Modal theory of long horizontal wire structures above the earth

Radio Science, Volume 13, Number 4, pages 605-613, July-August 1978

Modal theory of long horizontal wire structures above the earth, 1, Excitation

Edward F. Kuester and David C. Chang

Department of Electrical Engineering, University of Colorado, Boulder, Colorado 80309

Robert G. Olsen

Department of Electrical Engineering, Washington State University, Pullman, Washington 99163

(Received July 13, 1977.)

The current excited on a horizontal thin-wire structure above the surface of a finitely conducting earth is derived, and is shown to be the sum of discrete modal contributions as well as a number of continuous spectrum (radiation) components for any source of finite extent. The wire structure need only be characterizable approximately by an axial surface impedance, and so particular cases previously treated in the literature can be recovered. In particular, an expression for the current excited by a transverse aperture of finite extent is given, and using orthogonality properties between different parts of the mode spectrum, it is concluded that any of the discrete modes can, in principle, be excited with arbitrarily high efficiency by an appropriately chosen aperture field of finite extent.

1. INTRODUCTION: HISTORY OF THE PROBLEM

As early as 1902, theoretical investigations were being made to account for the effect of the earth on electromagnetic wave propagation along a horizontal wire parallel to its surface. In spite of the many refinements to this theory (see the historical references given by Kuester and Chang [1976]), it has remained basically a low-frequency theory in which a static or quasistatic mutual impedance between the wire and the earth is calculated and

used in the telegraphists' equations to obtain a transmission line description of the behavior of the system. As such, it was sufficient to be applied to overhead power lines and for communication lines using long waves. Much of this work can be found summarized in the books [Sunde, 1968; Ollendorf, 1969].

At higher frequencies, the wave structure of the field becomes important and the low-frequency theory is, in general, no longer adequate to describe such problems as earth effects on elevated antennas or transient processes on overhead power lines. The wave nature of the discrete modes apparently was first analyzed by Pistol'kors [1952, 1953], who was followed later by many other authors of which we mention only Grinberg and Bonshtedt [1954], Wait [1972], and Chang and Wait [1974] (see also

Copyright ̧ 1978 by the American Geophysical Union.

Lavrov and Knyazev [1965]). There has been con- sideration of excitation by finite sources: the delta- function voltage generator [Layroy and Knyazev, 1965; Pistol'kors, 1953; Chang and Olsen, 1975], and most recently, a vertical electric dipole [Olsen and Usta, 1977; Wait, 1977], but no study of excitation by a general finite source, or more particularly by a given aperture field, has been made.

Strictly speaking, the general excitation problem should be dealt with before any investigation of the properties of individual modes is undertaken, in order to understand exactly the significance of expressing the electromagnetic field of an open structure in terms' of modes. The present paper is thus divided into two parts. This, the first part, will consider the excitation of an infinitely long wire located parallel to and above a homogeneous lossy dielectric half space by an arbitrary but finite distribution of sources. As a special case, the excitation by an aperture field of arbitrary form in the transverse plane is obtained, and the total current on the wire uniquely decomposed as the sum of discrete and continuous modal components. Orthogonality properties for the modes are established, and the significance of the modal concept is discussed.

Of particular interest to us is a higher frequency range (---100 MHz say), where any wire of length greater than 10 m or so can be considered long.

0048-6604 / 78 / 0708-0605501.00 605

606 KUESTER, CHANG, AND OLSEN

In this situation, the procedures usually employed for antennas of less than a wavelength become very inefficient. Thus, even though any real wire parallel to the earth will have only a finite length, it may be so long that much of its behavior can be studied by considering a wire of infinite length. One takes such an approach, for example, in the theory of waveguides, but because of the orthogonality and completeness properties of the modes of a given closed guide, effects at the end of a guide or at a discontinuity may be studied in terms of coupling between the modes or in terms of excitation of

the modes by equivalent sources at the nonuniform points (see, e.g., Collin [ 1960] ). For shorter wires, it will generally be more efficient to formulate an antenna integral equation of classical type, as done by Miller et al. [1972].

Unlike closed structures, the mode spectrum of an open waveguide consists of a continuous (radiation) component in addition to a number--usually finite--of discrete propagation modes [Shevchenko, 1971]. Instead of an infinite summation, the con- tribution of the radiation modes is an integration over a semi-infinite interval in a formalism analo-

gous to that of the guided modes, but describing essentially unguided fields. Unlike the higher-order modes in a closed waveguide, an individual radiation mode generally has no meaning by itself. However, if the structure is lossy so that the discrete modes attenuate exponentially along the axis of the guide, it is to be expected that the radiation component, which decays algebraically in free space, must eventually dominate the total field for a source of finite extent [ Wu, 1961]. This hardly renders the discrete modes useless, however, because of the catchword "eventually" in the last sentence. If a discrete mode can be excited to arbitrary efficiency to the exclusion of all other modes by a suitably chosen finite source, then "eventually" can be made an arbitrarily long distance.

In the second part of the paper [Olsen et al., 1978], the properties of the discrete modes of a thin wire over a losSy earth are investigated in more detail. The behavior of the propagation constants and some field plots for the modes are given.

2. FORMULATION OF EXCITATION PROBLEM; THIN-WIRE APPROXIMATION

Let us consider the arrangement depicted in Figure 1. An infinitely long thin-wire structure of

EXTERNAL SOURCES

\ I/

////

x

i i•/- APERTURE PLANE I i

I I

I L INTERNAL I SOUF•CES

I////

EO• •'•0

/// E=EO n •

•o

Fig. 1. Sources exciting wire over lossy earth: internal and external sources with infinite wire.

exterior radius a is located parallel to the z axis at a height h above the earth's surface (x = 0). Region 1 (x > 0) is taken to be free space, characterized by the permeability and permittivity •x o and s o respectively, and the wave impedance •qo = 0Xo/So) '/2. The earth, region 2 (x < 0), is characterized by the complex refractive index n, whose positive imaginary part corresponds to conduction and dielectric losses consistent with an implied time dependence of exp(-itot). For simplicity, the permeability of the earth is assumed to be that of free space. We suppose that there are external

-ext -ext

electric and magnetic current sources J e and J• in the region outside the wire which, in the absence

the known incident fields (E , of the wire, produce ' ' ' -• /•). The latter can be obtained in a straightforward manner using the solutions for electric and magnetic Hertz dipoles in the presence of earth [Ba•ios, 1966].

An exact formulation of the problem would proceed as follows. The incident field induces some

yet-unknown scattered field from the wire, which itself is rescattered due to the presence of the earth. The total field (incident, scattered, and rescattered) is then required to satisfy the appropriate boundary conditions at the wire. Since the rescattered field

depends in a known fashion on the scattered field, this results in an integral equation (more precisely, a system of equations) for the scattered fields at the wire surface, i.e., for the induced currents (actual or equivalent) on the wire. By expressing all quantities involved as Fourier series in the

LONG HORIZONTAL WIRE STRUCTURES, I 607

angular variable • of the cylindrical coordinate system centered on the wire axis, the integral equations reduce to an infinite system of coupled integral equations for the unknown Fourier coeffi- cients. An idea of the complexity involved already when the wire and earth are taken to be perfectly conducting, and the source a plane wave, can be gotten from Flareruer and $inghaus [1973].

If the wire is sufficiently thin compared to its height above the earth (a/h << 1), then the rescattered field at the surface of the wire varies slowly compared to the rest of the field, and can reason- ably be approximated by its average value at the surface. If, moreover, the wire is thin compared to a wavelength in free space (k, a << 1, where k, = o•0Xoeo)l/2), and those external sources not far from the wire compared to a are symmetric about the wire, then virtually only z-directed, •-independent currents will be induced on the wire [Timischl, 1971; Wait and Hill, 1975]. In this paper it will be assumed that the wire operates in a basically TM polarization with respect to the z direction, and may be characterized in accordance with the approximate symmetry noted above by a general- ized impedance condition

<E•(po,%,Zo)> =- 2. +

po----a

where Zs(z- z operator, while

z,(Zo - z')(H•(po,%,z'))dz';

') is an impedance convolution

(f(•o)) = 2,rr o (2)

denotes the average with respect to the angular coordinate on the surface of the wire. Imposition of (1) only (rather than the exact boundary conditions, which involve not only the angular variations of E z and H,, but H z and E. as well) corresponds to a truncation of the system of equations referred to earlier, and so we must likewise restrict the number of unknowns. Thus, the scattered field is taken to arise from the •-independent part of the equivalent axial electric current density at p = a alone. These assumptions together constitute the thin-wire approximation which will be the only approximation invoked in this analysis. A more quantitative treatment of the error involved therein has been investigated (for the case of a discrete

mode) by Pogorzelski and Chang [1977]. Let us now proceed to formulate the integral

equation. The incident fields Eiz •½ and •in½ .., at a point 2o = (a,•o,Zo) on the wire surface can, using the Lorentz reciprocity principle [Harrington, 1961, pp. 116-117], be expressed in terms of volume integrals containing fields due to unit sources on the surface of the wire:

E inc/- x • tXo• [•g•(•'•o)' y•x•_ v

(3)

Hinc• - x • tXo• = - [•$•(i ;•o)' j•eXt(•) v

_/•,()? ;•o) o •t(• )] d• (4)

Here, V is the volume containing the external ez ez

sources, (fro, •o ) are the fields produced at a point œ by a unit z-directed electric current •z $(• - •o) located at •o, and likewise (•'*,/-•'*) are the fields produced by a unit •-directed magnetic current t•,o$(• - •o) located at •o.

Since the media and boundaries of the problem are invariant with respect to the z direction, it will be convenient to deal with Fourier-transformed

quantities related by

f(x,y,z) = f (x,y,a)exp(iklaz)da (5)

where kl = to(IXoe,) l/2 is the wavenumber of the medium containing the wire (it is usual, though not necessary, to take e, = eo)- Taking the Fourier transform of (3) and (4) with respect to z o, and making use of the convolution theorem [Morse and Feshbach, 1953], we have

v

- J½,2t( • )' •g•()?t ;iot ;-a)] d• (6)

i .... ,Ct) = -- I e-'•"• [j•xt(•). •00(•,.•0 t , [Xot , ;-- v

-- jernXt(•)* •'(•t;•ot;--C•)] d• (7)

where we have put

e-•,, • (•-•o• •g•(•; •o)dz (8)


ez - ez j• •*, and/•'* all depend and so on, since/•o, Ho, on z and z o only through the difference z - z o. The subscript t on • denotes the part of • transverse to z.

Let us now consider the fields due to the yet-unknown equivalent electric current d zlz(zo)•(q> o - q>') 8(00 - P')/Po induced on the surface of the wire. Once again the fields are observed at •o = (Po, q>o,Zo) on the wire surface, while the source point •' = (p',q)',z'). These fields, denoted by j•scat(•0;• t ) and /•scat(•0;•' ), are the sum of what were called the scattered and rescattered fields above. The fields

due to the average part of the induced surface current are (/•scat)' and ( B scat) t, where ( )' denotes the av?age (2) taken with respect to the variable q>'. If/•scat and/•scat denote the Fourier transforms of /•scat and /•scat (with respect to z o only), the boundary condition (1) leads to the following equation:

(9)

where Z s (a) is the Fourier transform of Zs (z). Some particular cases of •,(a) for a variety of thin-wire structures are given by Kuester and Chang [1976].

In appendix A, expressions for the fields appear- ing in (9) are derived. For simplicity, let us define the fields /•o(•, ;a) and /•o(•, ;a) to be given by (A2)-(A4) with y' = O, x' = h, and Io(a) therein replaced by (kl/2,r). Then (9) can, in view of the results in appendix A, be solved for the current I(a) induced on the wire'

[(,•) = 2•(,•) J,(;J) ] 4 Jo(;J) - • •

ß 1o •

k ! ']oJo(•A)[Mo(o•) -- i• 2• (or) M,(a)

no [

ß I ½-a,,• [3•t(•)o •o(•t;_rx) -½xt - J,• (.•)' •o(•,;-a)]d• (10)

where

Mo(ct) = •2 [H(o,)([A) _ Jo(•l)H(ol)(2[H)]

+ Jo([A)[P(a;2H) - ot 2 Q(a;2H)] (11)

M,(ct) = [2 H(•')( [A ) + J, ( •l ) [- [a H(o• )(2 [H )

+ P(a;2H) - a 2 Q(a;2H)] (12)

where P and Q are defined by (A5) with Y = 0 with the remaining symbols defined in appendix A.

Several remarks can be made about (10). Upon carrying out the inverse transform of/(a), a contour deformation can be carried out to evaluate Iz(z) (or indeed any of the fields which are expressed in similar terms) as a sum of pole residues corresponding to zeroes of the denominator of (10), and branch cut integrals corresponding to branch line singularities of (10). As is usual for open waveguides, the pole contributions can be given the interpretation of discrete propagating modes, while each branch cut integral is the superposition of an infinite number of continuous or radiation modes

[Shevchenko, 1971 ]. Unless I•(z) is an actual, rather than equivalent, current (i.e., unless the wire is perfectly conducting: Z,(a) = 0), equation (10) indicates a set of poles at the zeroes of Jo([A). No physical interpretation is attached to this, however, since actual field quantities do not possess these poles. On the other hand, if an additional term - •eq (O•) is introduced into the right-hand side of (9) to describe sources internal to or on the surface of the wire, (10) is modified by the additional term:

4 veq (ot)

k, 'qo Jo([ A) [Mo(a) - i2•(a)M,(a)/'qo •1 (13)

which, in general, will introduce additional singularities. If, for instance, •,(a) = 0 and [7•eq(ot) = Vo(kl/2*0, we recover the case of a delta-function generator on a perfectly conducting wire [Chang and Olsen, 1975]. As is known for cylindrical antennas in free space [Wu, 1969], the presence of Jo([A) is then required because the generator "sees" the inside of the cylinder, and also in order to produce the proper singularity in I•(z) as the generator is approached. In fact, one may not, in general, replace Jo([A) or Jl([J) in (10) or (13) by their small-argument expansions (which is tempting in view of the thin-wire assumption) if sources are on or near the surface of the wire. This may be done, however, at individual values of a for which ]•A] is not large, as, for instance, when searching for poles of (10), the discrete modes. With the exact

forms of the Bessel functions retained, we recover the proper expression when the wire is excited by an annular current distribution and the earth is

removed (H--. oo) [Ledinegg, 1970]. We should finally note that if 2; xt or 2•m xt is set equal to a constant vector times 8(i -- i,), where •, is some point in space, the solution for an arbitrarily oriented Hertzian dipole (electric or magnetic) is recovered. The expression thus obtained for a vertical electric dipole agrees with the result of more direct deriva- tions [Olsen and Usta, 1977; Wait, 1977].

3. APERTURE EXCITATION

Of particular interest if the structure is to be used as a transmission line is the excitation by an aperture field of some kind. The problem can be variously restated by using the equivalence principle or the induction theorem [Harrington, 1961, pp. 106-116]. For example, the effect of the external sources in Figure 1 can be represented by an

-0

aperture field E t at an aperture plane (taken as z = 0) produced by the sources behind it. We may replace the sources and aperture field by a perfect (electric) conductor at z = 0 with an adjacent magnetic current sheet /•t ø x t• z located at x = 0 + [Harrington, 1961, p. 110]. This in turn is equivalent to removing the conductor and imaging the magnetic current sheet (as well as the wire and the media)

= 0 to give a total current J•n xt 20m •(Z) • 2ff t X 5zS(Z), concentrated in the aperture plane and exciting an infinite wire over the earth. Taking J e ex' and •eq to be zero, the solution for the wire current, from (10), is

8 Jo(•4) - i2.,.(a)J,(•A)/•qo • [(•) -

k, •o Mo(• 4) - i2,.(a)M,(a)/•o •

I t•z. [•,•o(•,) x/•o(g,;-a)]d•, (14) $

where S is the (infinite) aperture plane z = 0. Now, although in an actual excitation problem,/•ø t would have to be solved for from an integral equation over the aperture, for our purposes it will be assumed that it is known, for instance, as the field of the mode incident from a coaxial horn. We now

know/(a), and taking the inverse Fourier transform (assuming the order of integrations can be inter- changed), we obtain

LONG HORIZONTAL WIRE STRUCTURES, 1 609

L(z) - 8 -o - -- a•'E,(x,)x ß k,•o s oo

ß exp(ik, az)/M(a)l J(a)da d•,

where

(15)

M(a) = Mo(a)- i2.,(a)M•(a)/no• (16)

ff(•) = ffo(½4) - •2•(•V,([,•)/no[ (•7)

When the definitions of M(a) and Ho(.l•t;ot ) are extended into the complex a-plane, following the specifications

Im(0 > 0; Im ([.) > 0 (18)

as indicated in appendix A, several sets of branch cuts appear. As detailed in the second par=t of the paper [Olsen et al., 1978], M(a) as well as Ho(œ,;a ), whose singularities arise for identical reasons as those of M(a), possesses branch points at a = _+1, _+n, and _+a•,, where

a8 = n/(n 2 + 1) 1/2 (19)

so that, in addition to the branch cuts (18), we must also require

Im ([8) _> 0; where [• --- (Ot• -- Or2) 1/2 (20)

For z > 0, we may then evaluate (15) by deforming the contour of integration upward from the real axis of the a-plane over these three branch cuts, picking up pole residues from any points % where M(av) = 0 in the process (see Figure 2). As a result of the contour deformation, we obtain

/ /

ORIGINAL CONTOUR

Fig. 2. Original and deformed contours in a-plane. gB = (a2• - a2)1/2' im(gs) > 0; • = (1 - a 2) !/2; Im(g) > 0.


N 3

rn=l j=l

(21)

Each of the N pole contributions Ipm is given by

16-rri J(%,,, ) 1• (z) - • e'•'• •

kl 'q0 M'(%•)

ß I •' [•tø(•') x/•-m(•,)]d•, $

where

(22)

(23)

are defined to be the fields of the "transpose" (negative z.-j.ravel•ng) pth mode associated with %m; the fields œ + - + vm' Hvm associated with the direct pth mode are obtained by replacing --%m by +%m in (23). In a similar way, the branch cut contributions are given by

IBj(Z) -- kl •o o

ß j(a)e•k• g•dgj (24)

where g• = g, g2 = ga, and g3 = •n are taken to be positive and real. The fields of the transpose continuous spectrum modes are defined as

M(a) arg g• =0

•o(X,;-a) m(•)

M(a) arg •=0

larg (25)

larg •]= and as for the discrete modes, the direct modal fields are obtained by replacing -a by a in the numerators of (25).

Because of the choice of sign for the gj, it can be seen that the discrete modal fields (23), with the expression for /•o given by (A2)-(A4), decay exponentially as I•tl--> oo in any direction in the transverse plane if %,m does not actually lie on a branch cut (an exceptional case). The individual radiation mode fields (25), although they do not necessarily have any physical meaning and in fact violate the radiation condition as I gtl--> oo, are

nevertheless individually bounded at infinity. When excited by a finite source, they must, taken together, form a field which does satisfy the radiation condition, as follows from the derivation of the modal decomposition (21).

In certain special cases, it has been found [Olsen et al., 1978] that two of the simple roots of M(a) = 0 can degenerate into a single double root. In this case, (21) must be modified. In the absence of such an exceptional situation, it is shown in appendix B that the modal fields (23) and (25) are mutually orthogonal. Therefore, if œø could be made t

to duplicate the field of a given (nondegenerate) discrete mode, then the only nonvanishing contri- bution to (21) would be the corresponding discrete modal current. Since for a discrete mode with %, not lying on a branch cut, the mode fields decay at infinity in all transverse directions, then it follows that any of the discrete modal currents can in principle be excited with arbitrarily high efficiency by taking a large enough truncated replica of the modal E-field to be the aperture field. If such excitation can be realized, then the total wire current will essentially be that of the mode under consider- ation for wire lengths of practical interest.

4. CONCLUSION

In the first part of this paper, a proof has been given that the current on a thin wire parallel to a conducting half space may be uniquely decomposed into discrete and continuous (radiation) modal components. In addition, any given nondegenerate discrete mode may in principle be excited with arbitrary efficiency by appropriately choosing an aperture field. Thus, although the radiation currents must always eventually dominate the discrete modal currents on a lossy structure such as this [ Wu, 1961 ], the latter can in principle be made to dominate over an arbitrarily long portion of the wire, so that the concept of modal representation will still have important utility in practical applica- tions. Application of the results of this paper to sources of various types is presently being carried out.

In general, the dominant portion of the fields or currents excited on the wire can be obtained by a suitable deformation of the integration contour in the a-plane and using steepest-descent techniques (in performing the contour deformation, the advan- tages of orthogonality possessed by the proper mode


spectrum are lost). This can sometimes involve passing over a nonspectral singularity, i.e., a pole lying in one of the improper Riemann sheets. This "leaky" mode would then form an important part of the total field or current (in the mode spectrum representation, this shows up as a concentration of the amplitudes of a section of the continuous spectrum). Since such poles have been found in the case of a two-wire line in free space [Marin, 1975], and related poles are found for a single bare wire over a highly conducting earth [Olsen and Usta, 1977], it may be of interest for some problems to determine their effect.

We take up a discussion of the properties of the discrete modes in the second part of the paper [Olsen et al., 1978]. The question of expandability of the total field in terms of discrete and continuous

modes has not been addressed here, since there are, in the thin-wire approximation, substantial parts of the total field which do not correspond to currents induced in the wire (for the latter, the mode spectrum used in this paper has been shown to be complete). Such a question could be important in studying scattering and discontinuity problems on wire-over-earth structures.

APPENDIX A

We seek Sommerfeld integral representations for the fields/•,/•, satisfying Maxwell's equations

V x • = io•o •q- J,,, (A1)

V x/• = -ito• • + Je

It is convenient to express the fields as Fourier transforms in z (see (5)) and to make use of the z components of electric and magnetic Hertz poten- tials only, U = rgez , V = rgmz , SO that [ Wait, 1972]:

•, = ik!(oLV t 0- 'l]O•zX V, •r}[ Pz--- kl2(Er- OL2) 0

(A2)

I• t = ikl{oLVt •r •l- (Er/•O)azX Vt 0), J•z = k2 ß ! ([•r 0/?) •r

(A3)

where •7 t denotes the transverse "del" operator (0 / 0Z: 0), Er: El/EO(: 1) or E2/E 0 (:/,/2) accordingly as x > 0 or x < 0 respectively. The derivation of Sommerfeld integral representations for • and l? is by now classical [Batios, 1966] and generally follows that of Wait [1972]. For a line source of

electric current Je = a: /•o(Z)8(•, - • •), the result for x > 0 is

no

4k, [2 {•2 [no(l)( •R ) -- H•o'•([R,)]

+ P(a;X + X',Y- Y') - ct 2 Q(a;X + X';Y- Y')}

(A4)

4k, [2 {P*(a;X + X', Y - Y') - Q*(a;X

+ x',Y- Y')}

where

Y= k,y; R= [(x-x') 2 + (y - y')2]'/2' X=k•x;

R, = [(x + x') 2 + (y -y,)2] ,/2,

2 Iøø exp(-u,X+iXY) P(a;X, Y) = -- dX l'lT U 1 -[- U 2

2 Iøø exp(-u,X+ihY) Q (Or[ X, Y) --- -- 2 dX l'rr n u I + U 2

(A5)

2 I oo exp(-u, X + iX Y) XdX P*(a;X,Y) = -- l'n' tl, + U 2

2 Iøø exp(-u•X+iXY) XdX Q*(a;X,Y) = 2 i•r n u• + u 2

and [o(a) is the Fourier transform of Izo(Z ). In the integrals we have defined

U, -- (•k 2 -- •2)1/2; U 2 = (X • -- [•)'/•; Re(u,,u:) > 0

in order that the integrals converge, and

(A6)

[ = (1 - 0t2)!/2; •n = ( n2 - øt2) 1/2' Im([,[,,) > 0

(A7)

for reasons discussed in section 3. We have made

use of the representation

i oo e--UlB +i•c dX = i-rr Ho(')[[(B 2 + C2)1/2] } Im[ _> 0

(A8)

in obtaining (A4). To obtain the averaged fields around the wire

surface, we first let p > p' = a, use the expansion of a plane wave in a sum of Bessel functions


[Harrington, 1961, pp. 231,232], and average over •'. Then the observation point is brought to the wire surface, and we average over • to obtain, after some algebra:

( ( t; cat) )! : --(k I 'I10 / 4)io(a)Jo([A) {[2 [H(ol)([Z) --

ß H{ol•(2[H)l + Jo(;Z)[P(a;2H) -a • Q(a;2H)]}

/ •scat )t {x--, ) = -(ikl/4)io(a)Jo(•A){•H[l)([ •) + Jl(•)

ß [-[H[ø(2[H) + (1/[)P(a;2H) - (a2/0 Q(a;2H)] } (A9)

where H = k, h, A = k, a, and P(a;X) and Q(a;X) are obtained from the corresponding functions in (AS) by sett•g Y = 0.

The evaluation of ' ginc•- - inc - ß and {H (Xo•;a)) • •Xo, a)) for (9) reduces to the averaging -• S e• '•e• Ol •o• •o • and •* by reason of (6) and (7). •o and are found from (A2)-(A4) by taking [o(•) • k,/2•, and so the averaging process can be carried out

= ez - - . as above. The result is that {Eo (x,; Xo•, + a))

= ez - - . and { H o (x• ;Xo•, + a)) are obtained from (A2)-(A4) except that y' = O, x' = h, and/o(a) • (A4) is replaced by (kl/2•V•([A). To carry out this proce- dure for •* and •*, it can be noted that the averaging process will give the fields for a loop of magnetic current of radius a' Jm = •, g(z - Zo)g(0 - a)/2•a. It is easily shown that for 0 > a this loop generates the fields given by (A2)-(A4), if we now replace /o(a) by -ik•J,([A)/2•o [ and once again set y' = O, x' = h.

APPENDIX B

In this appendix we demonstrate the orthogonality relations for the mode fields (24) and (26). The arguments resemble those given for related problems in [Collin, 1960, pp. 483-485; Bresler et al., 1958; Manenkov, 1970].

From their definitions (24) and (26), and (A2) and (A3), the transpose mode fields can be related to the direct mode fields by

(•)

It is easily verified that both the discrete and continuous mode fields satisfy (sourceless) Max-- well's equations outside the wire, the boundary conditions at the air-earth interface and, in our thin-wire approximation, the boundary conditions at the surface of the wire.

Starting from the Lorentz reciprocity relation, we have the following equation:

--' --ikl(01.1 -- O[2)• z ' (/• X A; -- /• ]• (B2)

where et• and a 2 can be either propagation constants for a discrete mode, or a value of a on one of the branches of continuous spectrum. Let us first suppose that one of a• and ot 2 corresponds to a discrete mode. Integrating (B2) over a'cross section $ of large radius in the xy plane and applying the divergence theorem to the left-hand side, we obtain a line integral around the boundary C of $ (whose outward unit normal vector is •n). Because of the exponential decay of the discrete mode fields, and the boundedness of the remaining fields, this integral tends to zero as C recedes to infinity. The remaining part of C on the circumference of the wire vanishes because the fields satisfy homogeneous boundary conditions there; thus, when S becomes the entire xy plane (excluding the cross section of the wire), we have

I ß + 2 2 d• •i- x I•f d2, = 0, 011 • 0[2 (B3) $

In view of (23) and (25), this is precisely the form of orthogonality we require.

It remains for us to show orthogonality between two continuous spectrum modes. For i, = (p,½b), p • m• a steepest-descent evaluation of the Som- merfeld integrals (AS) which eventually describe the continuous mode fields shows that (see, for example, Chang and Olsen, [1975]):

+

. .l(6;•)e '•'•0 + •..(6;•)e -'•.•")

(•4)

for x > 0, and are exponentially small for x < 0. An expression similar to (B4) holds on the third branch for x < 0, while • and • are exponentially small for x > 0. Finally, the modes on the second branch are associated with the interface and give

•:(x,'•)- {•}:(•)e +'•'•'y + d' -'•.•

(B5)

for x > 0, where


u,,, - [- 1/(n • + 1)] '/•; Re u,,, > 0 (B6)

and a similar expression, decaying with Ix I, for x<0.

It can be seen that for any pair of continuous spectrum modes from different branches, the contour integral obtained from (B2)again vanishes as C recedes to infinity and (B3) follows at once. For a• and a 2 both on the first branch, (B2) gives, for C a circle of radius R:

s

k,(a, - p=R

From (B4) and known forms for the Dirac delta- function [Shevchenko, 1971, p. 141] we obtain, on letting R --•

I a•. (• •- x/-•- - •- x/•)dS s

= [•(,• + ,•:)/2• •1 N(,•,, - ,•:)a(•,- •:) (B7)

where N(a•,a2) is a norm related to the far-field patterns of these modes. A variant of (B7) similar to (B3) can also be obtained. The orthogonality between modes on the third cut is established in

like fashion; that for modes on the second branch uses a contour C which is a large rectangle rather than a circle. The result in all cases resembles (B7).

Acknowledgments. This work is supported by the Rome Air Development Center (RADC/ET) under contract no. AFF19628-76-C-0099. The authors are indebted to J. R. Wait

and L. Lewin of the University of Colorado, and W. Rotman of RADC/ET for many interesting and profitable comments and discussions on the present work.

REFERENCES

Barios, A. (1966), Dipole Radiation in the Presence of a Conduct- ing Half-Space, 245 pp., Pergamon, New York.

Bresler, A.D., G. H. Joshi, and N. Marcuvitz (1958), Ortho- gonality properties for modes in passive and active uniform waveguides, J. Appl. Phys., 29, 794-799.

Chang, D.C., and R. G. Olsen (1975), Excitation of an infinite antenna above a dissipative earth, Radio $ci., 10, 823-831.

Chang, D.C., and J. R. Wait (1974), Extremely low frequency (ELF) propagation along a horizontal wire located above or buried in the earth, IEEE Trans. Commun., COM-22, 421-427.

Coilin, R. E. (1960), Field Theory of Guided Waves, 606 pp., McGraw-Hill, New York.

Flammer, C., and H. E. Singhaus (1973), The interaction of electromagnetic pulses with an infinitely long conducting cylinder above a perfectly conducting ground, Int. Note 144,

Air Force Weapons Laboratory, Kirtland AFB, New Mexico. Grinberg, G. A., and B. E. Bonshtedt (1954), Foundations of

an exact theory of transmission line fields (in Russian), Zh. Tekh. Fiz., 24, 67-95.

Harrington, R. F. (1961), Time Harmonic Electromagnetic Fields, 380 pp., McGraw-Hill, New York.

Kuester, E. F., and D.C. Chang (1976), Modal representation of a horizontal wire above a finitely conducting earth, Sci. Rep. No. 21, RADC-TR-76-287, Department of Electrical Engineering, University of Colorado, Boulder, Colorado.

Lavrov, G. A., and A. S. Knyazev (1956), Prizemnye i Podzemnye A ntenny (in Russian), 472 pp., Sovestskoe Radio, Moscow.

Ledinegg, E. (1970), Die Sturfenhornanregung der Eindrahtlei- tung als Transformationsvierpol, Arch. Elek. (}bertrag. , 24, 244-252.

Manenkov, A. B. (1970), The excitation of open homogeneous waveguides, Radiophys. Quantum Electron., 13, 578-586.

Marin, L. (1975), Transient electromagnetic properties of two infinite, parallel wires, Appl. Phys., 5, 335-345.

Miller, E. K., A. J. Poggio, G. J. Burke, and E. S. Selden (1972), Analysis of wire antennas in the presence of a conducting half-space, 2, The horizontal antenna in free space, Can. J. Phys., 50, 2614-2627.

Morse, P.M., and H. Feshbach (1953), Methods of Theoretical Physics, pp. 464-465, McGraw-Hill, New York.

Ollendorf, F. (1969), Erdstr6me, Chap. 7, Birkhausen, Basel. Olsen, R. G., E. F. Kuester, and D.C. Chang (1978), Modal

theory of long horizontal wire structures above the earth, 2, Properties of the discrete modes, Radio Scœ, 13, this issue.

Olsen, R. G., and M. A. Usta (1977), The excitation of current on an infinite horizontal wire above earth by a vertical electric dipole, IEEE Trans. Antennas Propagat., A P-25, 560-565.

Pistol'kors, A. A. (1952), On the theory of a wire near the interface between two media (in Russian), Dokl. Akad. Nauk SSSR, 86, 941-943.

Pistol'kors, A. A. (1953), On the theory of a wire parallel to the plane interface between two media (in Russian), Radio- tekhnika (Moscow), 8(3), 8-18.

Pogorzelski, R. J., and D.C. Chang (1977), On the validity of the thin-wire approximation in analysis of wave propagation along a wire over a ground, Radio Sci., 12, 699-707.

Shevchenko, V. V. (I971), Continuous Transitions in Open Waveguides, 176 pp., Golem Press, Boulder, Colorado.

Sunde, E. D. (1968), Earth Conduction Effects in Transmission Systems, 370 pp., Dover, New York.

Timischl, W. (1971), Ober die Bestimmung der auf einer Zylin- derantenne durch ein elektrisches Feld induzierten Stromver-

teilung, Acta Phys. A ust., 34, 251-263. Wait, J. R. (1972), Theory of wave propagation along a thin

wire parallel to an interface, Radio Sci., 7, 675-679. Wait, J. R. (1977), Excitation of a coaxial cable or wire conductor

located over the ground by a dipole radiator, Arch. Elek. Obertrag., 31, 121-127.

Wait, J. R., and D. A. Hill (1975), Propagation along a braided coaxial cable in a circular tunnel, IEEE Trans. Microwave Theory Tech., MTT-23, 401-405.

Wu, T. T. (1961), The imperfectly conducting coaxial line, Quart. Appl. Math., 19, 1-13.

Wu, T. T. (1969), Intrucuction to linear antennas, in Antenna Theory, Part 1, edited by R. E. Coilin and F. J. Zucker, pp. 306-351, McGraw-Hill, New York.

Documents

Modal theory of long horizontal wire structures above the earth