NUSC Technical Report 6511 (EL18 September 1981

Image Theory EM FieldsOf Horizontal Dipole AntennasIn Presence of Conducting Half-SpacePeter R. BannisterSubmarine Electromagnetic Systems Department

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NOV 2 1981

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Naval Underwater Systems Center0 Newport, Rhode Island I New London, Connecticut

Apveeed for public rolosp; distribution unlimited.

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Preface

This report was prepared under NUSC Project No. A59007, "ELF Propagation"(U), Principal Investigator, P. R. Bannister (Code 3411); Navy Program ElementNo. 11401 and Project No. X0100, Naval Electronic Systems Command, Com-munications Systems Project Office, Dr. D. C. Bailey (Code PME 110), ProgramManager, ELF Communications, Dr. B. Kruger (Code PME I 10-XI).

The Technical Reviewer for this report was Dr. Rene Dube.

Reviewed and Approved: 18 September 1981

David F. DenceHead, Submarine Electromagnetic Systems Department

The author of this report is located at the New LondonLaboratory, Naval Underwater Systems Center

New London, Connecticut 06320.

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OIMAGE T6EORY F FIELDS OF HORIZONTAL gIPOLE'NENAS IN PRESENCE OF CONDUCTING HALF-SPACE.

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It KI1? WORDS11 #Cuudsus on r awl fnfem and kdsaifr by bbwh mHorizontal Dipole AntennasImage TheoryFirtl Conducting Earth

21F. AWM04M .. uasu . . W.f mw. ai Wow by Wh. amebwoIt is the purpose of this report to extend the use of quasi-static

range finitely conducting earth-image theory techniques to the nearfieldand farfield ranges. Simple engineering expressions for horizontalelectric and magnetic dipole antennas have been derived for the air-to-air,subsurface-to-air, air-to-subsurface, and subsurface-to-subsurface propaga-tion cases. These expressions have been compared with previously derivedanalytical (or numerical integration) results. For the air-to-air propaga-tion case, the image-theory exressionsare valid from the Quasi-statice

00,00N 1473 (J

_________ ~~~1 T_______ __ ~ >

--

1 2. (Cont'd)

to the farfield ranges, as long as the square of the index of refractionis large and the Somnerfeld numerical distance is small. For the sub-surface-to-air, air-to-subsurface, and subsurface-to-subsurface propagationcases, the additional restriction that the measurement distance be greaterthan three times the burial depth of the source and/or receiver must bemet.

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TABLE OF CONTENTS

Page

LIST OF TABLES........................ .. .. ...... . .. .. .. .. . . . ..

INTRODUCTION. .. ..................... .. .. .. .. ....

LLOCATION OF IMAGE DEPTH .. ......................... 2HED AIR-TO-AIR PROPAGATION. .. ....................... 5

HMID AIR-TO-AIR PROPAGATION. .. ....................... 9ISUBSURFACE-TO-AIR PROPAGATION. ........... ........... 11AIR-TO-SUBSURFACE PROPAGATION. ........... ........... 11

SUBSURFACE-TO-SUBSURFACE PROPAGATION .. ......... ......... 17

COMPARISON WITH EXACT SOMMERFELD INTEGRATION RESULTS .. .......... 17

EXTENSION TO LARGE NUMERICAL DISTANCES .. ............ ...... 22

CONCLUSIONS .. .............................. 23

REFERENCES. .. ............................ .. 24

LIST OF ILLUSTRATIONS

Figure Page

1 Image-Theory Geometry .. .......................

2 Replacement of a Finitely Conducting Earth With aPerfectly Conducting Earth at Depth z

. . . . . . . . . . . . . . . . . . . .

3 Comparison of Image-Theory and Exact SommerfeldIntegration Results for Cr = 0 Z I S/rn,i= 10 mn, e = 100, *=00, and f = 3 to 30 MHz .. .... .. 20

4 Comparison of Image-Theory and Exact SommerfeldIntegration Results for er = 10, al = 10-2 S/rn,R= 10 m, e - 100, *=00, and f = 3 to 30 MHz .. .... .. 21

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Table ~LIST OF TABLES Pg

1 MED Air-to-Air Propagation Equations .. .... .......... 10

*2 END Air-to-Air Propagation Equations .. .... .......... 12

*3 HED Subsurface-to-Air Propagation Equations[R p 2 + Z2, R p2 + (d + Z)2 ].. ............ 1

*4 HMD Subsurface-to-Air Propagation Equations[R 2 =P 2 + Z2 R =2 ( ) 2 ]..... ......... 1R+ h2 P 2 + (d + ).. .. .. .. .. .... 15

6 MMD Air-to-Subsurface Propa ation Equations[(I2= p2 + h2 , (R!)f = p2 + (d + h)2] .. ......... 16

7 HED Subrc-to-Subsurface Propagationain

8 HED Subsurface-to-Subsurface Propagation

Equations (p? p 2 + d2). .......... ........ 19

F Acc *

-i3o For

f. L iVoo

if

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IMAGE-THEORY ELECTROMAGNETIC FIELDS OF HORIZONTAL DIPOLEANTENNAS IN PRESENCE OF CONDUCTING HALF-SPACE

INTRODUCTION

During the past several years, finitely conducting earth-image theorytechniques have proved quite useful in determining the quasi-static fields ofantennas located near the earth's surface for both single-layered and multi-layered earths. (For detailed references, see Bannister. 1 ,2 ) The quasi-staticrange is defined as that range where the measurement distance is much less thana free-space wavelength.

Physically, the essence of the quasi-static range finitely conductingearth-image theory technique is to replace the finitely conducting earth by aperfectly conducting earth located at the (complex) depth d/2, where d = 2/y,and y1 = [iWU 0(al + iIel)]I/2 is the propagation constant in the earth. (Seefigure 1 for the image-theory geometry.) Analytically, this corresponds toreplacing the algebraic "reflection coefficient," (u1 - X)/(Ul + X), in theexact integral expressions by exp(-Xd), where X is the variable of integra-tion.3 For antennas located at or above the earth's surface, the generalimage-theory approximation is valid throughout the quasi-static range.1,2

RECEIVER

SOURCEo

h 7 AIR

V EARTH

'""

IMAGE.

Figure 1. Image-Theory Geometry

d/2 .

~1 * - * - _ _ _ _ _ _ _ _ _.... _ _ _ _ _ _ _ _ _

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The major disadvantage of the finitely conducting earth-image theorytechnique is that mainly it has been applied only in the quasi-static range.However, because this range includes the critical launching area, one wouldexpect to be able to extend it into the intermediate range where the principalfield propagation proceeds as though it were over a perfectly conducting plane.

Recently 2 ,4 we have shown, for horizontally polarized sources, that 6finitely conducting earth-image theory techniques are not limited to the qua-si-static range alone. That is, by replacing the horizontally polarizedalgebraic "reflection coefficient," (u1 - u,)/(ul + uo), by exp(-uod), wedemonstrated that finitely conducting earth-image theory techniques can beutilized at any range from the source. Mohsen5 has validated and extendedthese results to include higher order terms that correspond to multiple imagesat the same location. Mahmoud and Metwally,6 employing discrete and discrete-plus-continuous images, have computed satisfactorily the change in the inputimpedance of a vertical magnetic dipole (VMD) due to the presence of the earth.

It is the purpose of this report to show that nearfield and farfieldrange finitely conducting earth-image theory techniques also can be employedfor determining the fields produced by horizontal electric dipole (HED) andhorizontal magnetic dipole (HMD) antennas (which are a combination of verti-cally and horizontally polarized sources).

In this report, the HED and HMD are situated at height h with respect toa cylindrical coordinate system (p, ,z) and are assumed to carry a constantcurrent I. The HED (of infinitesimal length Z) is oriented in the x directionwhile the axis of the HMD (of infinitesimal area A) is oriented in the ydirection. The earth, which is assumed to be a homogeneous medium with con-ductivity a1 and dielectric constant I(= crco), occupies the lower half-space(z < 0) and the air occupies the upper half-space (z > 0). The magnetic per-meability of the earth is assumed to equal p., the permeability of free space.Meter-kilogram-second (MKS) units are employed and a suppressed time factor ofexp(iwt) is assumed.

LOCATION OF IMAGE DEPTH

Basic antenna theory tells us that the fields produced by a current-car-rying wire of any length, when placed over a perfectly conducting earth, canbe represented by the combined fields of the wire and its image. 7 If thefinitely conducting earth could be replaced with a perfectly conducting earthat some specified depth below the surface of the finitely conducting earth,we then could use standard image theory to locate the antenna image and theresulting fields.

Several methods are available for deriving the depth of a perfectly con-ducting plane that can be used to replace a finitely conducting earth. One ofthe most general methods8 is to equate the wave impedances for grazing inci-dence at the surface (z = 0) for the two cases shown in figure 2.

2

~ -- - - M-RON

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CASE A CASE B[

AIR AIR ____

FINITELY CONDUCTING EARTH AIR Z1

...............I.I.I.I.I.I......................... I . ..................

PERFECTLY CONDUCTING EARTH

Figure 2. Replacement of a Finitely Conducting Earth With aPerfectly Conducting Earth at Depth z1

For case A, for transverse electric (TE) propagation,

where y2 =2 iwu 1 Yo and

a W 1 e'op Y2 00 iwe1),an

=n ' (2)For case B, we can write

z no tanh(yoz1 ) ,(3)

where no = Ioe - 1207r

For small values of z1 (i.e., jy~zjf < 0.5), tanh(yoz1 ) -yoz 1, and

ZB - n~yz iajij z1 (4)

Equating the two impedances results in

zi - 1 F77 1 ~.(5)

I -.. 1oy3

..., "-, .' . . .. . l

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Since the image depth is equal to 2z,, we see that, for TE propagation,the image depth dTE for a wire on the surface of a finitely conducting earthcan be expressed as

2/Y

dTE - - 2/yn (6)

subject to the condition that IYozI 0.5 (i.e., I n2--il > 2, where n2 =2/Y2Y1/ 0).

Similarly, for transverse magnetic (TM) propagation, since

ZA =i4- y0/y2 , (7)

then,

dTM ' - O/2 = V 2 l i/n2(8- Vo/JYi7 , (8)

0y1 Y1subject to the condition that IyoZiI 1 0.5 (i.e., Jn2/ V T -11 > 2).

For normal incidence, or if'In 21 > 15, equations (6) and (8) reduce tothe well-known resultl,2,3,8

dTE -dTM -d -2 /yl (9)

where y, - i-o for a, >> WEo0 r and y1 " 60wal/ _r + iChoJ o-r for1 > 1, equation (10) reduces to

4---- '!L~1

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x 4i p3 ( + y0p)e (12)

while equation (11) becomes

fix 47riwt 0p (1 + y0p)e0 (13)

Equating equations (12) and (13) results in

Td T 2 (14)

which is identical to equation (6).

Following Wait and Spies,3 another way to determine the image depth isto expand the function

f(uo) = eUod(u (15)

in a Taylor series about u0 = 0, resulting in

u I .Uo e - d j+ u2\d/ + (16)

where d is given by equation (6).

The introduction of exp(-uod) into the nx integral equation yields animage at a distance h + d from the earth's surface (see figure 1), whilehigher order terms would correspond to multiple images at the same location.

5'6

HED AIR-TO-AIR PROPAGATION

When h and z are > 0, the Sommerfeld integral expressions for the HEDHertz vector are

9 -I -

It e12 y JoRo 0 *-y 0 R1 +f(j2uo \ uo z+h)ox)~j (7i i R.(

and

ItCos~ I 2(u, - uo) -u0 (z+h)1z iOSw ' ., J0 (Xp) XdX ,(18)z = 4iwe---- 7P~ Y 2u u2 00 f Yju0+ 0 u 1

S

ITe

-K 4. - In

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where R2 P ( h)2, R p2 (z h)2, U2 A2 y2, and u2 A2 +y

From equations (17) and (18),

eY-R -O Y 2 e -U 0 ( Z +)1Z zCos 3 e-' R e-~4ritw0 dp eX00 RI +/X .u .(z Xp (19)

0 1J

If y0 Z1 1 < 0.5 (i.e., n 2 - I > 2), which is applicable in most prac-tical cases (for example, In21 > 81 at all frequencies for the sea/air case),

u - U

ul +u0 e (20)

and

- 2u 1 e- uOd (21)u1 + 0 U 1 + U 0

For small Sommerfeld numerical distances (i.e., IYoR I n - l/(2n 3)1

2 1)2~u +y 2u y2u (22)

Utilization of the identity (u1 - uo)(u 1 + Uo) = y2 yO, equations (21)and (22), and Sommerfeld's integrallO

= 00e-o h Jo0 (XP)-"d X = ey 0 , (23)u0 R(

0

results in

Y -yR -yR yR 1 -y R2IZ Le 0 - e- e- 0 - e02x 41tiwe o F Ro R R1 R2

it fe- Y O R0 e-YOR2

(24)

47ritweO F R0 R2

4. ItCos f e- 0 R 0 2 e'tO R1.r ii 0[ 0 -V(w 0. o, R (25)

IZp cos (1 - (1 YoR4 kmE + YoRo) e R0_T R

-- 4iTiwE: 0 y0R0 R3 2)1 + 1 y R30 n

64- - - - . - i----

. nu 1 u um um m, u lm a=l, n un n

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and

a . Cos (I - l/n2 ) - e e-uD j (X)dXEz 4i 0 u0 U 0

(26)

1- cos (I - lln2)f (1 - e -u0 u0 (z+h) )20 i 0 u 00

where R2 = + (d + z + h)2.2

Because of the u2 term in the denominator of equation (26), it does notreadily appear that the equation can be expressed in closed form, except forthe quasi-static range (y0 - 0), where equation (26) reduces to

I2 cos o - (z+h)

z 4wriwE e

27)II cos (d + z + h)- ( + h)4e i 0f P RR8 2J

However, if we replace X2 by u2 - y2 , equation (26) can be broken up .ntofour integrals, two of which are of the type

e Jj(AP)dA = 1-e -ke ,(28)

where R2 = p2 + Z2, and two of which are of the type

fe (Xp)dX (29)0 0

Since1 2

f J(x)d = I- e" Y OR!u0 I-. 1 ~ X p, l (30)

0

then

7

TR b311

2 U Z e-Y 0 Z Y Y yRJ f )dX e + e-0dz (31)0 u0 1 z

Inserting equations (28) and (31) into equation (26) results in

It cos (1 - i/n2) (d + z + h) -YOR 2 (z + h)e-YOR1 +1 (32)Hz ~_ 4riwE OP R2 R I32

where

e-Y OR I [ - dz (33)When (z + h) >> p, R- (z + h), and R2 - R - d, resulting in

I - e-yOd)e-yO(Z+h) (34)

For IY0dI < 0.5 (i.e., Jn 2 1 > 1S),

I e-y 0(z+h)I - 0de (35)

Comparing equations (27) and (32), we see that for the quasi-static range(1yoR 1I > Idl, equation (33)becomes

I - d - Rc , (36)

which is identical to equation (35) when (z + h) >> p.

Therefore, we can approximate the HED Hz vector as

It cos *(1 - 1/n2 ) d + z + h)eYR 2z 47riwe 0p R2

(z + h)e -y 0R yR (37)

R1 + Y0de ,

subject to the restriction Jn 2j > 15. This equation should also be valid forJn2 l > S if y 0 d is replaced by (1 - e-Yed).

Now, it seems that we have gone to a lot of trouble to derive the lastterm of the Rz expression. However, it is this term that yields the verticallypolarized farfield (for small numerical distances) produced by a HED located

8

. -___

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very near or below the earth's surface. This is the so-called quadripole term(i.e., the quadripole moment is Ity 0 d = 21Z/n). Because of the close spacingand opposite sense of the dipole and its image, direct radiation from thedipole is not the prime mechanism. Rather, the quadripole consisting of ver-tical conduction currents in the lossy medium representz the prime source. 13

If the first two terms in equation (37) are ignored and z = h = 0, equa-tion (37) reduces to

S t *Cos0 yde (38)z 41riwc 0 P 0

When jy0P ! >> 1,

H --iw0 z It 2 ey2)p 0 (39)00 ap 27ry 0 0

which is the correct farfield result for small numerical distances when In2j>> 1.

Since we have now derived expressions for the HED Hertz vector (equations(24), (25 . and (37)), the fields in air can be obtained from

-.y2-)' + _ * . )

H. . (40)

The 3sulting HED finitely conducting earth-image theory field expressionsfor the ai -to-air propagation case are presented in table 1. They are validfor small numerical distances and In21 > 15. When IyoR 1I > 1, they reduceto Norton's14 ,15 farfield results.

These results easily can be extended to a multilayered earth simply byletting d = (2/y1 )Q, where Q is the familiar plane-wave correction factoremployed to account for the presence of stratification in the earth.16,17

HMD AIR-TO-AIR PROPAGATION

Four of the six HMD expressions valid for h and z > 0 can be obtainedcompletely from reciprocity consideration (En, E01 Ez, and Hz). The remainingtwo can be obtained from Maxwell's equations (Ho and HO). Alternatively, theycan be obtained from

9.!I

... .. - n u- n -n n |

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Table 1. HED Air-to-Air Propagation Equations.Cos - 2 2 [1 1,[(0-o _ _ _ _ _ " - v Ro~ 2 1 e Y[R-

47r 0 R 00 R0

f,-yRo e-yO>.2(I +

-j PY

0 4 i~ 0 R 3 0 [ YR 0-((1 y0R ) 22 eYR -) 2[20 - oR

I -y0 R0 -yORjE i n R + -(1 -L(1 + yRR2)e

o47Riw1 ( + yo) R3 n2 " R30 0

+ 2 R -oRe

0 2R I2

E -h) + 3yR + yR ) YO-O

2 - (z + h) (3 + 3yR1 +

y2YR 2 ydR (z + h)-y]

42 RD3 Y ~ R3 eoRH 1I sin (d + z + h)_ 1 + (l+yRleYOR2P 4Tr R2 [ 2 R 2 2

Hz 4 t1 ( + ]YoR 1 -Rz0.yR) o o

2 10

O. os *I - .+h) ] -yR - h) z...h+0R, (1 + 0 e ( 1 + yR )e

1(d + z + h) Y0 R2 (z +. h) e-y0 1 dYOR [ + RiiY2

02j R- p2R, 1 0J

li i n G2f ~( + Y R 0) + 0yR YR 2z ~ 7 R3 -(+yR) R310 2 1

10

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H -y2 y sin + iH)0 QY

(41)-y 2n cos + x - )

Following the same procedure outlined in the derivation of the HED fieldcomponents results in

- y OR0 -yoR1y IA R 0 + eRI (42)

andIAp sin @t " YORO -y

3 1 + YR) + R (I + YoR (43)

The resulting HMD finitely conducting earth-image theory field expressionsfor the air-to-air propagation case are presented in table 2. They are validfor In21 > 15 and small numerical distances. When 1y0R11 > 1, they reduceto Norton's 14' 15 farfield results.

SUBSURFACE-TO-AIR PROPAGATION

The HED and HMD image-theory expressions for the subsurface-to-air propa-gation case (h < 0, z > 0) can be obtained from the air-to-air propagationequations (tables I and 2) simply by setting h = 0 and multiplying each expres-sion by exp(ylh). The resulting equations are presented in tables 3 and 4 andwill be valid 18 for R = Np2 7 > 13hl, jn21 > 15, and small numerical dis-tances. It should be noted that by following the procedure outlined by Bannis-ter and Dube, 18 the restriction R > 13hl can become less stringent.

When ly1RI >> I, the HED and HMD subsurface-to-air propagation equationsreduce to the nearfield and farfield range results presented in tables 3.1 and3.3 of Kraichman19 (see also Bannister 2 ).

AIR-TO-SUBSURFACE PROPAGATION

The HED and HMD image-theory expressions for the air-to-subsurface propa-gation case (h > 0, z < 0) can be obtained from the air-to-air propagationequations (tables J and 2) simply by setting z = 0 and multiplying each expres-sion by exp(ylz). (Both Ez components must also be multiplied by 1/n2 tosatisfy the boundary conditions.) The resultinequations are presented intables 5 and 6 and will be valid 18 for R' = Npl + > 13zl, In21 > 15, andsmall numerical distances.

11

_ L m m mml m m mgm maml mN m

. .... . -

i .

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Table 2. HMD Air-to-Air Propagation Equations

I w01A cos 'z - h)(I )e-YOR 0 + (z + h)( l -yRE 4 R3 (+ yoR0 R ( yoR 1)

l[(d + z + h)o-Y z + h)_-Y0R _ deY0ip 0 2 -_L - R ,

+ y2p2

iwiu 0IA sin C (d + z + h)[.+ --- (1 + YR) e-YOR 247 r2 R 2 02

fYO dR1 - ( z + h ) ] -YR (z - h) -Yo Ro0S+ + (1 + y R )e

p2R R3 0

iwvl0 IAp cos ( -oReYR

z 4 yR 0 ) R3 + ( 1+ YoR 1 ) R3e0. 0

H IA sin [ (z h)2]j r1 + YRo) - h)2 e 00

-oR ^-YoR R20R

[.0 .J[ 0+ 1I[2 3 (d + z + h ] [1 + YO R + y 2 p2 e 2 R 1

R2 2] 0 R 0 R

Hz 4eY0R2 .(- YR 022 2H A s I + hYe 0 + R

z4 r 05 (3 0 R3 0

0

R 12

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Table 3. HED Subsurface-to-Air Propagation Equations[2=p2 + Z2, R2 = p2 + (d + Z)

2 1.

(y R-y h) -1(1 yR - -. jI* R.i )lE~ 12 co e/~2

2 - yO(RiR

2i~a * i )R3 JR 2 d2i

I si e-(Y0R-ylh) - o CR.-R)1E 3z sin +, R) + R2 R - Yo2rc,+ iWE 1)R3 I 1 YOR d2L1 j

IziwU0 cos se(YR~) - y(Ri-R)

E Rd3 3y + -ygR+2)~

Ii sinrpe- (yRyh [ R

+ p~'( + R y(R- YORi)112R4 co0eIdy

-(y 0R-y~h)H Ii sin .1 0 Rdz222( yR

+ ~ 2 (d- + z) + + z 0Re

IZ cs e-(yR-y 1 h) y(iR

H lip sinf 4irR1 + y0 R) R ~-14 a )e oIL 41R1.

tI

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Table 4. HMD Subsurface-to-Air Propagation Equations[R2 = p 2 + z2 , R? = p2 + (d + z)2 ]

3.

iwj o IA cos -E 04IAos 0 - (Yo R- Ylh) d(YoR + y2p 2)

2zp2 + d+ R -Y0(R i-R)R2 G + Y0R) -

iw 01IA sin 0 -(yoR-ylh) 2E dy R - z + (1 + yoR)

4irp 2R 10 R

+ (d + z) __ I1(1 y YRi)] eY(

iE U 0 IAp cos --(Y R-y h)Ez - 2,rR

3 (1 + yoR)e 0 1

IA sin 0_-(y 0R-y h) 3z2)(1 + YoR) - 2y2z2 2 2H 47R 3 /2 2 0

+ (R -yo(Ri-R)

H IA cos -(YOR-Ylh) + yR + 2y2 R2 + (R (I + YRieyo(RiR)

H IAp sin -(yoR-ylh) (3 + 3YR + y2 R2)

z 4rR5 Y 0

+ (d + z) +53 + 3yR i + y2Ri)ey(Ri)

14, i4

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Table S. HED Air-to-Subsurface=Propagation Equationsp(I2 2 4. h2, (R')2 =p 2 +(d + h)2

27 (a +e~ _____RI~ 0 ~ TR2 2 2(Rt)Z 2 R' -Y0(R-R

d2 [E - 2 snS(YRT-)1 + y R' + 2(' 2 i '- 0(

S 21r (a + iwe I)(R' )3 1 0 d ~lJ

U 2 p c o s o e {o ~ - l i[3 +. 3 y R I + y R P 2

y 2(RI)L4 r R'2p0 j dR' h + L-d+ h)e -YO(RRI ]

H I-t sin oe(YRlZ (RI /Rt\2 /R\2 1Y(RR'H1(R) R( d + h)1y ~ 4. 1-. + y RI)e l

+ h(1 + y RI) + ( 2 )(dy 0R' - h)J

IZ cos o_ ___________!v_ RI -yo(RI-R')

d yR' + Y2 p - h-2 + yI

-(y R'-y IZ) yR-IUP Ipsine (R0 3 0~ 1 R'

1H I15 R yRz1 41 ( 30 1

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Table 6. HMD Air-to-Subsurface Propagation Equations

[(R p2 = 2 + h2 , (R!)2 = 2 + (d + h)2]

iwvo0IA cos e - (YOR - 1Z ) R' -yo(R!-R')E (d + h)R.--eP Asn 41rp 2R' i

E@~ 0 ioIA sin Oe- (YoR'-yIz) h

E 2R ' 0dR' - h (1 +y0 R)t (R,) 2(i ~ -yo(JR!-RI)

(d + h)KI[ + P(I + y0 R!)eY(R1)

iWUosIAp (os + -(y0R,-y 1z)E "3 (1+ YoR')e

z 2rn2 (R I) 3

IA sin e-(yOR '-y z) 3h2 i1 + YoR' - 2y2h2 y2p2r 41r_(R_) 3 2 (R) 2 0

3(d + y2 2 - 0 1R')]

21- "[~ ~ 0 ' R' 01] I )eH@ IA cos e-(Y O R'-y 1 z) +

47 (R) 1 + y 0 R' y0 (R')

H 41r, (R 5d +4 h)() [3 +e 3y 0R!

2 Y(R!)2]ey(RR) h [3 + 3yR' + R- '

4 16-

w

TR 6511

When IyiR'1 >> 1, the HED and HMD air-to-subsurface propagation equationsreduce to the nearfield range results presented in table 3.5 of Kraichman.19

SUBSURFACE-TO-SUBSURFACE PROPAGATION

The HED and HFID image-theory expressions for the subsurface-to-subsurfacepropagation case (h < 0, z < 0) can be obtained from the air-to-air propagationequations (tables 1 and 2) simply by setting both z and h equal to zero andmultiplying each expression by exp[yl(z + h)]. (Both Ez components must alsobe multiplied by 1/n2 = yO/y1 to satisfy the boundary conditions.) Theresulting expressions are presented in tables 7 and 8 and should be valid 18

(for most cases) for p > 3 z + hi, jn2 j > 15, and small numerical distances.When lY0P1 > 1, they reduceto the subsurface-to-subsurface nearfield and farfield range results presentedin tables 3.2 and 3.7 of Kraichman.19

COMPARISON WITH EXACT SOMMERFELD INTEGRATION RESULTS

Mittra et al. 20 have presented some exact Sommerfeld integration resultsfor the HED 0fHx and nz vectors (i.e., the correction terms to the perfectlyconducting ground solution) for frequencies of 3 to 30 MHz. For their case,Ri = 10 m, a = tan-l[p/(z + h)] = 100, = 00, and the quantity I0 = Ii/(iweo)is normalized to unity.

The image-theory solution of Hz is given by equation (37), while 0i isgiven by the last two terms of equation (24); that is,

1 0 [eRR e~ 21ox - Z[ R1 R2 J . (44)

A comparison of the image theory and exact Sommerfeld integration resultsfor the situation where E r = 40 and a1 = 1 S/m is presented in figure 3. Forthis situation, n2 varies from 40 - jlSO at 3 MHz to 40 - j1S at 30 MHz. Since1n21 > 15, d - d - d = 2/y Note that the agreement between the two solu-

TM TE 1,tions is excellent.

Presented in figure 4 is a comparison of the image-theory and exact Som-merfeld integration results for the case where e r = 10 and al = 10-2 S/m. Forthis case, n2 varies from 10 - j6 at 3 MIz to 10 jO.6 at 30 MHz. Sincen2 , < 15 and (z + h) >> p, we have replaced yod by 1 - exp(-yod) in the imagetheory a z expression (equation (37)). Note that the agreement is excellent(within I percent for the ox component and very good (within 5 percent) forthe 1z component.

17

TR 6511

Table 7. flED Subsurface-to-Subsurface PropagationEquations (p? = p2 + df2)

O- fy~p-y 1 (z~h)) 2 IP

E X Cos T -e 12 + 2yP+ 2 2 2 -e Y (Pi-P

E - sin Oe[ YOP yl(z+h)I o 202[ -Pe-OP2ir(al + iw,0 Ji* yPi ~(jP]

H Z I Cos [ 0 Y1(Z -h)] +h) + P i)e YO(i + p

It sin e - [ y~p-y 1 (z+h)] + 2-Y piPH Trys~ P 3 ? 0pypi + yOPj

IZcs- y op-y (z+h)] - 3 YY 0(PjiP)

H I;. sinf Oe -[Opy ( (1 +~ y p) - 3~(l + y~pi)e-y('P

181r 0 P

TR 6511

Table 8. HMD Subsurface-to-Subsurface PropagationEquations (p? p2 + d2 )1

IAy1 cos *e - [YPY (z+h) I

I 1:

IAy1 sin *e - P) [yopylO~h) -P) ypiwo1-p 3 [I + -1 + yOfpi)3- [p -y1 ~p (yo

-iw 0 IA cos e- ypy(~ )

21rn~p P1+Yp

-[yOp-y1 (z+h)]H~-IA sin Oe L) 2p2

3 ( Y3 'O(Pi-P) f[ d2J [I+Y~] YPJ

+A co 1[2O~lZ~ I _ + y + yyp 2 Pe~ 2 p.e1]~ ~4~p2 0 0'/ l

I~psinOe-[y 0p.i -yI (z+h)]Hl~ (3 + 3y P Y0p )

19

TR 6511

7

S61

0 0.2 0.4 0.6 0.8 1.0______________R_

EXACT* IMAGE TH :ORY

dTM d drE -d =2/y 1

11

10,

Ln 9

6.

0 0.2 0.4 0.6 0.8 1.0

Figure 3. Comparison of Image-Theory and Exact Sommerfeld IntegrationResults for e r =40, a =1I5/n, R, = 10 m, e= 10*,

0 0, and f = 3 to 30 MI-z

20

_T

TR 6511

37

36 c

35..

34,

- 33

-x 32.

31

30.. dT 2/

29 ~1 -1/n 229.,

280 0.2 0.4 0.6 0.8 1.0

-EXACTa IMAGE THEORY

50

49..

48..

b47

-. 4645 dTTIn2

1 444.

43-0 0.2 0.4 0.6 0.8 1.0

RIx

Figure 4. Comparison of Image-Theory and Exact Sommerfeld IntegrationResults for r = 10, aI = 10-2 S/m, R I = 10 m, e = 100,

* = 00, and f = 3 to 30 MHz

21

i

TR bSll

EXTENSION TO LARGE NUMERICAL DISTANCES

Image theory also can be utilized to determine the fields at large numer-ical distances (i.e., 1p[ = 1-y0R 1/2n21 >> 1). For the sake of simplicity, wewill let z = h = 0 and In2! >> 1. For this case, equation (18) reduces to

IZ 1Cos co I_ 2uo, n2 - 1 (X p)..LdX (45):41riwe0 TO ao UI + Uo)\U + n2UO) 0 uO0

Since yip! > 1, u1 - y1 = 2/d, and

(u - i- eu - u0 d (46)

Furthermore, because !pl>> 1,

n2

- ! n2 d -n2du /2+n

2u 2- (47)

Therefore,

Id cos 0 4 e- n2 du Jo(Xp)dX- 4nriwE: ( \ 2, )f e04 i 0

ICos 12d2 _Y0D

(48)

21ry 'i 0) 2

where D2 P2 + (n2d/2) 2.

Because p2 >> (n2 d/2)2 ,

H -i~~e a cos 27 y ) __0_n2L-S- (i"" ) - 20 10) (49)

I cos O(2)e-y op

which is the correct farfield result for large numerical distances.

22

TR 6511

CONCLUSIONS

Simple engineering expressions for HED and HMD air-to-air, subsurface-to-air, air-to-subsurface, and subsurface-to-subsurface propagation have beenderived by employing finitely conducting earth-image theory techniques. Forthe air-to-air propagation case, the expressions are valid from the quasi-staticto the farfield ranges as long as In2 l > 15 and the Sommerfeld numerical dis-tance is small. For the subsurface-to-air, air-to-subsurface, and subsurface-

* to-subsurface cases, the additional restriction that the measurement distance. be greater than three times the burial depth of the source and/or receiver

must be met. We have also demonstrated that image theory can be utilized todetermine the fields at large numerical distances.

We have compared successfully image-theory and exact Sommerfeld integra-tion results for four cases, yielding agreement within I percent for threecomparisons and within 5 percent for the other.

It should be noted that the two media can be inverted and the airreplaced by the earth's crust (of conductivity a2 and dielectric constant E2)"The same equations (tables I through 8) can be utilized as long asI2/2 1 and n2 = y/y > 15 simply by replacing iwe0 by G2 + iWE 2.

The results presented in this report should be particularly useful forsea/air and sea/earth's-crust propagation. They should also be helpful togeophysicists engaged in determining the electrical properties of the earth:The simple, yet accurate, formulas obtained from this theory make it a verystrong tool and a promising one for determining the coupling between antennaslocated above or below the earth's surface.

23--

TR 6511

REFERENCES

1. P. R. Bannister, "Summary of Image Theory Expressions for the Quasi-StaticFields of Antennas At or Above the Earth's Surface," Proceedings IEEE,vol. 67, no. 7, 1979, pp. 1001-1008.

2. P. R. Bannister et al., Quasi-Static Electromagnetic Fields, NUSC Scien-tific and Engineering Studies, Naval Underwater Systems Center, New Lon-don, CT 06320, 1980, 515 pp.

3. J. R. Wait and K. P. Spies, "On the Image Representation of the Quasi-

Static Fields of a Line Current Source Above the Ground," Canadian Journalof Physics, vol. 47, no. 23, 1969, pp. 2731-2733.

4. P. R. Bannister, "Extension of Quasi-Static Range Finitely ConductingEarth-Image Theory Techniques to Other Ranges," IEEE Transactions onAntennas and Propagation, vol. AP-26, no. 3, 1978, pp. 507-508.

S. A. Mohsen, "Earth Conductivity Effect on the Field of a Long HorizontalAntenna," IEEE 1980 International Symposium Digest, Antennas and Propa-gation, vol. 2, 80 CH1557-8AP, 2-6 June, Qudbec, Canada, 1980, pp. 440-443.

6. S. F. Mahmoud and A. D. Metwally, "New Image Representation for DipolesNear to a Dissipative Earth, Part I: Discrete Images; Part II: DiscretePlus Continuous Images," Radio Science (in press).

7. E. C. Jordan, Electromagnetic Waves and Radiating Systems, Prentice Hall,

Inc., Englewood Cliffs, NJ, 1950, pp. 408-410.

8. A. D. Watt, private communication, 1962.

9. J. R. Wait, "The Electromagnetic Fields of a Horizontal Dipole in thePresence of a Conducting Half-Space," Canadian Journal of Physics, vol.39, 1961, pp. 1017-1028.

10. A. Sommerfeld, "On the Propagation of Waves in Wireless Telegraphy,"Annalen der Physik, vol. 81, no. 25, 1926, pp. 1135-1153.

11. A. Bafios, Dipole Radiation in the Presence of a Conducting Half-Space,Pergamon Press, NY, 1966, 245 pp.

12. A. Erdelyi, ed., Tables of Integral Transforms, vol. 2, McGraw-Hill BookCompany, Inc., NY, 1954, 451 pp.

13. R. C. Hansen, "Radiation and Reception With Buried and Submerged Antennas,"IEEE Transactions on Antennas and Propagation, vol. AP-ll, no. 3, 1963,pp. 207-216.

14. K. A. Norton, "The Propagation of Radio Waves Over the Surface of theEarth and in the Upper Atmosphere," Proceedings IRE, vol. 25, no. 9,1937, pp. 1203-1236.

24

L

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IS. R. J. King, "Electromagnetic Wave Propagation Over a Constant ImpedancePlane," Radio Science, vol. 4, no. 3, 1969, pp. 255-268.

16. D. J. Thomson and J. T. Weaver, "The Complex Image Approximation forInduction in a Multilayered Earth," Journal of Geophysical Research, vol.80, 1975, pp. 123-129.

17. J. R. Wait, Electromagnetic Waves in Stratified Media, Pergamon Press,NY, 1970, pp. 21-31.

18. P. R. Bannister and R. L. Dube, "Simple Expressions for Horizontal Elec-tric Dipole Quasi-Static Range Sub!,urface-to-Subsurface and Subsurface-to-Air Propagation," Radio Science, vol. 13, no. 3, 1978, pp. 501-507.

19. M. B. Kraichman, Handbook of Electromagnetic Propagation in Conducting

Media, U. S. Government Printing Office, Washington, DC, 1970, pp. 3-8 to3-16.

20. R. Mittra, P. Parhami, and Y. Rahmat-Sainii, "Solving the Current ElementProblem Over Lossy Half-Space Without Sommerfeld Integrals," IEEE Trans-actions on Antennas and Propagation, vol. AP-27, no. 6, 1979, pp. 778-782.

4

2S/26Reverse Blank

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