modal theory of long horizontal wire structures above the earth - part 2

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Radio Science,Volume 13, Number 4, pages615•623, July-August 1978

Modal heory f longhorizontal irestructuresboveheearth,2, Propertiesf discrete odes

Robert G. Olsen

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

Edward F. Kuester and David C. Chang

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

(Received July 13, 1977.)

The characteristics of the discrete propagation modes on horizontal thin-wire structures located

above a dissipative earth are investigated. In addition to the well-known transmission ine mode,

a new root of the characteristic equation is found which is identified as a surface-attached

mode becauseof its close connectionwith the Sommerfeld pole (Zenneck wave) in some parameter

ranges. Under many conditions the surface-attached mode suffers substantially less attenuation

along the propagation direction than does the transmission ine mode. Numerical investigation of

the propagation constants of the two modes is made, and field plots for the modes for a variety

of wire parametersare presented.

1. INTRODUCTION

In the first part of this paper [Kuester et al.,

1978], the excitation of current on an infinite thin

wire over a finitely conductingearth by an arbitrary

source was considered, and it was demonstrated

that a unique decompositionof the currents nduced

on the wire into discrete and continuous modal

components is possible. In this part of the paper,

we examine the •properties of the discrete modes

more closely and show that, due to the singularities

of the function M(a) (where M(a) = 0 is the

characteristicequation for these modes), there are

generally two modes in the neighborhoodof ot =

1, and that for some parameter ranges the new

surface-attached mode can have substantially

lower attenuation that the well-known transmission

line mode. It is also found that, for some particular

values of the physical parameters, the two modes

have identical propagationconstants. Such a double

root of the characteristicequation s called a degen-

erate mode, and under these operating conditions,

conventional transmission ine theory breaks down

entirely.

Most of the existing literature on the wire-over-

earth problem has been in the context of ordinary

or modified transmission ine theory. However, at

the beginning of the present authors' investigations

Copyright 1978by the AmericanGeophysical nion.

(a preliminary announcement of whose results

appeared n 1974 [Olsen and Chang 1974]), it was

discovered that the behavior of the transmission

line mode and of a single spectrum of radiation

modes was inadequate to describe the input con-

ductance of an infinite antenna over a conducting

half space [Chang and Olsen, 1975]; it failed to

oscillate about the free-space value as the height

of the wire above the half space continued to

increase. Only by including the effect not only of

a second discrete mode, but also of an additional

branch of radiation modes was it possible to obtain

the correct behavior. Regardless, therefore, of

whether or not it proves easy to excite the new

mode individually or not, its effect must be ac-

counted for in order to achieve a proper under-

standing of wire-over-earth structures at high fre-

quencies.As in Part I of this paper, the frequencies

of interest are such that wires longer than 10 rn

or so are electrically long. Therefore, end effects

and discontinuities can be analyzed independently

(e.g., by Wiener-Hopf methods) and used to study

wires which are long but finite [Olsen and Chang

1976].

2. SINGULARITIES OF M(a)

In equation (16) of Kuester et al. [1978], it was

shown that the discrete propagationmodes satisfied

the characteristic equation M(a) = O, where

0048-6604 78 0708-0615501.00 615



616 OLSEN, KUESTER, AND CHANG

M(•x)= Mo(•X- i• s o0M (o0/'qo• (1)

Mo o = •2 H•o) (•A) - Jo •)H• • (2•H)]

+ P(a;2n)- a:Q(a;2H) (2)

Mi (•) = •2n(ll••) -- J• •) {•2n•l' 2•n)

- e(a;2n) + a:Q(a;2n)} (3)

• = (1 -- a2)1/2; n= (n2 - a2) /2; Im •, Im •n• 0

and H = klh, A = kla are the heightand radius

of the wire normalized by the free-space wave-

number , w•e n is the complex efractivendex

of the earth. A propagationactor of exp(iklaZ --

i•t) is associatedwith the mode.

The surface mpedance (a) of the wire will

be analytic for all finite a since it is associated

with a structure bounded in the transverse plane.

Only the analyticityof P and Q, which were defined

• Part 1. as Fourier •tegrals with respect to a

variable •, remains to be determined. We recall

that P and Q are actually special cases of a set

of functions which determine the potentials n terms

of which all electric and magnetic ield components

above he earth canbe expressed,and are the values

at Y = 0 of

2 •ø xp(--UlX+XY)

(•;x, ¾)= •

lqT U 1 -[' U2

dX (4)

2 o xp(--UlXX )

(a;X, Y) = • •

lqT n u I -[- U2

where

dX (5)

Ul _- (•k2 _ 2)1/2} U --' •x -- •2n)1/2'Re Ul,U >' 0

(6)

Here X - klX and Y- klY are normalizedcoordi-

nates in the plane transverse o the wire.

It is required that branch cuts be taken in the

complex a plane to ensure hat fields and currents

given by Fourier integrals in the a plane decay

properly at infinity (since the sourceof excitation

considered by Kuester et al. [1978] is finite),

regardlessof how the a-plane integration path is

deformed. These cuts will define our proper Rie-

mann sheet. Any roots of the modal equation ound

in this sheet will correspond to discrete modes

whose fields decay as the distance from the wire

approachesnfinity in any transversedirection,and

will be denoted proper modes.

X-PLANE

+•

Fig. 1. Integration aths or Sommerfeldntegralsn X plane.

Let us consider irst P(a;X, Y). If [ variesaround

a semicirclein the X plane) = eiø,0 <_0 <_

,r, then the branch cuts of its integrandat _+[ in

the X plane move as indicated n Figure 1. Now

the integrand s the same along the contour of

integrationC for 0 = 0 and 0 = ,r except or the

segment etween [ and +[, where the signsof

u are opposite, nd thereforeP takes different

values for arg [ = 0 and arg [ = ,r. However,

the semicircle racedby [ correspondso a complete

loop n the X-plane roundoneof the branchpoints

a = + 1 (Figure 2). Since+ [ cannotactuallycross

over the real axis of the Xplanewhile the integration

contour C remains on the axis, the lines Im [ -

0 shown n Figure 2, whichare alreadybranchcuts

for the Hankel functions, are likewise cuts for

P(a;X, Y). By similarargumentst can be shown

that P possesses ranchcuts along the lines Im

(l - PLANE

/•- Im n=O

Fig. 2. Locations f a in complexplane;branchcuts m • =

0 and Im •n = 0.



LONG HORIZONTAL WIRE STRUCTURES, 2 617

[, = 0. It is easily seen that the integrand for P

has no poles, and so we have completely charac-

terized its singularities.

Precisely the same argumentsas for P show that

Q also possesses he branch cuts Im [ = 0 and

Im [, -- 0. There is, however, also a pole in its

integrand, which may or may not appear in the

Riemann sheet in which the integration is taken.

The pole locations (there are two, symmetrically

located) are given by:

X= +hi,---- •B= +[n2/(n2+ l)-a 2] /•

-= [,• - ,•1 '/• (7)

where, or definiteness, e take X•, thusalso

a a, and the square roots to have nonnegative

imaginarypart. The notation a for the pole ocation

is chosen to emphasize he similarity to [ and [,,.

For a lossy earth, with 0 _< arg n _< 'rr/4, both

poles are indeed in the proper Riemann sheet

for all values of a. The intepretation of this pole

when 0/Oz = 0 (i.e., a = 0) is well-known [Batios,

1966]' it is the Sommerfeld-Zenneck urface wave

associatedwith the lossy nterface. When a differs

from zero, it is clear that the meaning of the pole

is that of the Zenneck wave travelling at some

(complex)angle with respect o the z axis in the

yz plane (the plane of the air-earth interface) so

that the z componentof the normalizedwave vector

is +_a,and they components +X•, recallinghe

expressions or the fields from Kuester et al.

[1978]). We emphasize hat in spite of the fact

that this portion of the total field may not dominate

in any given range, neither can it be neglected,

for it is often at least the same order of magnitude

as the entire field, even for dipole sources [Barios,

1966].

Let us then tracethe semicirclea = I[•l eiø•

0 _<0• _< rr.The polesmove as indicated n Figure

3. If n is not real, so that [ is not real simultaneously

with [a, then the values of Q corresponding o

0 = 0 and0 • = 'rrdiffer precisely y the difference

of the pole residuesat _+ [a. Since the semicircle

traced by [a corresponds o a complete oop in

the a plane aroundone of the points +-a, it follows

that in addition to _+ and +_n, he function Q(a;x, Y)

must also possessbranch points at _+a . For Y

_>0, Q may be evaluatedby deforming the h plane

contour of integration upward over the two branch

cuts; for Y _< 0 by deforming downward; in both

cases the residue contribution from the pole can

•-PLANE

+•j s (0= •.)

ß

+/JB 0 =

+/jB (O=O)

Fig. 3. Integration paths for Sommerfeld ntegrals n X plane

relative to pole location.

be computed explicitly to be

exp(--Ulp + iX•, rl) <8>

•B n4- 1

where l• = i/(rt q- 1) /2 -- i//•, the ootbeing

chosen o that Re(Ulp) ->0. It shouldbe noted

that if the integrationcontourwere indented o allow

the poles o come between t and the real axis (i.e.,

if we allow 0a > 'rr) the residue contribution 8)

would give wavesgrowingexponentially s I YI •

0%and would correspond o values of a no longer

on what we have designated the proper Riemann

sheet. inceor large n , (8)containsfactor 3,

it is tempting to drop it in approximatingQ, but

it should be noted that this term contains an inverse

square oot singularityas a --->+_ a• and cannot,

in general, be neglected.

Using these or analogousmethods,all the func-

tions of a defined as Sommerfeld integrals with

respect o X, whichwere encounteredn connection

with the excitation problem of Kuesteret al. [1978],

are found to have branch-cut singularities with

branch points at +_1, +_n,and +_et. In closing his

section, we should note that an inverse square-root

singularity imilar o that n (8) wasalsoencountered

recently n an analysisof open microstrip [van der

Pauw, 1976], and s due in that case o the influence

of a pole corresponding o the lowest-order TM

surface wave of a groundeddielectric slab.

3. NUMERICAL SOLUTION OF CHARACTERISTIC

EQUATION

In a numerical search for solutions of M(a) =

0 it is very time-consuming o repeatedly compute




the Sommerfeld integrals P and Q by numerical

integration, since this must be repeated for each

different value of ot tried. We have developed a

number of accurate analytical forms for P and Q

which employ series expansions aking much less

computer ime. Detailed derivationswill be present-

ed elsewhere; we give the most useful expressions

in appendixA. An essential eature is that the proper

singularitiesnear [ - 0 and [B - 0 (where the

most important modes are expected to be found)

are retained.

We consider first the bare, perfectly conducting

wire, for which s(a) = 0. Since or Inl- 0%

P and Q vanish, a solution of M o •) = 0 should

be expectednear • = 1 if Inl is large. This solution

should correspond to the well-known result of

transmission line theory first given by Carson

[1926]. Since, however, near • = aB, Q could

conceivably dominate the characteristicequation,

it also seems possible that a mode could exist in

this region, as well, which would not be found if

Q (or more specifically (8)) were neglected. Using

these rough estimatesas initial guesses, he charac-

teristic equation was solved usingNewton's method

to converge to the exact roots.

In Figure 4, results are shown for a relatively

highly conductingearth. The refractive index n used

here corresponds o a relative permittivity e/e o •-

10 and a conductivity of • •- 2.8 millimho/m at

0.02

O.OI

_ 0'2

0.3

0.2

• 0.3

1.00 I.O •

Re a

Fig. 4. Mode propagation constants or bare wire above lossy

earth as function of h/h: n = 5.52 + i4.53, a/h = 0.01.

a frequency of 1 MHz. In this case, as indeed was

found in all but one of the cases we studied, two

modeswere found. At large wire heights,one mode

is found in the vicinity of a a, while another is

found nearer • = 1. The latter can be shown by

perturbation arguments to agree quite well in this

parameter range with Carson's [ 1926] result, to

which a number of authors have made extensions

[ Wise, 1934, 1948; Kikuchi, 1957;King et al., 1974].

A mode closer to a a than to 1 will be designated

surface-attached because of the influence of the

pole in the Sommerfeld integral Q, while in the

oppositecase we use the general headingof struc-

ture-attached, indicating fields not spread out

along the surface. It is to be recognized that •

for the structure-attached mode approaches 1 as

h --> 0% in which limit it no longer functions as

a pole in the excitation problem, because of the

logarithmic singularity possessedby M(a) at [ =

0. This, of course, is known from the theory of

long antennas [Wu, 1969]. As the height of the

wire decreases, he structure-attachedmode gradu-

ally becomes surface-attached, while the surface-

attached mode becomes another kind of structure-

attachedmodewhosepropagationconstant for very

small h) is predicted by a result of Coleman [1950]

[see Wait, 1972;Chang and Wait, 1974] to approach

ot --- (n q- 1)/2] 1/2.The potential sefulnessf

the surface-attachedmode at low heights because

of its much smaller attenuation constant is evident.

Clearly, as a result of continuously varying the

height , a propagationonstant%,which s ocated

in the surface-attached region of the a plane can

move into the structure-attached region, and vice

versa. It is thus not possible to label a single mode

trajectory as having either of these properties.

Figure 5 examines a case where the earth is less

lossy. This refractive index corresponds more

closely o a higher requency, and somewhatdamper

earth, with e/eo -• 27, and • -• 56 millimho/rn

at 100 MHz. When the wire radius is 10 3 wave-

lengths (curves 2), the qualitative behavior of the

two modes s much the same as in Figure 4, although

the new location of a a has distorted the curves

somewhat. When the radius is decreased to 1.66

X 10 4 wavelengths,owever curves ), an inter-

esting phenomenon is seen to have occurred. The

structure-attachedmode at low heights passescon-

tinuously to a = 1 as height is increased without

ever attaining a definite surface-attached character.

The surface-attached mode, on the other hand,




0.02 0.1

/ o.,//

/ /

O. Ol

I

o| i I

0.99 1.00

Fig. 5. Mode propagation constants for bare wire above 1ossy

earths unctionfh/h:n = 5.3+ iO.95'Oa/h 1.66

X lO-4,Qa/h.-'-.OX 103.

retains this property to some extent for all heights,

remaining n the vicinity of as. At some value of

a between these two, there must be a situation

wherein the two curves touch for some value of

h (probably around h /h = 0.15). This phenomenon,

where two roots of the modal equation become

one double root, is known as modal degeneration.

Examples of it have previously been found in the

earth-ionosphere waveguide [Krashnushkin and

Fedorov, 1972; Budden, 1961 and in the earth-crust

waveguide [ Wait and Spies,1972]. The full physical

basis for such degeneracy does not seem to be

understood at present, but some discussion of it

will be given in the concluding section. It should

be noted, however, that the exact degeneracydoes

not have to be achieved for the effect to become

noticeable; if the phase difference between two

propagating modes over a length I of wire has not

become significant compared to 2vr, the results will

be much the same as if they were degenerate.

Figure 6 shows results for a dielectric coated

wire (Goubau line) over the same relatively less

lossy earth of Figure 5. The dielectric has radius

a and refractive index nc, and the inner conductor

has adius . The surfacempedances (a) for this

case is given by Chang and Wait [1974]. Once

again, this time as a result of varying the refractive

index of the coating, we see that modal degeneration

has occurred, at nc somewhat less than 1.1, and

h/h about 0.25. The general behavior is much like

that of the bare wire, with the exception that the

structure-attached mode for large heights ap-

proaches a value of a larger than 1 on the real

axis, corresponding o the TM surface wave of the

G-line in free space.

It can thus be seen that Kikuchi's description

[Kikuchi, 1957; Chiba, 1977] of the continuous

transition between an earth return mode and a

surface wave mode can be modified by the existence

of a degeneracy. For sufficiently large coating

index, a variation in height does indeed produce

the transition between the structure-attached mode

and the surface wave. As the coating becomes less

dense, however, the continuous transition is inter-

rupted by the acquisition of surface-attached

properties.

4. FIELD DISTRIBUTIONS OF THE MODES

The fields of the discrete modes, the expressions

for which can be found by Kuester et al. [1978],

have been plotted for some special cases. For the

surface-attached modes it was found that much

more of the field shows up above the wire than

below it (and hence in the earth), as comparedwith

a structure-attached mode, so that the reason for

0.02

0.01

/5

0.2•

0.99 1.00

Fig. 6. Mode propagationconstants or Goubau ine above 1ossy

earth as function of h/h for various coating indices: n = 5.3

+ i0.95,/h = .006, /h= .005;h = 1.O, nc = 1.04,

Qnc= 1.1,Qn= 1.11,Ca= 1.25.




the smaller attenuation is simply that a smaller

percentage of the mode fields actually travel in

the earth.

In Figures 7 and 8, cross-sectional ield plots

are given for the two modes on a Goubau line above

the earth. The greater transverse spread of the

surface-attached mode as compared with the struc-

ture-attachedmode (in this case, predominantly he

G-line surface wave) can be seen. The fields of

the latter are all nearly linearly polarized, with the

exception of points very far removed from the wire.

On the other hand, the fields of the surface-attached

mode are elliptically polarized near the wire, but

more linearly polarized near the earth's surface,

where the E field is also more vertical, and less

attenuated with distance from the wire. Plots such

as these can be used in conjunction with the

excitation discussion of Kuester et al. [1978] to

design an efficient excitation scheme to select one

of the discrete modes in preference to the other.

This is a necessarydevelopment f practical utiliza-

tion of these modes is to be achieved.

Some further insight into the characteristics of

the modes can be obtained by examining their

asymptotic behavior in the transverse plane. Once

again, since the procedure for all field components

4.7

- I

y/X -0.2

9.0

8.5/13.0

I

-0.1

WIRE

0.2

0.1

00

Fig. 7. E-field lines for structure-attachedmode on Goubau ine.

Arrows give field strength along direction of major axis of

polarizationellipse. Numbers A /B are ratio of major to minor

axis, n = 5.3 + i0.45, h = 0.24h, a = .007h, b = .0lb, nc

= 1.25, ot = 1.0123 + i0.01134.

9.8

4.3

2.0

1.5

2.2

6.7 I0.0

I

WIRE

0.2

A/B=•o

. I

y/x -o.2 o

Fig. 8. E-field lines for surface-attached mode on Goubau line.

Arrows give field strength long directionof major axis of

polarizationllipse.Numbers /B are ratioof major o minor

axis,n = 5.3 + i0.45, h = 0.24h,a = .007h,b = .0lb, nc

= 1.25, ot = .9854 + i0.00577.

is similar, we shall consideronly the integral (5)

for Q(tx;X, Y). The asymptoticbehavior of Q is

foundfrom the propertiesof the functionappearing

in the exponent:

f(h) = -i(h cos • + iulsin •))

(9)

where 4) = arctan (X/Y), and the exponentof

(5) becomes(X 2 + y2)l/y(•k). Sincewe have

0 _• 4) -• 'rr for the integral (5), it is clear that

the saddlepoint h• for f(h) (where '(h•) = 0) is

located at • - • cos 4). As is well known [Felsen

and Marcuvitz, 1973], by deforming he integration

path into a steepest-descentath (f(

2

= s, -oo ( s (+oo), the major contribution o

(5) is made to come from the vicinity of the saddle

point, and s dominated y a factor exp[-(X 2 +

y2)l/•f(h•) = exp[i•(X2 + y2)1/2].Some om-

plication arises from the pole in the integrandof

(5), however, and different cases can arise.

Let us consider mode ocated t %1 such hat

I•al is smaller han I•[, and thus we identify this

as a surface-attachedmode. The singularitiesat

•, and • are shown n Figure 9, and the saddle

point lies on the dashed ine between +• and -•.

At 4) = 'rr/2 (observation point directly over the




(•) xPLANE

ß

.

•' « ARG2 •'/4

Fig. 9. Steepest-descent aths in h plane for fields of discrete

mode%,.

wire) the steepest-descent ath (SDP) is located

at curve 1 in Figure 9, and no influence of the

pole s felt. At some ntermediate ngle b, he SDP

is locatedat curve2, andthe pole at h•,hasbeen

picked up (the detailed structure of the field in

this vicinity must be describedby Fresnel ntegrals

[Felsen and Marcuvitz, 1973]). Finally, near {b =

0, curve 3 is reached (observation point near the

surfaceof the earth), and, depending n the mutual

dispositionf • andh•,, he residueermcan be

as large as, or actually arger than, the saddlepoint

term, and thus the ground wave term due to this

residue forms the major part of the field of the

mode. Note that it is the difference in relative

locations of pole and branch cut that make this

possible, n contrast o the situationof spacewave

and ground wave for a Hertzian dipole. source

[BaHos, 1966].

A mode%2 such hat I<l is smaller han I<l

would be designated structure-attached, and the

situation in the h plane is shown in Figure 10. The

SDP's for the same angles as in Figure 9 are again

shown. In this case, however, not only is the pole

not picked up until a much smaller angle, but its

contribution is much smaller than that of the saddle

point, and it never forms the dominant part of the

field.

5. DISCUSSION AND CONCLUSION

A numerical solution of the characteristic equa-

tion for a thin-wire structure located parallel to

a lossy earth has shown the existence of a second,

heretofore unknown, propagating mode in the

proper Riemann sheet of the •x plane (i.e., whose

fields decay at infinity in all directions normal to

the wire). Since the only approximation involved

in the derivation of this equation was a thin-wire

assumption [Kuester et al. 1978], it must be verified

that the location of the second mode is not simply

due to this approximation. Pogorzelskiand Chang

[1977] have investigated the error involved by

including angularly distributed and azimuthally di-

rected currents around the wire surface, and show

that their effect is negligible for all •x except within

very small circles surrounding (x = •x and (x =

1. It can be verified using these criteria that none

of the data presented n this paper fall within these

circles, and so proximity effects contribute a neg-

ligible correction to our results.

The possibility of modal degenerationwhich has

been demonstrated by our results warrants some

additional discussion of the phenomenon. The de-

generacyof the kind found here is, in the language

of mathematical linear operator theory [Kato,

1966], algebraic but not geometric. This should be

distinguished rom the related phenomenonwhich

occurs, for instance, in closed circular waveguides

between TE and TM modes. The latter degeneracy,

which is both algebraic and geometric and could

be designated semi-simple following Kato

[1966], is accidental, a result of the high symmetry

of the waveguide. The fields of these modes,

however, due to their distinct polarizations, remain

mutually orthogonal even at degeneracy, and

sources can be constructed to excite one of the

modes at the expense of the other. There remain,

X-PLANE

G:) :

(•)/•/2-•

•/2-

'\\

Fig. 10. Steepest-descentpaths in h plane for fields of discrete

mode%,2.




in this case, two orthogonal ½igenfunctionsmodal

fields) correspondingo the•degenerate igenvalue.

From the construction of Kuester et al. [1978],

however, it is seen that as two modes of the present

structureapproachdegeneracy, heir fields tend to

become identical, and that at such a nongeometric

degeneracyonly onemodal field remains. However,

an adjoinedmodal ield appears ecauseheGreen's

function resolventkernel) possesses doublepole

at the degenerate igenvalue; his can alsobe viewed

as the interference produced by two distinct but

almost degeneratemodal fields.

Other researchers [dos Santos, 1972; Levitt and

Marx, 1974] have found a second mode on an

improper Riemann sheet (though their definition

seems to be different from ours). In Figure 11,

we compare our results (this time for fixed, un-

normalizedphysicaldimensions,dielectricconstant

of the earth, and conductivity)against hose of dos

Santos [1972] as a function of frequency. What

dos Santoscalled a proper mode agrees ather well

with our structure-attachedmode (the data point

at 12 MHz appears o have been misprinted n dos

Santos paper, and we have made what seems to

be the correct modification to the real part of a).

0.06 -

0.05

o.o4-

0.02 - AI2

148'-:'e'-'•x/3'

o.o, z

ol %,7o . ,

0.97 0.98 0.99 1.00 1.01 1.02 1.03

Rea

Fig. 11. Mode propagationconstants or bare wire above 1ossy

earth as functions of frequency. The dashed curve indicates

the motion of the branch point otB with frequency. a = 2.5

x 103 m, h = 1.0m, e/e = 15, = 10-2mh_o/m;

Carson-Colemanode,Qdosantos'roperode,{•)surface-

attached mode.

The discrepancy, which was largest at 6 and 12

MHz, can be attributed n part to readingdos Santos'

results from graphs, and from tables of numbers

having only three-decimal-place accuracy, as well

as to possibledifficulties in his numerical procedure

for integrating P and Q. Our surface-attachedmode,

however, has no analog in dos Santos' results; his

improper mode is even further removed from the

branch point at a• than is the structure-attached

mode, and does not show up on the portion of

the a plane exhibited in Figure 11. It is well worth

noting that at around 215 MHz, the surface-attached

mode disappears nto the branch cut Im [ - 0 and

passesover to an improper Riemann sheet. Although

by this time the wire is about 2/3 of a wavelength

over the ground, and the ground effect might be

expected to be small anyway, it is of interest that

the surface-attached mode does not always show

up as a proper mode, as in Figures 4-6. An exhaus-

tive search of all possible Riemann sheets for

improper modes has yet to be made, as does a

study to determine which, if any, of these can

contribute significantly to the field of a finite source.

APPENDIX A

The most useful formulas for P and Q valid for

the range of parameters examine in this paper are

these: For P,

P(ot;X) • (2•2n/N2)(l/[.X){- [(IX) 2/[.X]Ho(I)([X)

+ [i[XH(l ) (IX)- 2/-rr] 1 - i/[.] + al(--i[nX )

-- Yl(-i[nX)- 2i/'rr[nX } (A1)

for I /In I and IXI not too large, where H• is the

Struve function of first order [Abrarnowitz and

Stegun, 1965, p. 496]. For Q,

Q(a;X) = Qo(a;x) + Ql(a;X)

under similar conditions, where

Qo(o•'X)2rt2//•N2)[H•ol)(•X)'1-ig/h)e•x/a

2arcsin(X•,/•

He l)(1 h•,•X) + --

• (x,,/0

QI (oL;X)- (ih [/N 2h2n2)H(l) IX)

and

He(l•(a, ) = e 'H(o) t)dt

o

(A2)

(A3)

(A4)




is known as an ncompleteLipschitz-Hankel ntegral

or Schwarz function [Agrest and Maksirnov, 1971;

Luke, 1962] for which seriesexpansionsare availa-

ble. In all the above, N 2 = n2- 1, h: = n: +

1, with Im (N) and Im (•) taken positive.

Results or solutionsof the modal equation using

these approximations have been checked against

those presented n this paper, which were obtained

using exact values of P and Q (computed by

numerical integration). For wire heights up to a

third of a wavelength, with n = 5.3 + i0.45, the

exact and approximate values of a differed at most

by 10 5 in either eal or imaginary art, andwere

frequently s close s 10 7 When he wire height

approached half wavelength, he error approached

10 4 (whichcouldgive substantialrrors n the

attenuation constant); however, alternative repre-

sentationswhich are accurate for larger heights can

also be derived. The solution of the characteristic

equationo withinan accuracy f 10 s tookabout

5 sec on a CDC 6400 when the Sommerfeld integrals

P and Q were integrated numerically, while only

0.1 sec was required when the approximate expres-

sions above were used.

Acknowledgments. This paper represents results of research

carried out since 1970 under partial support from the following

organizations: he US National Oceanic and Atmospheric Ad-

ministration (NOAA) under grants E22-58-70(G) and N22-126-

72(G), and Rome Air Development Center (RADC/ET) under

contract no. AFF19628-76-C-0099. The authors are indebted to

J. R. Wait and L. Lewin for many interesting and profitable

comments and discussion on the present work. Some of the

numerical results were obtained by S. Plate.

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modal theory of long horizontal wire structures above the earth - part 2

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