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AD 13 9 0 57 NUSC Technical Repot 6883 11 Januury 1084 New Formulas That Extend Norton's Farfield Elementary Dipole Equations to the Quasi-Nearfield Range Peter R. Bannister Submarne Electromagnetic Systems Department 9 I Naval Underwater systems CenterDTIC " Newport, Rhode Island I New London, Connecticut E! L~~j S AR1 3184 Aptxd fmor p roeease; distribution t'nlmlted 8 .4 03 16 066

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AD 13 9 0 5 7

NUSC Technical Repot 688311 Januury 1084

New Formulas That Extend Norton's FarfieldElementary Dipole Equations to theQuasi-Nearfield Range

Peter R. BannisterSubmarne Electromagnetic Systems Department

9

I

Naval Underwater systems CenterDTIC" Newport, Rhode Island I New London, Connecticut E!

L~~j S AR1 3184

Aptxd fmor p roeease; distribution t'nlmlted

8 .4 03 16 066

I-

Preface

This report was prepared under NUSC Project No. A59007, "ELFPropagation RDT&E" (U), Principal !nvestigator, P. R. Bannister (Code 3411),Navy Program Element No. 1140 N and Project No. X0792-SB, Naval ElectronicSystems Command Communications Systems Project Office, D. Dyson (Code PME110), Program Manager ELF Communications, Dr. B. Kruger (Code PME I 10-XI).

The analysis and write up of this report was performed while the author wasoccupying the Research Chair in Applied Physics at the Naval Postgraduate School,Monterey, CA. The author would especially like to thank Professors Otto Heinzand John Dyer and Dean Bill Tolles for recommending him to occupy this post andNAVSEA (Code 63R) for sponsoring the Chair.

The Technical Reviewer for this report was Anthony Bruno.

Reviewed and Approved: 11 January 1984

D. F. DeuceHead, Submarine Electromagnetic

Systems Department

The atithor of this report is located at theNew Lordon Laboratory, Naval Underwater Systems Center,

New London, Connecticut 06320.

-!- -

4 BREAD rNSTRUC11ONSREPORT DOCUMENTATION PAGE BEFORE COMPLETING FORM

I, EPORT LGOVT ACCESIION NO. 3. ECrIEN'$ CATALOG NINSM

TR 6883 ___A__I or)_4. TITLE mlad 4 ale S. riPIOFPtl POORT a PieOO COVMEDs

NEW FORMULAS THAT EXTEND NORTON'S FARFIELDELEMENTARY DIPOLE EQUATIONS TO THEQUASI-NEARFIELD RANGE a [VWM OL48

7. AU H N S. ItTR C OR GR N I I I I

Peter R. Bannister

S. PRPOMI ORGANIZAON • AM• AO AAKUE o•. ., MNT . o=.TANNaval Underwater Systems Center AEA t WORK UNIT UM

" New London Laboratory A59007New London, Connecticut 06320

11. CONTROING4 OCE NAME AM AOCU Ia2. ROT'OATI

11 January 1984

I&. NINEW OF PAGE

14. MONITORIN AGENCY KAME A, A0065 fiI diff~mt ftma Cnaamlii Offlkw, IS. SECMMT CMEW ftf 414spm,

UNCLASSIFIED1IL OECLAS ICA11ONI I OWINGRA

SCHIKIMS. 0ISTRIGUTION STATEMENT W this R@pWlD

Approved for public release; distribution unlimited.

17. OtS1TRUTION STATEMENT ff rho shMacl eWwed in Bloc X. if differmlanu RO.P I.1*

ISL SUPPLEMENTARY NOTES

•VIH iKE t al wo c trin D .i pe if V ert and asnet byc DIp ol

Air-to-Air Propagation Horizontal Magnetic DipoleElectromagnetic Fields Vertical Electric DipoleHorizontal Electric Dipole Vertical Magnetic Dipole 12L. AISTRACT•Casmi on ww s if meNom. and idina* by. baek anain

New formulas for the electric and magnetic fields produced by the fourelementary dipole antennas have been derived for the air-to-air, subsurface-to air, air-to-subsurface, and surface-to-surface propagation cases. These Iformulas are of rather simple form and reduce to previously derived resultswhen either (1) the measurement distance is much less than a free-spacewavelength, (2) the Sommerfeld numerical distance is small, or (3) themeasurement distance is much greater than a free-space wavelength. Zbey-•are valid at any frequency and at any range beyond a certain minimum distance I

DD, •,1,473

-i,,

21120. (Cont'd Al)4

for the flat-earth cas The main restrictions on these formulas are (1)In2 1 10, (2) the measurement distance is >10 skin depths from the source,and (3) the measurement distance is >5 times the depth of burial of the trans-mitting or receiving point sources.

XIn terms of computer time, these new formulas can be evaluated in fractionsof a minute compared with hours for the complete numerical evaluation of theexact Sommerfeld integrals.

These formulas are intended to supplement the author's recently derivcd

subsurface-to-subsurface, subsurface-to-surface, surface-to-subsurface, andsurface-to-surface propagation formulas.

S"-

TR 6883

TABLE OF CONTENTS

Page

LI •-r OF TABLES ............. ....... .............................. ii

GLOSSARY OF SYMBOLS .............. ... ........................... . 1.

INTRODUCTION ............. ... ............................... . ... 1

AIR-TO-AIR PROPAGATION DERIVATION PROCEDURE .......... ............... 2

SUBSURFACE-TO-AIR PROPAGATION ................ ...................... 9

AIR-TO-SUBSURFACE PROPAGATION ............ ...................... ... 10

SURFACE-TO-SURFACE PROPAGATION ............. ...................... 10

DISCUSSION ............. ......... ................................ 10

CONCLUSIONS ............ ........... ......... ......... ............ 12

REFERENCES ............. ......... ................................ 43

I3Accession For

NTIS GRA&IDTIC TABUnanpounced E:

just if icat ion

]Distribution/

ý Availability Codes

ist Special

11LI-a .ado

S~TR 6883 gI

LIST OF TABLES

Table Page

I Electric-Field Air-to-Air PropagationFormulas (In 2 l > 10, RI > 106) ...... ................ . I.S.

Magnetic-Field Air-to-Air PropagationFormulas (n2j > 10, R1 > 106) ....... ................ ... 17

Electric-Field Air-to-Air Propagation Formulas forthe Farfield Case (2 10, yR .......... 21

4 Magnetic-Field Air-to-Air Propagation Formulas forthe Farfield Case (In21 L 10, IYoRI >> 1) ... ........... ... 23

S Electric-Field Subsurface-to-Air Propagation

Formulas (1n 21 > 10, R > 106, R > 5h) .... ............. .. 25

6 Magnetic-Field Subsurface-to-Air PropagationFormulas (In 2 I > 10, R > 106, R > 5h) .... ............. .. 27

7 Subsurface-to-Air Propagation FormulasWhen p >> z ({n2I > 10, p > i06, p > 5h) ...... ........... 29

3. Subsurface-to-Air Propagation FormulasWhen z >> p (n 21 > 10, z > 106, z > Sh) ....... ........... 31

9 Electric-Field Air-to-Subsurface PropagationFormulas (In 2I > 10, D > 106, D > 5z) .... ............. .. 33

.10 Magnetic-Field Air-to-Subsurface PropagationFormulas (In2j > 10, D > 106, D > 5z) ..... ............. 35

11 Air-to-Subsurface Propagation FormulasWhen p >> h (In 21 10, p > 106, p > Sz) ...... ........... 37

12 Air-to-Subsurface Propagation FormulasWhen h >> p (In2I 1 10, h > 106, h > Sz) ...... ........... 39

13 Surface-to-Surface Propagation Formulasfor p > 106 (1n21 > 10, z = h 0+) ..... .............. .. 41

3ii

TR 6883

GLOSSARY OF SYM3OLS

sin + A F(w) (+ij\i-~A si +, A ~---- JF(w)sin~~~1 +A \ 2

•'• sin2 !-A21F(v,) [1 + r,,

B A2~w _ 1_i 1 _~ F(w) sn* j3ifl + A1 2 _ ~ 2

D (p 2 + h2)1/2 (meters)Ep Horizontal electric-field component in the p direction (volts/meter)

Et Horizontal electric-field component in the f direction (volts/meter)

Ez Vertical electric-field component (volts/meter)

F F(w) or F(wO), Sommerfeld surface-wave attenuation factors

h Height (h > 0) of transmitting antenna with respect to earth's sur-face (meters)

HED Horizontal electric dipole

HMD Horizontal magnetic dipole

Hp Horizontal magnetic-field componvnt in the p direction (amperes/meter)H• Horizontal ma-netic-field component in the ý direction (amperes/meter)

Hz Vertical magnetic-field component (amperes/meter)

I Current (amperes)

J 0o(p) Bessel function of the first kind, order zero, with argument)Lp

m Magnetic-dipole moment (&.npere-meters 2 -1

n y1/y0, index of refraction

n 120w, free space impedance (ohms)

nI n(l - A2 COS 2 *1i)1/2

p Electric-current moment (ampere-meters)

P ai2t it e tegral

R (P 2 + Z2)1/2 (m~eters)

TR 6883

R0 [p2 + (z - h)2]1/2 (meters)

R[p 2 + (z r h) 2 ] 1 / 2 (meters)

S 1 Exp(-yOR1 )/R 1 , Sommerfeld integral

t Time (seconds)

UO (X2 + y2)1/2 (meters- 1 ) (air)

ul ()2 + y2)1/2 (meters- 1 ) (earth)

VED Vertical electric dipole

VMD Vertical magnetic dipole

SWO or w', Sommerfeld numerical distances

w0X I - A2 COS2 •

Height (z > 0) of receiving antenna with respect to earth's sur-face (me-ters)

sin Ai -1

s1n , Fresnel reflection coefficient for verticalsin •+ A, polarization

sin ý1 - nl•-. , Fresnei reflection coefficient for horizontal

sin *1 + n1 polarization

Y0 (-W2c /2 = i2r/A0 , upper half-spoce (air) propagation

constant (meters- 1 )

I 1(iwu 1 o1 -w2ui£i)1/2, lower half-space (earth) propagation

constant (meters-l)'

A /7= 1/n

A A(l - A2 cos 2 p1)1/2

2 1/.2rjW2e2 -/2/2I )/ I -+ 1 --- .skin depth in the water or

LX J/earth (meters)

E0 10-9/36w farads/meter, permittivity of free space

C •I Permittivity of lower half-space (earth) (farads/meter)

x Pummy integration variable in the basic Sommerfeld integrals (Meters)1

X 0 Wavelength in free space (meters)

iv

TR 6883

p (x2 + y2)1/2 radial distance in a cylindrical coordinatesystem (meters)

aI Conductivity of the lower half-space (earth) (Siemens/meter) I

tan-I1 (y/x), azimuth angle in a cylindrical coordinate system

-0 = 41 10-7 henries/meter, permeability of free space

San-I (z/p) or tan-I (h/p), elevation angle

WO tan-1 [(z - h)/p], elevation angle*1 tan-1 [(z + h)Ip], elevation angle

2nf radians/second, angular frequency

I] -1

ii

Re s Bv/viS~Reverse Blank

IR -6883

NEW FORMULAS THAT EXTEND NORTON'S FARFIELD ELEMENTARYDIPOLE EQUATIONS TO THE QUASI-NEARFIELD RANGE

INTRODUCTION

It is the purpose of this report to present new formulas for horizontalelectric dipole (HED), horizontal magnetic dipole (HMD), vertical -!zctricdipole (VED), and vertical magnetic dipole (VMD) air-to-air, subsurface-to-air, air-to-subsurface, and surface-to-surface propagation. The new air-to-air propagation formulas extend Norton's results,",2, 3 which are stated to bevalid for measurement distances greater than a free-space wavelength (A0),down to the quasi-nearfield range, which is defined as the range where themeasurement distance is much less than a free-space wavelength but much greaterthan an earth-skin depth (6). The new subsurface-to-air and air-to-subsurfacepropagation formulas reduce to previously derived results4 -1 0 when the meas-urement distance is much less than, or comparable to, X0 . Norton'sl, 2 , 3 (cor-

rected) air-to-air propagation formulas are summarized by King,ll whileKraichman1 2 has tabulated the subsurface-to-air and air-to-subsurface propa-gation equations derived by various authors. 4 -I0

In the past, many investigators erroneously have believed that the field-

strength equations tabulated in Chapter 3 of Kraichmanl 2 are only valid whenthe conduction currents in the water or earth are much greater than the dis-placement currents (i.e., , >> we1). Indeed, as long as In2 j = jy2/y•j >> 1,the displacement currents can be included simply by replacing a, by 0, + iwein the field-strength equations. Thus, Kraichman's tabulated results are con-siderably more general than they are stated to be.

The formulas presented in this report are intended to supplement theauthor's recently derived subsurface-to-subsurface, subsurface-to-surface,surface-to-subsurface, and surface-to-surface propagation formulas.15 Theyare valid at any frequency and any range beyond a certain minimum distance forthe flat-earth case. The main restrictions on these formulas Pie (1) In2! >10, (2) the measurement distance is >106 from the source, and (3) the measure-ment distance is >5 times the depth of !'urial of the transmitting or receiving

pont sources.

For the air-to-air propagation case, the four dipole antennas (VED, VMD,HED, and HMD) are situated at height h (h > 0) with respect to a cylindricalcoordinate system (p,o,z) and are assumed to carry a constant current, I. Theaxes of the VED and HED (of dipole moment p) are oriented in the z and xdirections, respectively, while the axes of the VMD and HMD (of dipole momentm) are oriented in the z and y directions, respectively. The earth, which isassumed to be a homogeneous medium with conductivity a, and dielectric con-stant e (= er--), occupies the lower half-space (z < 0) and the air occupies

the upper half-space (z > 0). The magnetic permeability of the earth isassumed to equal p0 , the permeability of free space. Meter-kilogram-second(MKS) units are employed and a suppressed time factor of exp(iwt) is assumed.

.P1

TR 6883

AIR-TO-AIR PROPAGATION DERIVATION PROCEDURE

"As an example of our derivation procedure, consider an HED source. Whenh and z are >0, the Sommerfeld integral expressions for the HED Hertz vectorare4 , 5 ,1 4

-R 0 e-YOR1 2u z+h)euu __ __ + U Uo 0 d

and

00 2x ± - U0) -_0p os yuI + T~u e J0o(Xp)XdX , (2)

where

R2 P2 + (z - h)2,

R2 p2 + (z + h)2

1

U2 =2 + y20ooU12 X2+y

Y2 =..W 2 O10 e 0 and

Y2 0 io(al + iwe 1).+J

From equations (1) and (2), and utilizing the identity (u1 -U 0)(u1 + u0)

pe-R0 e-YO R1 f _w2ye-UO ) 1Z• •• •= co @x 8. + Jo(X`p)Xd . (3)

=41riwe0 ap [ R0 Ri 0 1 u0 + y~ 1 J0 ) J. (3

When In2 l >> 1 and Re(y1 Rj) >> 1 the function uI in the exact integralexpressions can be replaced by yli the propagation constant in the earth .4

Therefore,

D1r-YR -Y0R1 ____- 0 zh

11r 2 0fO (y1" Uo)e JoAp)XdX . (4)L:'00 e- 2 R 1 - 0 ( A)

4ir0 1 1R0

2

_'R 6883

Since Sommerfeld's integral, S1, is equal to

S u. ( z+h)je(X _d e e-yOR1

S =fe .u R 2 (5)0

then,

S(z+h) as sin 4.1 -yR (6)

e-u 0 _____X(1 y~~eYR 6je J 0 (aP)XdX (z + h) = R ( yoR1)e 01

and

00 2S1

UoeUo(z+h)Jo(Xp)XdX = 1f (z + h)2 • :

(7)e-YoR I

R I [(1 ÷ yoRI) - sin2 p1 (3 + 3yoR1 + y2R2)]•,

where sin 1= (z + h)/R 1 . Therefore, from equations (4), (6), and (7),

S p [eYO R 0 e-YOR1 2e- OR1X 41iwe0o RO R1 (y2 _23

Vo)RI(8a)

[(1 + y1R1 sin ý,)(l + yoRI) - sin2 ,i(3 + 3yoR1 y2R2) . 8a

Alternatively, we could have replaced the quantity-u in the exact inte- ggral expressions by yl[l - (y21y2)COS2 iý1]1I2 instead of Y1~.3,11,15 This0 1change is equivalent to setting the normalized surface impedance, Al, equal to

A= A( - A2 cos 2 ýI)1/2 (9)

where A = yO/y1 = 1/n and cos = p/Ri.

Since Iy2I >> Jy•l, the factor in parentheses in equation (9) is very

near unity (i.e., Al - A). Thus, the resultant modification is usually small.The retention of the parenthesized term is justified by the fact that theexact form of the Fresnel reflection coefficient is recovered from the asymp-totic solution. 3 ,1 5 For example, if we utilize the identity (u1 - uo)(u 1 + Uo)

S Y• , substituting uI = l(l - A2 cos 2 *i)1/2 into equation (1) and eval-uating the Sommerfeld integrals results in

3

TR 6840j5Y)

Ie-yR__o e-YoR 2e-YOR14w•ic Ro R y• - y2)

(8b)

[(I. +V~yift 1 si 01'~~)(1 * y0Rj) -si * + 3y0RI +Y Y~ ~III (b

where X =-A 2 cos 2q 1 "

When IyoR1I >> 1, equation (8b) reduces to

i~x - p [e-YORo e-YOR1

47riwcO r RO R1

x 2y sin • ( 10)

The quantity

2y0 sin *i(yjiFX - yo sin ýi) (yV•1 + Y0 sin *i)x

y2 YO CyJ(yjf+ y0 sin *i)(11)

2y0 sin Ij 2 sin *1= = = 1 + r•Y1V3U+ YO sin *1 n + sin "

wheresin n- 1(

sin *1 + nl

is the Fresnel reflection coefficient for horizontal polarization and

nI = n(1 - cos 2 *1)1!2 (13)

when Iy»l >> Iy~l, n1 - n, and A1 - A = 1/n.

Therefore, from equations (10) and (11), when IyoR 1 1 >> 1,

R° eYOR.X - 1 .)+ (14)

and the exact form of the Fresnel reflection coefficient is recovered.

When 1n21 >> 1 and Re(yIRj) >> 1, equation (2) reduces to

"4 4

TR 6883

nz p Cos u O [ ( . - u/Yj) _Uo ( +h) ofl x e fWJ u0 ()eA0ZhjQp)AkdA (15)

2ffiwcoYl y Po Uo + YOA 0

Since. 1 = U ____ = ____"__6)I Uo ~~~~+ y OA - 0 \u 0 u0 + •0 0 U ( 0 + ¥ A

- U 06 u u0(u+ fOA) (16)U

then, for JA2 1 << 1,

( 1 - u o/y 1 ) 1 _ __ __• - 1 - - . (1 7 )

u + YOA u o u o(U o + Y OA) Y,

Therefore, equation (15) reduces to

p Cos 0 a [F uo(z+h) d

(18)-.- f - ee-U(Z+h)J(P)xd ,

where

oAe-UO(z+h)

P J 0 u0(u0 + Y) Jio0 p)XdX ()

From equations (5) and (6),

S [e - + YoR1)e JY (20)2 I Ti 1 1 1R

Because In2 j >> 1 and Re(y 1 Rl) >> 1, the third term in equation (20) will

be of importance only near the source. Furthermore, since JyOAI << 1, the

integr-al P will be of importance only when 1yoR 11 >> 1. Wait 4 ,15 has shown

"that, when iy41 >> jy j and oRil > ,>- 1,1/2 -YoR1

P ~ /2iCrw)1/2e-werfc(iw1/2)e (21)P R1

: wl -Y0RiA2/2 (22)

- . • - A -,.,I- ••,. - z '

TR 6883

and

YoR1

w + - ~~.sn 1 A)2 (23)

If we replace A by A3 ,lllS (equation (9)),

w(•)1/2 (1-r)= A 1Ssin • + •1(24)

where

sin 1 Ar,, =(25)sin + A1

(is the Fresnel reflection coefficient for vertical polarization. Therefore,equation (21) can be expressed as

/1 -',,i2 e-_OR1

p [I F(w)], (26)

where

F(w) - 1 - i(rw)1/2eWerfc(iw1/2) (27)

is the Sommerfeld surface-wave attenuation function and

0oR12 4w'w 2(sin *1 + A1) 2 (1- 2 (28)

is the Sommerfeld numerical distance. For small numerical distances F(w) - 1and for large numerical distances and negative arguments F(w) - -1/(2w).

Since

[) - F(w)] = )+ CW) (29)

sin *1 + A1 F(w)

sin + A1

equation (20) reduces to

p cos * . [_Ae YOR1 sinHiz p Cos € x 1 (30)

21iwe0y1 Bp R YR2

6

TR 6883

Another factor that we will encounter in the derivation of the field-strength components is the factor B, which is equal to

B = sin A1 - 1 A = (1n - \ A F(w)(31)

sin2 - A2F(w)

sin ' 1 +A1

For small numerical distances (i.e., F(w) 1 1), A 1, and B sinAl. Furthermore, for sin * >> IAJ, A - 1 and B - sin *j" For sin

comparable to or less than A1 , the horizontal distance p will be mutch greaterthan the sum of the transmitting and receiving heights (z + h). In the limitas *1 approaches 0, A - F(w 0 ) and B - -AiF(wo), where i

-WF(wo) - 1 - i(,rw0)1/2e-Oerfc(iw 0 1/2) (32)

and

Ynpw = A2 (33)0 2 1

When p2 >> (z + h) 2 , Wait 4 ,15 has shown that

F(w) - (1 + y0 Aj(z + h)]F(w0 ) = [1 + yoR1 A1 sin *1 ]F(w0 ) (34)

and

aF(w) - (35)

3z A1 F(w 0 )

Extensive numerical results for the function F(w0 ) have been provided byWait15 and King and Schlak.16

Since the factor A (equation (29)) is different from unity only when (1)the angle *1 is very small and (2) the Sommerfeld attenuation function F(w) is"different from unity, A is only a farfield surface-wave term. Therefore, wecan discard all derivatives of A that are not farfield terms. For examp'e,when I »I> I>yI and "A11 sin «1,

•.(Ae•RI.) =- A(1 + y0R1)cos *1 e0R1 e-R;ap/i'

80si *1 << 1, 2 ÷ 0•

e RY0

- A(l + y0 R1 )cos *1 e-R (

e-O1

eY0R1- - (1 + Y0R1A)cos "1 e-R2

1

7

TR 6883

Therefore, from equations (30) and (36),

ip cos * cos *senYORSRz - 2[i(I 0y1R• 1 + - (3 3yOR1 yR (37)

When In2 1 >> 1 and Re(yIR1 ) >> 1, equation (3) reduces to••'g•_I - O RO e-Y0R

••- 4riwc0 3P[ R R0 R1

4; Oc: C• e~O O(38)

f+ u0 +.0 J 0 (Ap)Xd]

00

Since

=~Y A (16)U0 + Y0A u 0 kUO U0 + YoA) UO u0 (u0 + Y0 A)

then, following the same procedure as in the derivation of the HED Hz compo-nent results in

~p Coso X Ae-0R0 e-YOR1 2 e-YORI

4witis 0 BP Ro n 2\ R,

p Cos e [- 0R o 2Ae-YOR1

41Iiws L R0 R + n2 R1 J(39)

pcos *(1 + yoRo)cos '0 ee-YOR1 + eYORO - (1 + yoR1 )cos *1 1 R

2ww Y R 2 12"+ -•-(l + yoR1A)cos co2 R21-

nd I

Since we have now derived expressions for the HED Hertz vector (equations(8), (37), and (39)), the fields in air can be obtained from

_y=-vj + viyx 10= iw• 0 (• x f) . (40)

By following the same procedure as outlined above, we can also obtainsuitable expressions for the HMD, VED, and VMD Hertz vectors. The resultingHED, M4D, VED, and VMD field-component expressions for the air-to-air

8

Z . • .

TR 6883

propagation case are presented in tables 1 and 2.* They are strictly validfor In 21 >> 1 and Re(y 1 R1 ) >> 1. However, for most cases, the requirement thatIn2 l > 10 and R1 106 is sufficient. When IyOR 1I << 1, they reduce to Bannis-ter's quasi-near range results. 7 ,10

When lYoRi >> 1, they reduce to Norton's farfield range results1 ,2,3, 1 1

(for In2 1 >> 1). For convenience, the farfield expressions (lyoRil >> 1) arelisted in tables 3 and 4. In these tables, sin *0 = (z - h)/Ro, cos *0 = p/Ro,sin *1 = (z + h)/Rl, and cos *1 = p/R 1.

The air-to-air propagation results presented in this report can be extended jto a multilayered earth simply be substituting yI/Q for y1, where Q is thefamiliar plane-wave correction factor employed to account for the presence ofstratification in the earth. 15 For a homogeneous ground, the argument of theSommerfeld numerical distance w0 (equation (33)) is always between 0 and -90deg, resulting in the transverse magnetic (Th1) surface-wave fields varying as1/p2 as p + -. For a stratified ground, the argument of w0 can be positive,resulting in the TM surface-wave fields varying as 1/,V-. This fact is dis-cussed in further detail by WaitlS and King and Schlak.16

SUBSURFACE-TO-AIR PROPAGATION

The HED, HMD, VED, and VMD field-component expressions for the subsurface-to-air propagation case (h < 0, z > 0) can be obtained from the air-to-airpropagation equations (tables 1 and 2) simply by setting h 0 and multiplyingeach resulting expression by exp(ylh). (All VED components must also be multi-plied by I/n 2 to satisfy the boundary conditions.) The resulting equations arepresented in tables 5 and 6 for the general case, in table 7 for p >> z, and intable 8 for z >> p. In these tables, R = p2 + z2 , sin p = z/R, and cos i =p/R.

The expressions presented in tables 5 through 8 are strictly valid forIn2 1 >> 1, Re(yR) >> 1, and R >> Ihl. However, for most cases, the require-

ment that jn 2 1 1• 0, R > 106, and R > Sihi is sufficient.

When 1yoRi << 1, they reduce to Bannister's quasi-near range formulas, 7 ,10

which are tabulated in Kraichman1 2 (with a, replaced by al + iwc,). When F(w)= 1 (i.e., small numerical distances), they reduce to Bannister's nearfieldrange formulas, 9 , 10 which are also tabulated in Kraichman1 2 (with a1 replacedby aI + iWel). Furthermore, when h = 0 and IyoRI >> 1, they are consistentwith Norton's farfield surface-to-air results.l, 2 , 3 ,11

*All tables have been placed together at the end of this report.

4-

TR 6883

AIR-TO-SUBSURFACE PROPAGATION

The HED, [MD, VED, and VMD field-component expressions for the air-to-subsurface propLgation case (h > 0, z < 0) can be obtained from the air-to-airpropagation equa.'ions (tables 1 and 2) simply by setting z = 0 and multiplyingeach resulting expression by exp(ylz). (All Ez components must also be multi-

plied by 1/n 2 to satisfy the boundary conditions.) The resulting equationsare presented in tables 9 and 10 for the general case, in table 11 for p >> h,and in table 12 for h >> p. In these tables, D = p2 + h2 , sin p = h/D, andcos •=p/D.

The expressions presented in tables 9 through 12 are strictly valid for1n21 >> 1, Re(yVD) >> 1, and D >> Izi. However, for most cases, the require-

ment that 1n21 > 10, D > 106, and D > Sizi is sufficient.

When lyoDi << 1, they reduce to Bannister's quasi-near range formulas, 7 , 10

which are tabulated in Kraichman 1 2 (with al replaced by al + iw 1). When the

numerical distance is small (i.e., F(w) - 1), they reduce to Bannister's near-field range formulas, 8 which are also tabulated in Kraichman 1 2 (with a1replaced by aI + iwe 1). Furthermore, when z = 0 and yo0Di >> 1, they are con-

sistent with Norton's farfield air-to-surface results.l, 2 , 3,11

SURFACE-TO-SURFACE PROPAGATION

The simple-form HED, HMD, VED, and VMD field-component expressions forthe surface-to-surface propagation case can be obtained from the air-to-airpropagation equations (tables I and 2) simply by setting both z and h equal tozero. The resulting equations are listed in table 13. They are strictly validfor In2 1 >> 1 and Re(ylp) >> 1. However, the requirement that jn 21 > 10 and

p 1106 is sufficient. When either (1) ty0pI << 1, (2) F(wo) - 1, or (3)

lyopi >> 1, they reduce to previously derived results. 4 -1 2 Note that the

function F in this table is equal to F(wo).

DISCUSSION

Since In21 >> 1 (i.e., 1A2 1 << 1), the factor nj - n and A1 ~ A. Fur-

thermore,

sin 1- "1 (41)

sin € + A

"and

sin *1 - n 2 sini+• ~ - (42)

sin .e+ n n

10

TR 6883

For small numerical distances (i.e., F(w) - 1), A - I and B - sin 1 - A.

Furthermore, for sin ýI >> JIA, A - I and B - sin *I. When sin *1 is compa-

rable to or less than A, the horizontal distance p will be much greater than

the sum of the transmitting and receiving antenna heights (z + h). In the

limit as *1 approaches zero, A F(wO) and B - -AF(wo), where

F(Wo) - 1 - i(7wO)1/2e-WOerfc(iw1/2) (43)0?

and

w °OP A2 (44)wo = - -- (44)

0 2

For this case (i.e., p2 >> (z + h) 2 ), Wait 4 ,1 5 has shown that F(w) can be

replaced by

F(w) [1 + y0A(z + h)]F(w0 ) , (45)

and we can make use of his tabulated resultslS of the function F(wo).

When we were confirming the validity for the subsurface-to-air and air-

to-subsurface propagation equations derived in this report (which were derived

from the air-to-air propagation equations), we discovered that they could also

have been obtained by combining previously derived results. The derivation

procedure can best be shown by example.

From table 3.3 of Kraichman, 1 2 the nearfield range subsurface-to-air HED

H. component is

HHE - _ p cos ehe-R(I + yOR +2R2) (46)

2?ny 1R3 0

When IyoRI >> 1, equation (46) reduces to

HHE y0 p cos *e 1he 0HE 2rR (A) (47)

If we take Norton's farfield H equation (table 4), set h 0, and multi-

ply by exp(ylh), we obtain (for 0yot >> I1)

Top COs elhYR/ - r111 -_1[sin, + AF(w)] . (48)21rR 2/

Since

(l-r,,) " •r49)ncr sin + A

TR 6883

then,

'i j,\Ajsin + A

[ )sin + AF(w)] A AAw] M (50)

Therefore, equation (48) reduces to

h -Y Ryop cos e Yih e 0

HH* - - 2R (A), (51),

which is equivalent to equation (47) except for the factor A. When eithersin »j >> I or the Sommerfeld numerical distance (w) is small, A - 1. For

1n21 >> 1, the range of validity of equations (46) and (51) overlap when IwI<< 1 and IYoRI >> 1 simultaneously. Therefore, we can simply combine equations(46) and (51) to obtain an expression for the HED Hý component valid from thequasi-nearfield to the farfield ranges. Therefore, for In2' >> 1, Re(1,R) >>1, and R >> IhI, T,

co eYjh -Y0RHE ~_pcos •e le-YO( 2RA)(2

21ryiR 3 (1+ y0R + 2A) (52)

which is the result listed in table 6.

In following this procedure, we discovered a small error in the HED andHMD E. and Hp farfield subsurface-to-air and subsurface-to-subsurface propa-gation equations tabulated in tables 3.1 and 3.2 of Kraichman 1 2 (and also inequation (14) of Wait 4 ). hlese equations should be multiplied by the factor[I + F(w0 )]/2. They will then agree with Norton's farfield surface-to-surfacepropagation results. This correction is unimportant for small numerical dis-tances, since, for this case, F(w.) - I and [I + F(w 0 )]/2 = 1. For largenumerical distances, the difference is a factor of 1/2. however, for thiscase, Ep >> E and >> lip.

CONCLUSIONS

New formulas that extend Norton's farfield elementary dipole equations tothe quasi-nearfield range have been developed for the air-to-air, subsurface-to-air, air-to-subsurface, and surface-to-surface propagation cases. They arevalid at any frequency and at any range beyond a certain minimum distance forthe flat-earth case. The main restrictions on these formulas are (1) In21 >10, (2) the measurement distance is >10 skin depths from the source, and (3)the measurement distance is >5 times the depth of burial of the transmittingor receiving point sources.

These new formulas reduce to previously derived results when either (1)the measuremont distance is much less than a free-space wavelength, (2) theSommerfeld nmerical distance is small, or (3) the measurement distance ismuch greater than a free-space wavelength.

12

SM. i

TR 6883

It should be roted that the two media can be inertt:d and the airreplaced by the earth's crust (of conductivity a2 and dielectric constant E2).The same equations (tables 1 through 13) can be utilized as long as In21=Iy2/y21 > 10, R1 > 1062, and R > Shi (or D > Szi) simply by replacing iwc0

by q2 ' iWC2

We have recently employed finitely conducting earth-image theory tech-niques to derive field-component expressions valid at any range from the sourcefor the air-to-air propagation case. The only restriction on these new image-theory formulas is that in 2 j > . When the measurement distance is >10 skindepths from the source, they reduce to the air-to-air equatio s derived in thisreport. Th.ese new image-theory results will be the subject of a futurereport.17

I

I

I3/11

I

I.

I-

•: .••... . . .. _ .: ., .- , - -- - Reverse 131Blanik

--

Table 1. Electric-Field Air-to-Air Propagation P

Dipole Ep'E

- voc 0 -y0R03

~ I~ 3~f 0 yR2)Sin e0 0 10

2y, Cos -yR --n

2R2 [ +(I-rl F~w)Fcy0R11

--(~1 + y0 R)cos *0 R

VIED 0 2 cost 0I4 (y ~ r~(l + y1R, $in *1)(3 4

-sin2 #1(15 + l5y0R, + 6y2OR2 3R

P os2 0 .1( +(12R+in e-Y0R0(3 2 * -l( co yR 0) YO y0 sin R3Y~ p sin * 1- OO+ yR +P400 4wiwe 0l 0]3

I n2R3 +yR(

R I

+n 2 R3 Ir1,) 0 1

1 ý + y1R1 sin +O' y0 1

1( e 0 0 sn 0 (LLpom sinyR #-( YO R 3 0R)in#1 +cosR 4wYOR2

" 34D ( I + r,,y 0R lsin #,

0 1"r ,-, 1I+ -eR3 [2 + y0 R1 (1A) sin2

*'l yO + 2 10

TR 6883;Ar Propagation Foimulas (In2 I>1,R 1 >16

Ef EZ

j1 LJ [- 3 sin2 *O)Cl + ORO0) + y2R2 COS2 *OeYR

0ww~ 0 0( 3R3"'-D

00

0 R

+j2R) (1 -4.FwyR2C

00 10 R

e-O1~Si *#0 R2 y(1-+ A + y1R1 2~ 3+3 0 1 +1jyR)i 1cs*

+ 2yR cos **YO(l 1 -yR y

+ 2R2

+y + 0 1)i *1R 2 ioRmco *-y~) 0 R (1

0R 1 e OR

YO A) +n ~ Y0R1 2 YR2)e 4wwc (1 0. R3yR)cs

4. (1 r,1),)SinR COS *1-yR

1S1

0 1)] 2y IRCverse eBlank

2R - F~-y'

Table 2. Magnetic-Field Air-to-Air Propagation)I

Dipole H

0

4(1 - l,)F(w)y0Rl o 1 oR

1(3 + 3y0R0 + y2R2)sin *0 Cos *j0

2R2 co *1 e-Y0R1(3 +3yR, y 0 psin *1 R3~---

2M cos R 0 '4

2 Cos # [(3 + 3y0R, + y2R2)

-sin2 #11 + l~y0Rj + yR2 + y3R3)]1

TR 6883

-ArPropagation Formulas (jn2j >10, R, > 106)

H# Hz

*0 ~+ (1 + r,,y 0Rj)cos *1eYOR

00

RCos 1 1

eYOR1- Mi 2

4 2rR2 Cos *0

T[(O1 3 yR sin- 0( *OR)[( Y 0 3 ~~

-~~~~~~~~ sine 0~1 1 S 0 1 +6~I+yR)

[sl- s1 in?- + 15y01 + y j)+ y22 Co01 01 3 II

-IIT

1 0

Table 2. (Cont'd) Magnetic-Field Air-t~o-Air Pro~

D~ipoleHp#Type

P Cos *1(A

p~~-O 4wn la + Y R Y~ YORO)sinl*

psn (I + y0R0)sin *'o (1 + Y0R,)sin *1' eyR_

HED (I + r1,yORj)sin I~ e 2

2e-YOR1 [2 + y0Rj(l + A) -sin2 01(3 + 3y0Rj + y2R2)] 1I

+ 2e 0 1l + Y 0RI +(

1(sn 2(1 + y0R0) -sin2 *0(3 +yR + y2R2)] 4w 0 (0+yR ',yR)R

eR1%0 + ~(1 +- y2.2 e y-ROR

-N 0__1__ _ _ _ _1)_R3

2 (3 3y R 2R2)

* [2l + ORA sin 1 0 + Y I-R

F(W~y2 -YO R1

ITR 6883

dAir-to-Air Propagation Formulas (jn 2 j > 10, RI > 106)

SH# Hz•,H,

t e-YORo YORI

+ y0 Ro)siins 4w 1+ yoRo)cos *0 R-2 (1+ yeR=)COS 10 0

)sin YoR1 TOR)sin 2 cos *1e-0R

1 2 ( - y)R (I + yIR1 sn * 1)(3 + 3yoR + y2RZ)[0 1y0 R1 ÷ (1 -)Fv•R~]I - sin2 ,1(15 + lSyoR 1 + 6y2R2 + y3R3)]

m sn +(3 + 3y0 R0 + y2R2)sin *0 cos #O e-YoRo"2oR2 4w 0 R3--

R RI + (3 + 3yoR1 + y2R2)sin *, CoS *1r •2R2 e-0O 10 1R3

2 cos t1e-0O11[(3 + 3y÷R, + y2R2)0 1 R1 - sin2 #1(15 + 15yoR1 + 6y0R2 + 3•R3)]

19/20

V Reverse Blank

Table 3. Electric-Field Air-to-Air Propagation Formulas for the4,

DipoleEE

TypeO~o.~+~ P ~ o~

(I2R (1 ,)AFy 1R1 s *

11

im~~o co0 e 0 ______ _

14D 0 0 0~11 +' +,R sin in

(1 r WeYR1 2eYOR1

iWPYO Co e-OR1iWnROMsin~

7tR6883

, agation Formulas for the Farfield Case (In2 I 10, IyOR 1I >> 1) I~

Ic 0

+ (1 - I,)F(w)cos2 ~ R

pY-Y ORO -y0R1

0

Cos #ie-YOR1

-- (I~co yj1 siRj

n 2R2 1 +A +yR i 1)e O~Yon_ I

+O (1p Co #j)F~~o * R1

p ~ s i n &O Y 0R O 4y R wi y m [S i *0 [o eY O R o 0_ _

R0-0 IAi sin2

O *1 eYR4y0

1 A+ R sin)1-an 1 -3 (1. I*)F(w)cos1J

Yom in # *0 eYOROe-YOI iwoyomCos1/22Rin~~Rves +Blnank r o

41r R 4w R0 1 [O 0-

-YOR'[ Si 2 *1n2 + (1 -rRI.U-)~~cs*

Table 4. Magnetic-Field Mir-to-Air Propagation Formulas for the Farfield Cast

DipoleHp#Type

0 ~-[co .~eYOR + OS~e-YOR1

VED0

+ -( P,1)F(w)cos*1e IRII

... fin * o4wL 0O~ v~ ~

VMD -sin 1 cos 0RIR

2 cos I-RI

- - i n2j

Top sin -Y0 R -*YRi o o y

sin2 --in 1 An - 3ssin

nR1 y0R1

_ _ iU 1 2_#

-- -I0 - 3 sin2 + ( ,, ,F w

F R I S3 COSI RIO a T

[sOn *1- j(2 -3 si 2 3J~ Li2# -Y i

+~~~~ (1-l)FwyR,,

HNN

+ [si2 *1- + 1 ;,F~w)21-.

TR 6883

~rPropagation Forinzlas for the Parfield Case CIn 21 > 10, IyORI >> 1)

H# Hz

P [oslO +0 cos 1#1eyR

+ ( -P 1 )F~w)cos *

0 _ _ ____ý2 8yOO-yR

[2 . 2OS #I-

+ fnR 2---S (1 TI R1 sin .

Y COS -YOR 1 Yop sin 4 YORO e-YOR1

-,)FW e*Ol + 2 cos *ie-YOR 11 l R i

on__e'0 __ sin # i # cos 0e

2O aCos ,r -YR 0 0 1 IS

(1 si2 ,cos*1 #,O 1 R-

4 ~23/241

Reverse Blank '

Table S. Electric-Field Subsurface -to-Air Propagation Fornu~

DipoleType P#

-Y1 h -y0 RP Cos *e e +(3yO~i2ir(cy + i0C1)R

3 +3 0 ~i

-Ay 0R + yOR2B]

0 Co #e -0Rsin )(3~

2 yR 2) -sin 2 *(IS + l~y0R + 6y2R2 + --

-Yjh -y0R ____________

3 Y1 R sin~ 3~R6iT [2 +yR sin +y 0R,HD2vj+ iwc1)R

3 2w(al + iME 1) 3

+ -O y2R2nB) + Y1 R sin *)-sin2 #(3 + 3y0 R + yR

-y h -y R -y h -yORy~m cs #e 1.e 0 riYinsin #e 1e

131) 2w(a1 + iwe1 )R3 (1-y 1 R sinf * wa w 1 r3 2 + yjR sin $+y

+ YOR - y2R2ns) + y1 R sin ~)-sin2 +(3 3y0R + y2R

t-ArTR 6883

toArPropagation Formulas (In2 I > 10, R > 108, R > Sh)

-Y h -YOR

0w 2(a + iac 1)R3( -3 Sin2 #)(I y0R)

+ y2R2A cos 2 4]0hy0

+)R L YR sin W,)3 +3y 0R

*(15 +15y R +6y2R 2 +y3R

3)]

r 2 y j s i n + y O ~ l A- y h -y RR3 L y~p cos cos *e 1e0

2z~q + ~a~c)R2 (1 +y 0RA)sn#(3 4+ 3y R +y2R 2)] 2~l ic)

I0 052

03 (1 -OA

Table 6. Magnetic-Field Subsurface-to-Air Propagation Formulas(I

Dipole Hp Ho -Type 2 _ _ _ _ _ _ _ _ _ _ _ _ _ _

-y h -yRVED 0 P R e-I)Of 2in 2R2

HEDo -eYj e + Y0 R r + 3y R2 y2R2 1R (1stR+yRA

1I i 2iR 01 + ly 0R + y6R2A) + y33]S

2-yR +A~ YO yOR2)]

sin2*(3 3yO + -2R2) +m si #e-lh -YORa S

HN tR OR( )mcsO Yhe-O

TR 6883

face-to-Air Propagation Formulas (jn2 I 10, R > 106, R > Sh)

H+ 0

R2 +Y 0RA)

-Yjh -YaRjme~y - y~R (1 +yjR sin *)[(9 +9y 0R +4y2R

2 +y3R

3)

0

-sin2 *J(75 + 7Sy0R + 33y2R2 + 6y3R3)

+sin4 *(1O5 + lOy 0R +45y2R

2 6 63R3)1

-y h -y R

-y~~h -y0R ~ ~ sin *cos *e 1e 0 (_ 1Rsn~(3+3 0e yR 2A)1 0

wR3 -(1I y0R y -

+g ..2R2) sin2 *(1S + 15y0R + 6y2R2 + y3R3)]

-Y h -Y R-yhm0 msinlcos *e I e

eYj eO -(I1+ yjRsin *)(3 + 3y0 RleR (1 + 0 + R2A) 2ry R4

+ y2R2) -sin 2 4i(15 +lSy 0R +6y2R2 *y3R 3)]

27/28Reverse Blank

Table 7. Subsurface-to-Air Propagation Formulas When p >>

DipoleE

Type E#z

VED 27e~ylhY+ icp3[(3 3yop)sin * 1( 1 +i~)3

-AyOP + y~p2 B] + I+YP+ypA

0M 0

-y~ ~ ~ p sn e-y1h eYOP 4%-p coli T h-)HED 0i~1 +ic) 2ir(a 1 OC) 3 [2+ 12r + iwel 1)p0

+ y~p - y~p~nB) + y~(3 + Ayo + y2 2)]x( ypAX

-Yjh-yop -y h -'rop yy~m co ehyopp~ cos #e e 0 3-' sI p in 4e Ie 311 ipcsO

I34D 2ir(a1 + i (1 - 2ir(a1 + iWE)3 2 2ir(a~ + iwe 1 )PT2

+ *y OPYOP nB) + yop(l + A + ylz)] x (1 + y~pA)X

TR 6883

Formulas When p >> z (In2 l > 10, p >106, p > Sh)

Hz Hp HO Hz

-y h -Y p4ee

+ iWEI)3 0 pe-Yjh e-Top02 r 2 2 (1 + YOPA)0

2 2A)

-y h -yOp -yh-y pme e 0me Ie 0

2wylp4 21r(y2 - 2) 5(1+% z

x(9 + 9yop + 4y pZ

s in 4eI- p Cos Oe ep sin *e 1e

~;+ iwl~. 2wyjp 3 2rvlp3 2wr(y2 I .F1 ypz

OypA) x (2 +yop(l A)] + Ye. ' y0 2+ A (3 + 3'rop y ypj

1-j e yo -Yi h -Yp aYj co e hyop -yi h -YOp.

4iwmc1)p2 2irP3 2p3 2rylp~

bpA) x (2 + yp(1 A)] x (I +YOp + Y2 2A) x (3 +3yop y2 y 2 )

29/30Reverse Blank

I Table 8. Subsurface-to-Air Propagation Formulas

Dipole E1Type

p sin cos#e~ leYO pen leiO - i 2

VED 21r(a1 + iwc1)z 3 o2ir (a + ie 3( i

x(3. + yz+y)x (I + YZ) + y 2 co2 *jS

Yi m sin* cos *eY eYO

VMD0(3+ l iwel)z 3 0

(33yoz + y z2)~7

[~ pcos sin *e IhefO 0yh-~ y i i e eY~p cos *cos ýe ~eTOk lED 2w(al + iwca,)z 2 27!(al + ie)2 2w( Ii + jwlz

+ (lYOz) + (lYOZ) + (YOZ)

y2m cos sin *eYh e-o y2m sin sin *e-h e-o y 2m cos co* j~j e_4Z

HhID 2w(al + iwcl)z 2 21rCG1 + iwel)z2 ira+ w 1 z

+ 1YOZ) + (lYOz) + (lYOz)

TR 6883

Formuulas When z >>p CIn 21 > 10, z > 106, z > Sh)

32 sn p pcos te _I eY

m Cos*e-Ylhe-yoz -yh -y z-(12 +12y~z msn*-1e

21y+ 5 z4 0 Iry I z

+ 0yZ y3z3) x (3 +3yoz + y~z2)

h ,-o n# *8-Yjh e yoz- p cos OeJ~ he-yoz psin *sin icos *e-Ylh1e yoz

k2~ 21rylz 3 2irylz 3 27ryjz 3

x 1YOz + y~z2) x (1t~ + Y2 2) 2 (33~~~ 2)

2wz 3 2wz 3 21rz 3

xlyoz + y~z2), x (+ YOZ+ y~z2) 2 (33~~~ 2 )

31/32Reverse Blank

tg

Table 9. Electric-Field Air-to-Subsurface Propagation

Dipole~

VED ~y1p cos *e~ leYO2u,+ iwe1)D

2 ( 0

0m Pfcs Te .Y1Ae YO v.

V1410 2nr(al + iwca)D

-sin2 *(15 + lSy0D + 6y2D2 +y3D 3)]

pcos *e jeYO 0HED Iwo . 1)

3 ( yjD sin i,+ yOD -y2D2flB) 2w(al + iwc1)D

+ Y0D(l + A + yjD sin *)-sin2 *(3

z -Y D ~y1m sin #e _Yze-Yy1m cos#e e0- 2+yDl A

+ o ~ 1) 3 ( + y 0 + y D 2A ) -

21r Ca l + iw e 1)D 3 2 + 0 D l + A

2w~u, + ioc,)D3 + 0- sin2 *(3 + 3y0D + y 2D2)I

1k ~-'---- - ---

TR 6883

-gubsurface Propagation Formulas (In2 I 10, D > 106, D > Sz)

E EZ

pe 1 e 0 73[(l 3 sin2 *i)(I + y0D)

0 21r(al + iwc1)DI,+ Y2D2A COS2 ~

)YlZ YOD

- +ý U yjD sin #)(3 + 3y0D +y2D2)

r 1y 0D +6y2D2

+y'D3)j

Z e - 0 Dp co os *-ylz e-Y0DL2sco1I e y (3 +3D

I)D r D sin 2~a + iwie 1)D3 3y y0 Dslin

+ Y1D sin *)-sin2 4i(3 + y0D + y2D2)] -Y0AD + y2D2BI

I eY O D-z -YOD

c1)D3 2 + yD( + A) m cos; *cos te1 e-Y20 1DOD

2ir(ai + %e1 D YO)I+ 0

43y0D + y20D 2)]

33/34Reverse Blank

Table 10. Magnetic-Field Air-to-Subsurface Propagation Formu1i,,

Dipole HType H

VED 0 p Cos,(1 *-y 0zeYD2iD2 (I y- yDA)

m~~2f Do 2e1eo

m iyD c(s *e e D sin *)C3 +3y D +y 2D2)0

- sin2 *~(lS l~y0D + 6y2D 2 + y3D3)]

sin Oe ee

-E s win tD3SJO [2 y;D si o +e 3Y0D eyD2 2-yODc Y1 sf. O

2'ITD3 (2 +J sin~ + A)+H _ _ __+A+_ _ insn ( yO yD)

s=if2 *(3 +- 3y0D + y2D?)] 2nD3 10

TR 6883

usraePropagation Formulas (jn21 10, D > 106, D > Sz)4

HHZ

z-Y0 D(1 +Y0DA

me e'l 2eYOD- f)D5(l y1' sin ~bl,(9 +9y 0D +4y2D

2 ~ 3

0-sin

2 *(1S + lSy0D + 6y2D2 + y3D3)] -sin 2 *(75 + 75y0D

+ 33y2D2 +6y3D 3) *sin 4 *(1tJS + l05y0D 45y2D2 + 6y3D3)~

z1iOD sin *cos 1e 1e O[(I yjD sin *)(3 +3yOD +y2D2)

Iz e*2.2w0 - zD0-- 1-I yjD sin q,+ y D yo y) 2nB

0 0

-YO sin2 co *(1 e ~ 0 yD D)

y0D ~ ~ ~ ~ ~ D U ~2)msn*cs*~e0[3 +3y 0D +y2D2)

-sin2 #(1S +lSyD 2 yD2

+y3D3)]

3S/36Reverse Blank

"VIM w M I

Table 11. Mir-to-Subsurface Propagation Formulas When:

4 Diple E,~+:i~( oA e

jE -YPp-YZ-O

0wc +.( + ydh)

(3 + yy 2 0 A

-Y ~~mez~ P e e __________

3(2 + -yo + co2 2e)~

liED 2w (r + iwe 1) P 32r( wl3 2wira + iciw 1)p3

x (1 - y1h + y - 2P2B o~ y1h)] x[3+3yop)sin Ayp y P 2B]

y1m cos #e Yle 0O y1m sin fe 1ecP m Cos #e I eYO0

H4D 2w(al + ib)61)p3 - 2ir(a1 + iwc1)p

3 2ir(a1 + iwsj)p2 -

2 (+ + p2A) x [2 + yop(l + A)] x (y 2)(l + y pA)

x (1 +YOP + OP 0-

L'

YINTR 6883

Formulas When P >> h (1n2l > 10, p > 106, p > 5z)

I JHpHf Hz

0 pe~-ylz e-yap00?p -( y~pA)0

-Y1Z -yip -Yiz -Y0PRe e . me e -q 1+ y

2wyl 0 2w(y1 - y0 )P'

2 )x (9 + 9YP+ 4yp2 2 y33)1x (3 + 3yop +o P O

psin fe 1e~ p cos #eeY1ZpO sin je2 e ~2irylp

3 [2 +y 1 h 21rylp3 2wy -2)4 +yh

yOp + Y p B] + y p(1 + A + y1 h)] x (1 - yh + y p - yp nB) x (3 + 3yop + y p)

m sin OeTyzeYOP a Cos *e-Yl e-yop m sin #e-YZe-o

Zirp3 2wp3 Zirylp4

x 2 + yop(l + A)] x (I + YOp + Y2 2 A) 2 (32)i'~

37/38

Reverse Blank

-~~~ee a - ' ,..--

Table 12. Air-to-Subsurface Propagation Formulas WhenI

DipoleEEType EP EE

ypcos *e-I *t1e-YOh pe__1j e&YOh +y

VED 2w1r@ + iwel)h 2 0 2ir(o + iwe1)h&

x (1 + yh) x (1 -3sin2 ,+ y~h2 COS24

y1m sin cos *e Y1Z eyoh

V14D 0 21r(a 1 + iwcl)h 3 0

x(3 +3yoh + y~h2)

y~ o i e-111z e-Yoh Yl i i e-y 1z e-y 0h p cos *sin.* cos *e -yze -

lIED 27r(ci + iwcl)h2 2w1r I + iwe1)h2 2w(a I + iwe1)h

3

x (1+ +~h x (l h) x LL 3yoh + y2 2)

Tim cos *e le~ Y1 m sin #e 1e~o YO Cos Cos*e T ZeYh

ID 21r (a1 + ieh32ir(a I + iwcl)h3 2ir(al + wlh

x (1+ Yoh +y~h2) x (I+ Yoh + yh 2) x (1+ Yoh)

......

TR 6883

-n Formulas When h >> p (In21 > 10, h > 106, h > Sz)

HP H, Hz

-y :z -yh

cos ,ee-I e0+ y0h) S0 2wh 2 0

*2h2 cos2 x (1 + y0 h)

1 YI-Yozyh eYlze-Yoh

sin 0 cos *e1e m sin #e- e2wh 3 0 tYlh 4

.x (3 + 3y 0h + y2h2) x (3 + 320h + y•h 2 )

V

e e ,, p2sin hsin2e e p Cos sin teYz- eh p sinfsin cos ef1Ze-eh

j)h3 2'rhz 2rh2 21ry 1h 3

x (1 + y0 h) x (1 + y 0 h) (3+ 3y0 h +yh 2 )

-y~0

-4Oh 3 -Yoh YZyha___. ___n _____ h sin # cos #e- lme-nc.°h,

I 2rh3 2wh 3 21rylh 4

Sx (1 + y0h + yh 2) x (1 + Y0h + y¥h 2) x (12 + 12yoh + 5Syh 2 + y"h3)

39/40Reverse Blank

Table 13. Surface-to-Surface Propagation Formulas for p5

DipoleE E EType Pi

y1peO -peTO

VD 2c + iwel)p 2 0 -2wiwe 0p 3

x(1 + y~pF) O 1+yP + y P2F

04

o1 e-O me'YOP

1iy -O 0 2wylp4

x (3 + 3yop + y2 2)-x(33y

p cos *eTYOP -yo iwuotp cos # *--rOPsnep sin_#e sin_6_!

PIED 21r(al + 3i)e 1)p 23 + iwC1)p3 2irylp 2 2wylp

x (+ y 0 P+ Y2 P 2 F) x [2+ y P(l +F)] )l+~F x [2 + y

ipmcos #e-yo o)I1o sinl #e -YPiPMCos # Pmsn#1

feD2wyp 3 2wy~p 3 2w2 2wp3 '

xl+ YOP+ y 2P2 F) x(2+ y 0 P(l F)] x (I + YPF) x[2 +

-~ _ __0

4

TR 6883

gainFormulas for p '106 CIn~2I > 10, Z h = O)

0 +e

2wp 2 l pF~2 2F)

2wy~p4 02~2 2p

x (3 + 3y y2p2 ) x (9 + 9yop + 4y~p2 + 4p 3 )

p sin *e~~0 p cos 4eT pSl21yjp3 2ry-p 3 2w(y2 -2p

x[2 + yop(l F)] 2~ + ~ ~ 2F) x (3 + 3yp+ y~p2)

pa sin aeyo mCos #e -YPmsn#-O

2wp 2rp3 2wylp4

ýX 12 +*e(l+ F)] + 1y p + y~p2 F) x (3 +3yop y~p2 )

41/42~1.Reverse Blank

f'

TR 6883

REFERENCES

1. K. A. Norton, "The Propagation of Radio Waves Over the Surface of theEarth and in the Upper Atmosphere, Part I," Proceedings IRE, vol. 24, no.10, 1936, pp. 1367-1387.

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

3. K. A. Norton, "The Polarization of Downcoming Ionospheric Radio Waves,"FCC Report No. 60047, National Bureau of Standards, Boulder, CO, 1942.

4. J. R. Wait, "The Electromagnetic Fields of a Horizontal Dipole Antenn• inthe Presence of a Conducting Half-Space," Canadian Journal of Physics,vol. 39, no. 7, 1961, pp. 1017-1028.

5. A. Banios, Jr., Dipole Radiation in the Presence of a Conducting Half-Space,

Pergamon Press, NY, 1966, 245 pp.

6. R. K. Moore and W. E. Blair, "Dipole Radiation in a Conducting Half-Space,"Journal of Research of National Bureau of Standards, D Radio Propagation,vol. 65, no. 6, 1961, pp. 547-563.

7. P. R. Bannister, "The Quasi-Near Fields of Dipole Antennas," IEEE Trans-actions on Antennas and Propagation, vol. AP-15, no. 5, 1967, pp. 618-626.

8. P. R. Bannister, Utilization of the Reciprocity Theorem to Determine theNearfield Air-to-Subsurface Propagation Formulas, USL Report 786, NavalUnderwater Systems Center, New London, CT, 22 November 1966.

9. P. R. Bannister, Surface-to-Surface and Subsurface-to-Air Propagation ofElectromagnetic Waves, USL Report 761, Naval Underwater Systems Center,New London, CT, 17 February 1967.

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

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

12. M. B. Kraichman, Handbook of Electromagnetic Propagation in ConductingMedia, U. S. Government Printing Office, Washington, DC, 1970, Ch. 3.

13. P. R. Bannister, New Formulas for HED, HMD, VED, and VMD Subsurface-to-Subsurface Propagation, NUSC Technical Report 6881, Naval Underwater Sys-tems Center, New London, CT (to be published).

43- .. -~' ~ - - ~ --

TR 6883

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

15. J. R. Wait, Electromagnetic Waves in Stratified Media, Pergamon Press,NY, 1970.

16. R. J. King and G. A. Schlak, "Groundwave Attenuation Function for Propa-gation Over a Highly Inductive Earth," Radio Science, vol. 2, no. 7,1967, pp. 687-693.

17. P. R. Bannister, Extension of Finitely Conducting Earth-Image TheoryResults to Any Range, NUSC Technical Report 6885, Naval Underwater SystemsCenter, New London, CT (to be published).

II

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