378 IEEE TRANSACTIONS OW ANTENNdS Ah‘D PROPAGATION, MA4Y 1969

where the unit vectors art and auz and the Fresnel coefficients RI , Rll, TI, and TI’ are all functions of the variables of integration k, and k2u. The superscripts 1 and 11 in the Fresnel coef6cient.s indi- cate perpendicular and parallel polariza- tions, respectively. Equations (3) show that m,(=ni+n,) as well zs Ih( =n,) in this formulation contain only, and al- ways both, horizontal components in describing the electromagnet.ic field every- where.

Using the stationary phase method, we can obtain the geomet.rica1 optics resulk of (3): II,= = { RL(ao) sin2 Q~ - R” (ao) coso + o }

I l l = ( I L , & , O ) . Needless to say the derived boundary relations remain com- patible with each other.

IV. CONCLCSIOX It has been shoan that Sommerfeld’s

resolution, n =(n,,O,n,) is not unique, and it is just one of four possible resolu- tions. This point. is ambiguous in Som- merfeld‘s work. If Sommerfeld was not mistaken himself on this, his analysis is certainly misleading, ahich apparent.ly led many others to believe that the use of the particular resolution of n, II = (lL, O,n,), was necessary. It has also been shown how the horizontal resolution n = (IIz,l&,,O) logically arises by using the elect,romagnetic plane-rave approach and

where RT = Ro+R2 and other symbols are shown in Figs. 1 and 2. The preceding expressions are new, and especially t,hose for n, are by far the simplest expressions obtained so far for t.he transmitted Hertz vector in the horizont,al dipole problem.

111. DISCUSSION ON THE NONUNIQUENESS OF n

The preceding results shorn that there exists a t least another resolution for the reflected Hertz vector, namely, n = (UI, lIu,O). We can shorn that actually there are four possible resolutions altogether. We first note that. the requirement on n, is that I1, be any solution of the homo- geneous vector equation

vxvxn, - V(v.n,) - k ? m , = 0. (5)

If we add to n, the gradient of any solu- tion Q of the homogeneous Helmholtz equation

V v + kt& = 0 (6 )

then in view of the relations between (E,H) and n, t,he (E ,H) corresponding to &+V@ do not change. The specific factor in n, must., of course, be deter- mined according to the form of the incident wave and the boundary con- dition.

Given any n,, by choosing t.he factor @ properly, we can eliminate any one component of n,, and hence of n,, but not two components simultaneously. Hence any two components or all three components are necessary in n,.

By an identical argument, it can be inferred that the resolution of nt is also not unique. It can be further observed that the resolution of nr and t.hat. of nt need not be the same in a problem; for example, we can let &= (&,O,II2,) and

t,he Fresnel coefficients. The form of n, and nt obtained in the preceding geo- metrical optics approximation are new.

A H X E D ERTEZA

Bureau of Engrg. Res. University of New Xexico

Albuquerque, N. Nex.

E. K. P a R K

REFERESCES

Phys lcs . New York: Academic Press, 1949, -1. Sommerfeld, Partial Di ferent ia l Equations in

pp. 258-264. L. AI. Brekhorskikh, W a v e s i n Layered M e d k . New York: Academic Press, 1960. pp. 239-259. A. Ba~ios. Dipole Ead id ion i n the Presence of a Conducting Half-Space. Ne\\- York: Pergamon, 1966. p. 9.

A Planar Creeping Wave The concept of creeping waves has

proved valuable in understanding and predict.ing the scattering from bodies all of whose dimensions are large compared with the wave1engt.h. Such waves are pre- sumed “born” in the vicinit.y of the shadow boundary and creep around the shadowed side of the body, being atten- uated as they go by virtue of energy spewed off in the tangential direction. rlsymptotic analyses of the known modal

research as sponsored by the U.?. Air Force Ariorucs Nanuscript received November 23. 1968. T@

Laboratory, 1% right-Patterson .br Force Base, under Contract hF33(615)5170.

expansions of the surface fields on spheres and cylinders have shown the spatial characteristics of these waves and in- dicated the dominant role played by t,hem in the generat.ion of the field wit,hin the geometrical shadow, and the validity of the resulting pict.ure has been confirmed by probe measurements of the surface fieIds for these same geometries. If the radius of curvature is a variable function of position, the geometrical theory of diffraction [ l ] predicts that the creeping waves will follow geodesics on the surface, wit,h propert,ies which are primarily func- tions of the local radius of curvature in the direction of the path and a net am- plitude determined by a line integral of the radius to the -2/3 power. The work of Hong [2] has also shown the effect of the transverse radius, presumed large, but only recently have we had any in- dication that a form of creeping wave can still exist when the t.ransverse radius is, in fact, zero.

In the course of an invest.igation of the edge-on scatt,ering from ogives and disks, both elliptical and circular, when illuminated by a plane wave at grazing incidence with the electric vector in the plane, Ryan and Peters [3] observed that the scattering behavior was compatible with the interference between a wave “reflected” from the front edge of the plate and one which had circumscribed the rear perimeter. A high-frequency est.imate of the former is trivally obtained using the scattering coefficient for t.he edge of a half-plane, taking into account the beam spread produced by the planar curvature at the specular point, and for the latter (creeping) wave the param- eters (including t.he attenuation coeffici- ent) were estimated partly in an empirical manner and partly by using as a reference measured data for an ogive. For a circular disk, a t least, the resulting expression for the backscatt.ering cross section appears to be in good agreement wit.h the mea- sured data that Ryan and Peters present.

During t,he past year extensive mea- surements of the edge-on scattering from flat plates of various configurations have been made. One such configuration was a circular disk for which data was obtained over an extensive range of ka values, where a is the radius and k =%/X is the propagation constant, and though the data does not support the theoretical expression for the creeping-wave con- t,ribution which Ryan and Peters have proposed, it is in agreement with the con- cept that the nonspecular cont,ributor to the scattering is a form of creeping wave. This has now been confirmed by probe measurement,s of the surface field.

A series of six circular disks with radii ranging from 4 to 10 inches were cut from a 1/32-inch thick aluminum sheet. Each was mount.ed on top of a Styrofoam sup- port and illuminated at grazing incidence by a horizontally polarized wave a t a frequency of 3.0 GHz for which k=1.60 inch-’. The circumferential component of t.he current density JQ was then measured at the center of the rim using a small loop probe whose position was varied

COMMUNICATIONS 379

I 1 I 0 7 4 7 2 3T4

C# (radians1

'i Fig. 1. Circumferential edge current &I =9. Bare (solid line) and with

absorber on opposite slde (broken line).

from 4 = O (the front) to 4 =T (t,he rear) and back to the front again. Due to a slight asymmetry in t.he probe response, there was a det.ectable difference between the readings taken at corresponding points on t.he t.wo edges of the disk, but the differences averaged less than 0.2 dB and the subsequent discussion is based on the averages of the readings for the left and right sides of the rim. It was also observed that the field varied by as much as k0.3 dB between t.he upper and lower edge, and care mas therefore taken t.o en- sure a probe t.rajectory such that the loop gap remained in the center of the rim.

A typical curve of 1 J+I versus +, pro- duced from memurements made every 2; degrees in +, is that, shown for the l l i - inch diamet.er disk (ka = 9) in Fig. 1. Sormalization is with respect t,o the incident field level, but because of the unknown inOegrating properties of the probe about the rim in a radial plane, the scale is relat.ive as regards the absolute value of the local field. Within the shadow, say, + > 3 ~ / 4 , t.he mean of this curve is an exponentially decreasing function of +.

There are several feat,ures of Fig. 1 that should be noted. For 37r/4&<~ the solid line can be closely fitted by a formula suggested by the above inter- pretation, viz.,

I J d l = A COB [(lis,, + -;n) (T - +) ]

where y1 is the apparent phase velocity relative t,o that of free space and y~ is the attenuation coefficient. The fit be- comes poorer as x e approach the shadow boundary, douhtless due to the influence of higher order creeping-wave modes which have not yet decayed sufficiently for their effect to be ignored, and the transition region appears to extend for about ~ / 4 on either side of the boundary. For decreasing 4 < ~ / 2 , i.e., within the illuminated region, the wave launched a t

$ = 3 ~ / 2 still contributes to produce an oscillation of decreasing magnitude, and beyond the transition region ( + 5 ~ / 4 ) the component wit.h which i t interferes has the &K& dependence deduced from the half-plane solution [4].

The relevant parameters in the pre- ceding equation can be estimated from t,he measured data wit.hin the deep shadow. It is found that y~ =1.06, in contrast with the case of a circular cylinder whose dominant creeping wave has a relative phase velocity less than unity (0.91 for ka =9), and y z =0.90 (c.f. 1.45 for a cylinder and 0.21 predicted by Ryan and Peters for a disk).

To obtain additional confirmation of the creeping-wave charact,er of t.he edge current in the shadow, a piece of absorber was wrapped around the rim in the vicinity of the shadow boundary at 4 = 3 ~ / 2 . T h e aim was to suppress t.he creeping wave originabing there, leaving only the wave which is born a t 4 = ~ / 2 and proceeds in the direction of increasing 4. The measured data for T/Z are indicated by the broken line in Fig. 1, and although there are still some small and irregular oscillations (doubtless due to reflect.ion from the absorber edges), the exponential attenuation of the creep- ing wave is without question.

THOMAS B. A. SENIOR IT. V. LIEPA

Radiation Lab. Dept. of Elec. Engrg.

University of Michigan Bnn Arbor, Mich.

REFERENCES 111 J. B. KeUer, J . A p p l . Phys., vol. 28, pp. 426444,

121 S. Hong. J . X&. Phys., vol. 8. pp. 1223-1232, 1957. 1 OC?

C. E. Ryan, Jr., and L. Peters, Jr.. "A creeping- r a v e analysis of the edge-on echo areas of disks," I E E E Tram. Antennas and Propagation

March 1968. fCommunwafzons). vol. AP-16, pp. 274-275,

J. S. Hey and T. B. A. Senior, Proc. Phys. Sac. (London), vol. 72, pp. 981-995, 1958.

LO",.

The Maximum Echo Area of Imperfectly Conducting Dipoles

Abstract-The question has often been raised as to when the effects of imperfect conductivity must be taken into account when studying the echo area of a wire scatterer such as the dipole. The conditions under which the echo area is sensitive to the conductivity are shown.

I. INTRODUCTION

It is known that the maximum or peak echo area of an imperfectly con- ducting dipole is dependent on the con- ductivity u, frequency W, and the dipole radius a, whereas the maximum echo area of a perfectly conducting dipole is about 0.86X9. Hence it would be useful to relate the maximum echo area to these parameters. It is the primary purpose of t,his work to obtain such a relationship. For purposes of definition we will t,ake the maximum echo area to be the peak echo area of resonant short-circuited di- poles, where the dipole length is adjusted (near X/2) so as t o achieve the resonant peak.

Alt,hough the maximum echo area does vary with the conductivity, fre- quency, and dipole radius, the reMion- ship between t.hese parameters and the maximum echo area is apparently t.ied to the penetration of the electric field into the dipole. For example, if the conduc- tivity of t,he dipole material and/or the dipole radius is such that the electric field does not penet,rate appreciably to the dipole axis, then the echo area wil l be nearly t.hat of a perfect conductor as for t.he two 80-mil wires in Fig. 1. However, if the electric field does reach the dipole axis without Iarge attenuation, then the echo area can be expect,ed to fall significantly below that of the corresponding perfect conductor. We will show that there is a relabionship bebeen the maximum echo area of imperfectly conducting dipoles and t,he amount of penetrat.ion by the electric field to the dipole axis. This is done in some detail a t frequencies of 300 and 3000 MHz for both bismuth and platinum dipoles. Then from the data for bismuth and platinum dipoles, a curve is deduced from which the maximum echo area may be very closely estimated for dipoles of ot.her materials operating at various frequencies without recourse t.0 the more involved process of actually comput.ing resonance curves.

11. THEORP

To obt,ain an expression for the total electric field in t.he dipole of Fig. 1, the scalar-wave equation may be employed. That is,

( V + kZ)$ = 0. (1)

TFBJ supported in part by the Ohio State University Manuscript received December 6, 1968. This work

Research Foundation, and was performed under a subcontract with the M.I.T. Lincoln Laboratory. which is operated with the support of the U.S. Air Force.