 210­216, Oct. 1981.

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Towards a general tbeoIy of induced electrical polarization 1n geo­physical exploration, IEEE Trans., Vol. GE-19, No.4, 231-234, Oct.1981.

S. F.~, S. H.~, cdJ. R. ~t, EUc~t~scattering fma a buried cyliPdrical ~eneity inside a lossyearth, Radio Sc:1ence, Vol. 16, No.6, 1285-1298, Nov. -Dec. 1981.Also: Wave ~ti.an J'heorI. (by J. R. Wait), Pergaam Press,l;ew York, 1981.

D. A. Hill and J. R. ,"lait, Graund wave propagation over a mixedpath with an elevatia:l change, IEEE Trans., Vol. AP-30, No. I,139-141, JaD. 1982.

D. N. Henrot and J. R. Wait, Analysis of DIXie conversion phenomena:in Dan mifODll parallel plate wsvegu1de, Radio Sc:iax:e, Vol. 17,No.2. 337. MEc:b 1982. ( .ummary of microfiche article).Paper is available On microfiche fromAmerican Cieophysical Union, ZOOO Florida Ave NW, WashingtonDC, Z0009. -

K. A. Nabulsi and J. R. ~t, Transient coupling between finitedrcu1cs on a anisotropic ca1Cb:dn8 balf-spacep GeophYs. Pro!p.,Vol. 30, 470-485, 1982.

ElectraJllFl!tic response of a III!d1un loaded with coaced c:oncb:ti.vepart:icles, IEEE Trans., Vol. GE-20, No.4, 500-504, OCt. 1982.Also: ~traI!B!n!t1sm, (by J. R. Wait), Academic Press, New York,1982. [Soviet E! dit~on "Nedra" 1987],

Effective electric:a1 properties of heee;eneaus earth mcde1s, RadioSc:1ent:e, Vol. 18, 19-24. 1983. (CorrectioJ:.: PI. 796, .ame VOLT

Electrical transient analysis for disseminated mineralization,~Science, Vol. 18, 25-27, 1983.

~lex c:cnductivity of disseminated spbe%oidal ore grains, GerlsndsBeitr. Geophxsik, Vol. 92, 49-69, 1983.

lesistivity respanae of • taopneous earth with. c:onta:lnedvertical c:mdPJctDr, IEEE Tr_. on Geoscience sod R81Dte 5eftsg,Vol. GE-21, 109-113, 1983.

~ coupl.q betw82 gm.mded c:1rcu1ts mel the effect of • thinwrt1c:al CClDductor in the earth, IEEE. 'l'rana., Vol. AP-31, No.4,640-644, July 1983.

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A .~le v1aw of the 1nduced-polm:izatiaD 1'88pCIrtM of 81\ 1ab:d.welactt·Hi.petic~ ayarea. IEEE Tans. an eea.dence melI81Dte S!n!b' Vol. GE-21, SOS-S07, 1983.

General fomtledm of the fDb:ticn l.oaU1R prcblem forccn::entric layEa • ...::~ the borehole, IEEE Tnas. an Geoadence_ I81Dte Sensing.;;..,l. GE-22, 34-41, 1984.

R elaxation phenom~na and induced polarisation, C eoexploration(Elaevler), Vol 22, 107-127, 1984.

Eiectromalnetic re'pon.e of a d18crete1y grounded circuit,Gcgpby.ic;., Vol. 49, 577.580, 1984

On ''Jcmo.pheric induced wry low frequellcy electric field wavetilt~haDle." ,Qcophy.iCl, VoL 49, 60-62,1984

On modeling a well ca.ing for re.18tivlty and induced polarization,Ceophy.iCl, VoL 49, 2061·2063, 1984A1.0: Electromagnetic Wave Tbeory, Harper and Row, New York.1984 # (3rd Printing 1986), "ill1ey Interhationa"l Edition. 1988

with P. Debrowe. Induced polarisation in electromagnetic iDductive"heme., Ceophy.ical Pro'pectins, VoL JZ, No. 6,1147-1154,1984

P. W. Fa_pn aDd J. R. Wait, laduced polarization relpon.e ofdh.eminated miDenlbation for .pheroidalgeometrie., RadioScience, Vol. 20, 147-148,1985 (IUmmary of microfiche paper) ..

Pager is available on microfiche fromAmerican Ceophysical Union, 2000 Florida Ave NW, Wa~hington

DC, 20009,.

with P. D ebroux, The induced malnetic dipole of a polarized.phere, IEEE Tran•• Vol, AP-33, 465-467, 1985

T. P. Cru.su and J, Il. Wait, DUution and d18tortion effect. inthe lDduced polarisation re'pon.e of a two layer earth, IEEETrail. Vol. CE-Z3, 606-609,1985

with P, Debroux, lacluctive coupliD. between loop. 1yin. on apoJarisable balf••pace, IEEE Trail•• Vol CE-Z3. 609-610.1985

Focu.ed heatiDI in cyUndrical tar.et., Part I, g:EE Tran.Vol. MTT-33. 647-649,1985

with J. T. WWiam., EM and IP re'pon.e of a .teel wen ca.ingfor a four electrode .urlace array, Part 1:theory, Ceophy.icalPro.pectins, Vol, 33, 7Z3-735,1985

J, T. WWiam. aDd J. R. Wait, EM and IP reapoD" of a .teel-eUca.inl for a four electrode 'urlace array, Partn: Numericalre.ult., C.ophx.ical Pro.pectins, Vol. 33, 736-745, 1985

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7'6

Ferrollagnetic implants in hyperthermia, IEEE Trans.vol.SME-32,707-708, 1985Interaction of electromagnetic and electrochemical effectsin geophysical probing, Electronics Letters,vol.22,no.7,393-394,27 March,1986 <with T.P. Gruszka )with M.Lullori,Focused heating in cylindrical targets,pMrtII, IEEETrans.Yol.MT.T-:14 ,3~7-3S9, 1986

Analysis of the radiation leakage for a four-aperturephased array applicator in hypertharmia therapy,IEEE Trans•.Yol.MTT-3.,531-S." 1986.Extensions to the phenomenological theory of induced polar­ization, IEEE Trans. Vol. GE-24, 409-414,1986

D.A. Hill and J.R.W8it,Ancn8lous vertical ..gnetic field for 'e1ectralBg.induction in a laterally varying thin conductive sheet,

. Radio SCience, Yol.21,617-621 ,1986 .Impedance conditions for a coated cylindrical conductor,Radio SCience,Yol.21 ,623-626. 1986with T. P. Gruszka. On electromagnetic coupling "removal"from induced polarization surveys, Geoexploration,Yol.2421-27,1986Propagation effects for electromagnetic pulse transmission,Prcc. IEEE, Vol. 74,1173-1 181 , 1.986Electromagnetic response of a thin layer,ElectronicsLetters898-S~,1986Gerardo Aguirre and J.R. Wait, Resistivity and induced polarizationresponse of a thin sheet, Pure and A~plied Geophysics, Vol.123,882-892,1985 ( actual publicaHon da e : aec.1986 ! )

Excitation of an enclosed lossy cylinder by an aperturesource, ~~E Trans.vol.MTT-35,210-212, 1987General solution for excihtion by slotted aperture source inconducting cylinder with concentric layering, IEEE Trans.vol.MTT-35,321-325,1987. Also:Introduction to Antenuas and Propagation,Peter PeregrinusLtd.Stevenage UK, lEE (available from the publisher and fromIEEE service Center, PPL Dept.445 Hoes Lane,Pisca~away,N.J.08854-4150)

G.Aguirre and J.R.Wait,Surface impedance of a three layerearth with a thin intermediate layer,Int. J. Electronics,vol.62.327-340,1987

with T.P. Gruszka,Resistivity and induced polarizationresponse for a borehole model,IEEE Trans.,vol.GE-25;369- ,372, 1987Physical model for the complex resistivity of the earth,Electronics Letters, vol.2J, 979-980, 1987

Com.en t s on' Correction of Maxwell's equations for signals, I and II',1m 1:ln5. vOl.EMC-29,256-257, 1987

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General cylindrical model for focussed power depositionfrom an external antenna system, Jour. of ElectromagneticWaves and Applications,vol. 2,77-84,1987

Determining the strength and orientation of an elevateddipole, Electronics Letters,vol. 24, 32-33, 1988A. Sezginer, T.M. Habashy and J.R. Wait, An image methodto compute the static magnetic field due to currents injectedinto a homogeneous,conducting, and .agnetically polarizablehalf-space, Radio Science , vol. 23, 41-45, 1988Determining transient dipole source from observed fieldwaveforms, Electronics Letters, vol. 24 ,282-283, 1988S.F. Mahmoud, S.C. Tantawi, and J.R.Wait, Interpretation of

muti frequency cOlllplex resistivity data for a layered earthmodel, IEEE Trans. vol.GE-26,399-608, 1988

Multipole expansion for the EM fields of a linear radiatorIEEE Trans., vol EMC-30,413-415,1988Electromagnetic RAdiation from Cylindrical Structures,(1959)Corrected reprint edition, Peter Peregrinus Ltd.UK, 1988Comment on .. Electric field of a transmission line over multi-layeredMedia", Prac. IEEE, vol. 76,284, 1988Comments on "Closed form solution for underqround impedancecalculation" Proc.IEEE, vol. 76,1396-1397,1988Let ter to the Edi tor, ( on VLF radiation from lightning),Geoexploration, vol. 25, pg. 173, 1988

Complex resistivity of the earth, Progress in Electromag­netic Research(P1ER),vol.1, No.1, 1-175,1989 (Elsevier)111 defense of J.A. stratton, IEEE Trans. ,vol.E~=30,590,1988J.L. Young and J.R.Wait, Note on the impedance of a wiregrid parallel to a homogeneous interafce,IEEE Trans. vol.MTT-37, 1136-1138, 1989T.P. Gruszka and J.R. Wait, Interaction of induced polar­ization and electromagnetic effects in borehole probing,Geoexploration, vol.25, 267-277, 1989J.L. Young and J.R.Wait, Shielding properties of an ensembleof thin, infinitely long, parallel wires over a lossy halfspace, IEEE Trans. vol. EMC-31, 238-244,1989Comments on "Electric field sensors in electromagneticsounding", IEEE Trans. vol. GE-27(Geoscience and Remote Sens­ing), 789-791, 1989M.L.O.Lumori, J.R. Wait and T.C.Cetas, Power deposition andfocussing in a lossy cylinder by a concentric phased array,Radio Science, vol.24, 433-442, 1989B.A. Baertlein, J.R. Wait and O.G. Dudley, Scattering by a con­ducting strip over a lossy half-space, Radio Science, vol.24,485-497, 1989

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7'2 Co••ants on -Magnetotelluric exploration for hydrocarbons",Proc.1iSI, vol.77,1'8., 1989 -

7'3 Radiation from vertical electric dipole located over laterallyanisotropic ground plane, Electronics Letters ,vol.26,74-76, 1990

7'4 A note on the Quasi-static approach in teaching electromagnetics,IEEE Trans. vol. ED-33 (Education),164, 1990

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