O vEREAJ!TH:

RADIOWAVEPRO P AGATION.

O vEREAJ!TH:

FIELD CALC ULAT IONSAND AN IMPLEMENTATIONOF

TilE ROUGH NESS EFFECT

o

^nARRY,JOHN

^~~WE,

^B.Eng.

•Athesis sebrmued totheSchoolof Graduate Studies

in partiaJruflfillment ofthe requirementstorthe tJ degreeor Mas ter of EGgineerint

,

facultyofEngil1eeringaod AppliedScience .~emorialUniversit.yofNewfouodJaod

AprilllJ88

St.John's Newfoundland

J

.r.:

'"

r :

P.e rmissionhas been granted "

to the Nation a,l Library of Canada to microfilm this thesis and to"l e nd or sell co p i es of the fUm.

The author (copyright owner l h es. reserved c e-ner .pUblication'rights , and neither the thesis nor extensiv.e extrllcts,'from i t may be prlnted or otherwise reproduced wi t hout his/her written permiee1oD. .

L'autorisatlon al!ite accord"6e it. Ie. Biblioth~que nationale' du Canada "d e eicrofi.lller cette these et de pr~ter.au de vendre des exelllplaires d\l

1;11m.. . '.

L'auteur (titul.~·ir-e.du droit d'aute'ur) ^SB 'r ~ s e rve .Le a autres dratts de public8tion 1 nr;18 these' n1 .de longfi

:~:~:;tt .8 s: .Ci:\,~~i';6~ ~~

autrementre pr od u ! ts sans lion autori8~t1oneCrite • .

ISBN 0-315-43333-7

Ii . ,

-,

ABSTRACT

Compute rprograms,are developedtocalculat e radio propag ationlossesover .- .theoceansurface. The effeds oftheocea~surfacetoughn~areevaluate d

·th ~ughnumui~al implementatio ns01modi6e~·surfaceimpedance expreeslcue.

The surfacetoughness is expressed

.

in terms of standard eeeanogt aphic models

,

rOt' the direct ionalocean wave height spectral density. The modifiedsur face impedancemaybeus~d ~jtbeith,eritplanarearthprop ag~lionmodel,rcrshort propagationdistances, or a spherkal earthpropagat ion model

r~r

longprop aga·

' . .

^~

.tion dis t an ces.

The planarearthsolution for the

e1~e"tric

6c1d'dislant:" rom

th.c~~u·rcc,

is derived using a specialdecomposi tionmethodandexprC5Se.~in the formofthe'

, - ~

.

.spatialFouriertranejcrmof the elc-;:hicfield.No assumedboundary,cb ndit ions ar_e.used"in the derivation;the

mct~od s~pplics

^{its ownboundary}conditio,os..

~

well,th~surface impedance and the choiceofsourceremains.a~bitra,y, For a highly conductivesurface,such as the oceansurfa~~L~l:\d !lndem,;ntary "vertical-"

electr ic'dipole source,

th~e

expressions.; educe tothe

~laki(!al

^{pla na r}

~artb res~lts"

Forlong propag ationdistances ,the effects of rad iowave" diffractionaround thecurvatu ;eof the

~arth'rsurface a~e

sigj;ificallt. 'A" computerprogr am hag been written using moderncompact compute rcodewhich i.mplemenl.3theela:'~i'. calresid ueseries resultsfo!;.ground wave sphericaleartbpropaga.tion.'~epro- gramaccounts forrough surfaceeffects using animplementa t ionof themodlfled":<

surface impedance for aroughoceansurface. Trans missionl?ssfC!lultsr~r-a vllJiety offr ueneies in theMFand HFbandsanda varietyofsea statesar~

presentedwhich compa re favo ura blyto previo,usresults.

\

A CKNOWLEDGEMENTS"

Thecompletionor tbisthesis would not~~~eb~Dpossible withouttbesup-. J?Oflolthe NaturalSciencesand Eaginteriog-if{csearch

CQu~c.il

^{(NSERC)inthe -.}

,~

tormof gradu ate studentsupp ort throughan NSERC Strategic CranltoDr.

JOb,DWab~.Theautborexpresseshis appreciation fort~epatience,u6derst.and- lng.nnd.sup~ortoffered'by bis supervisor,Dr.JohnWalsh\during theCaUT5tat- st udy.Finally,tlieauthor wishes to express.his sincerest thankslobi!triendeend Cbllcagues, in

parlicul~r D~.

^{S.K.;rivastavaandMr W,Winsor,fortheir}

~iscu;

^•

..

^{). ." '}

. . . .

. sienaand assistance/duri ng theprog~~of thisthesis.

d /

"/

\

\

.I .

..

-'_.~

. "

• .,J, '::

. " ,!l!

i.

'.~

. "

. ,.

~- ."

I _

TABLEOFCON TENTS

1. INTRODUCTiO N.

1.0Cenerallat rodaction

e .

U LiteratureReview..

1.2Scope of Thesis.

:;,'

\ .

••. rTWE£AR11ISOLUTION F<:lR TIlEELECTRICFIELD....

1

2.0Cenera.1: _ .

.2.1ltiitialAssumptions . 2.2BeslePart ialDifferentialEquation

2~;J ~tl$_Eidd n~eomposition

: . 2,4ReductiontoIntegral Equations....

'2.5Solvin~theIntegul 14uatio~s .2,6TheElectricField Above the Surface

2.7IncidentFieldFrom~nTJe'~ental^{DipoleSource .} 2.8Electricfieldfor Elementa ryElectricDipoleAtlteDD&S...

•3. ROUGHSURFACE EFFECTS ,.;~: .: .

3.0 C'eieral

11 11 I<

"

23

. ..

'0

57

'7

3.1TheModified Surface ImpedancJforthe Ocean Surface 58 3.2 OCf':lDSu!!a:~Bli ght Spectral Density...

3,3NumericalEval....ationofthe Modified SurlaceJm pedance....

•• .\

,:'.

~.:

•.2 Polesoftbe'RfSidu'e Series..::::...

.~.34.41 SphericalEvaluation or theEutb Pro(l'amAiry-r;;Dd~DS~truttuJe

s.

: .

s.

NUMERICALRESULTS _... . . 5.0TransmissionLoss.

5.1Spheriei l EarthTransmi ssionLossResults" . 6.0CONCLUSIONS

, ,

.

.

REFERENC~S .:.~

.•...•.

APP'ENDIXA.TWODIMENSIONAL SPATIAL FOURIER TRANSFORMOFGREEN'S FUNCTION..

APPENDIXB ROUGIISP Il EomCAL EARTII FORTRAN PRO-

GRMt USTING .

..

.'1

(

},~

:

.

.

~

<..: .

li,i;.~ -,: ..;..:' . -.,

^"~.

/ .

"'.:'\;"

_{. .}

^,

- . .

, . !.

-vi·-

\

\

LIST OF FIGURES

2.1 Geomet ryofPlanar EarthProp agationModel. 3.1 Real Partof Modi6,cd SurfaceIm pedanceversus Frequency and

Wind Speed (Barri ck's Model) ,.

- ' .

3.2ImaginaryPar t of Modified SurfaceImpedance versus Freq uency

and WindS~eed(Barrick'sModel) .

3.3 RealPartof Modified Surface Im pedance versus Frequen cyand Wind Speed (Srivas t ava'sModel) .

3.4"ImaginaryPart01Modified SurfaceImpedanceversusFreq~cncy

and Wind Speed(Srivastav~'sModel)....

4.1 GeometryofSphericalEart hEart hPropagation Model. 5.1'Trans mission Loss vers usDi5tance and Frequencyfor a Smoot h ..

Ocean SurfaceUsing theSph~fiealEarth;-rodel(in

. d .n

relati,to1.0 WattTransmittedPower)... ...,..../...

5.2.~ddedLossversu'~J?istaneeand Wind Speed for Propagat ion Over

a Rou'ghOceanSurJaeeat1.0 MHz.(Srivasta va SurfaceImpeda nce]. 5.3Ad,dedLossversusDistanceandWindSpeedfo r:Prop:l.gatiooOver aRough OeeenSurfaceat3.~MHz.(Srivaslll.vaSurface Imped an ce]. 5.4A~dedL~versus De teneeand Wind Speed forPropagat ion Over

~a f!-0ughOceanSurface at

sll

MHz. (Srivasta va.Su rface Impedanc e).

5.S.AddedLOss versusDistan~andWinllSp~dror Propaga.tion Over

r

13

ss

"

70

71

".

ee

••

^,~,

.a

'00

J

^{-, I.. \}

..

J - . ^\

^{\ /..t.}

,I

f

5.6Added Lossversus Distanceand WindSpeedfotPropaga tion Over a RoughOceanSurface at 10.0MHz.(Srivastava Su rfacelmped auee]

... .... ... .,... 102

. .

^'

, ,

5.7 Added Lossversus Distanceand WindSpeed~or Propa~ationOver .4RoughOceanSurface at15.0 MHz.(Srivas t ava Surface Impedan ce)

j '

5.8

Add~d L~

^{versusDist anceand}

~iDd

^$,peedfor Propagati on Over a Rough Ocean

S urrace

at~o.oMHi .(Sr ivastava Surface Impedan ce)

5.Q Added Loss vers us Dist ancearid Wind Speed forPropagati onOver a RoughOcean Surface"al 25.4-MHz.(Srivasta va SurfaceImped a nce)

5.10 AddedLossversusDista nce and,WindSpeedfor Propagation Over nRougho'~canSurface at 30;0 MHz.(Srivastava SurfaceIIJlpedance)

5.11 Added LossversusDistan ceand WindSpeed for Propagation Over a RoughOeeen Surfa ce at1.0 MHz.(BarrickSurfaceImpedance]..

5.12 Added Lossverses Distan ceaod Wind'SpeedforPropagationOver. aRoug~OceanSurface at3.0 MHz.(BarrickSurface Impedance).

.<.'

103

10'

105

\

106

108

lOll S.13Ad1.cd Loss versusDistance and' WindSpeed for Prop"ig atioDOver e Rough Ocean Surrac\.atS.OMHz.(Barrick SurfaceImpedance). 110.

S.14AddedLoss versusDistanceand WindSpeedfor PropagationOver a RoughOceanSurfaceat7.0 MHz.(Barrick SurfaceIm pedance). III 5.15 AddedLossversue Distance and WindSpeedforPropagation Over aRo ugh Ocean Surfaceat 10.0MHz.(BarrickSurfaceImpedance). 112

1 \ ,.,

l

.yiii-

..5.16

Mid~

^Louversus

D~taD~e

^{'eedWindSpl'f'dfor}

Propll.gatio~

^Over

aRougb Ocean~urfaceat15.0 MHz.lnatr ick Surface.Jm peda.DCl'} 113 5,

Ii

AddedLossversus Oistucl'and Wiod SpeedTorPropagAtioDOver a RoughOc:anSurface at20,0 MHz. (Bar'rickSurfaceImpedance]....,)... 114 5.18AddedLbssversus Oistaoce and WindSpeed forPropagation Over· ' aR~oughOceeeSurfaceat/5,4MHz.llbrrid,·S urfacelmpedanee]... 115 5.IQ Added Lossvers~sDistance and~d'S'peedtorPropagationOver aRoughOeeae Sarleeeat30.0lo.flh.(Barr ickSurfacelmpedanee]. 116

·0

·.

~.

..

TABLE OF S.YMBOLS'

Ilo: TbePerm..,..!»li',of'Jtee'Pau'II'.It)

-e e: Tbeconduc tivityof'r~eespace'(I"H).

f.: Tbe p.mniUi.it,01mfdia m 1(I"HI.

-cr.: TbecODd.~ti.ity01 rnfdiam I(I"IC~

h(:): The Ikavisider, nft ioo{I'.lSI.

E:

Th..~If«ri(^lIddilltensit yYf(~f^(I"lSI

if"

^{Them...e 'd'}

i .,...",

^vee

t.,(, .

"I.

11 :

Tbe;"a&.eticllJldflll;tY'fC'~lp.IS).

D: ' Tb" ."..,_",... ",...

^[p."I.

1:

Curr(ntd(nlity'~«(tor{p,15)

p:

'Ch"r&e deas~.

1 . :

COGductioo urreudfuit, YfC'kl r(I'.Ui l.

•

^{Tberadil.l(requ u, (p..•61}

j: ~lp.1 6).

', 1 s :

SourcecuIff Dldejlsit y.J tor (p,16 ).

(.: The~fm;u.iv ityofa lMdiam rela tiv"totree spll'eII',11).

flO: The,d,adiY"(complf1)iodrx(I"ll )

15. :

DjsplllCemutdus;t,Yecklr,See ( p.18).

f,': Cornplf1 pttmiUi..ity11'.18 ).

lJ.to: Tbe ..alaeor

D . immfdi~t

^{..lyaboy"thesa"x e}

i",

Ill)

...

.)

'1

6(z): Di,acd~lta^(...dto.(p.10)".

. .

i: U.it.edOIa~lIll ~hIui. (~.10) j: U. it.rdOI ....

'*'••

t h fwI (p.101.

1: Lt.i t uftofalotll l h l~ ( p.lg ).

E' , £1«"" ,.Id .«'""

;m~di.""

.00 .. ...

~."K.lp"~

1::.:

Al....raal 'Ilotalioo rOl

E+

(p.20)

D e-:

Th, ula,or

De immtdia~l}'

^{btolO'O' Ibe}

'UI(~~

^(p.211.

1::-:

Eltdlifhid:,dOI~m m~ialtl)'·btolowIb~...rr~r.1p.21)

Em:

AIteTnte '1o\),ion f~'£·lp.21).

TSe: T~t50UlcefUII,n!d'll~il)' Optl:t.I~1 (~Itdrkl(p.221 102; S~tpage22.

k; W).enumbo.·loIth~rUnd:t.n·ltnt:t.1(p 22).

,

'-

';

. . '

"

E.: C:t.r~i~1l1-)rompo••nlorR(p21) E.; C;ut llsilll (r)rompOflull01

E .

(p_2~).

E.: Car t.ni1r,l(I Ifompoll"~o.E(p_2.1).

E..: Car~iu(I IfompoDu t 01

E.

(p.21)

£;. :'

^Carlntu111t~pOfltDl~

E .

^(p_21) E..: Caltuiu(I)tompoflt. l

oiE.

(p.24)

E....:

CartllsianI'll

_.

fompoDt DI

0 '· E...

(p.

_, 2$'

E....: Calt.ui antrlrompollffltor

E",

(p- 1;!!>1

e.:

)Cark's i:u'l

f l; r«~POIl'1lI

^or

^Em

^[p.2!>1',

" . ,

. " ^{,. :}

.

⁽

Partialduinti.e or

E

i~mtdiak'lr..bo.rth~IU,rll:rlp~!»

i

"

....

^...

l'

-.: "" ...•

... ...

-xi-

" [_ • •r ,J"'

^{PaUlalderiulin01}

^E

immedi..loellbe~the , .. r(au1,.%~)

'It

Su (p)I'll;

Ko• C'....n. F..."'tiollo_'

K.";;

'C 'eea , F"lKtio.

'bd uJd ~"lKe (p.27 ).

...

-1:.., lP(z,lI)' .n·( z..v)·

6'(l):

Seoolp_31).

Sr r } p 31j

'V'r:.Tho "ad ;,.",·bI ..' ..artioll ...itllfn~ttollJ(p.32 ~. Veo:to, h .clian.Hdloool.... t .. eilltelt' a1 O'f) ....lio.rorl !loe

elKtI,-,ae ld(~~"2).

l1(r.II); V•.,a. ,fulll'lIonuwdtoeoln"t ..rllIte ltU J..qllllhoatorIh..

"lr r t Tlrf1rl ~(p.33).

K.: S~atiaJ..a~eDumb;,( p.34).

K.: SPat ial..·u e..m,,: ,(p, '4) U£;, S Ip.34~

U: S (p_35)

),,:: S (p.34).

-/1:

S~.(p.341

.E,,: lnr id' lIt.I•.,a,icllejdill .. pi",...immfil iatelyklo...the

• ,r""e

Ip.3~). .

E~.

: Cartoni talr; "POllu t01tit (p.39)

E_iir: e.,tHi:....171«>mP01lut0I"e..(p.39) .

~i'. : e'l&~in.i

^{__ (a)}rompolI. ll torE;;.(p:39).

·xii ·

E_l: EI.dricB. ld.bo.~tb••urf...(p_41)

E

i•: [odd~Dl.lr"trit~ddina pI....immt<li"t..IJ iabo..-.lbe

lu, faee (p.42) )

E,...: Car~.;... (x)tompoOOPlof.S...(p.43) Ei.,. C.r~li all(f)"om po llOPtof

E ,.

(p_43) E....' C.r~.ian(. 1<omp onutorE_{i •}(p.43 )

El•: Clut.O'.ia.n.ompo ll~nlorEl (P(3)

E I1/>: Cal.w. iantompo~"lI l^ofE,(p.43I.

Ell: C:ut.O'.ian"ompolI~lIlof £1[p.431 lo(w); Dipo lecu""nt(p. 405)

df: Dipol" I" Ds th(p.405)

h o: Hdgh tordipole abo vethe ",rra.( e(l'_405r The X pa.llalde.ivali ve, ol th"Diu<ddta

rUllcti oo'(p.47).

Thefpa,ti,,1d..iva ti veof th"Oir.."delta runction(p.47)

6,(z) : Th". p...,lial d",ivati"" oftb..Oi...., ddta' functiolllp·47)

Ei.:"-':"'1el ian (x)"om po nen toflb"io"id"Olel«l ri., 6"Jd (p_48) F:_i,: C"'~5b.b(1)"ompo nutoftb~illdd " nt"l"ot.;"field(p 48) Eiz: C••~s.iUl "om poo~ lltoflll"iodd"nloI«lric,h ld(p.48)

rl: Radial di6laoce(p.48,ro j.

_r.: Radial <lillall" " (p,48,&0), C4: Dipol"cODltall t(p,4~1;

R. : Radial dill aD""(p,052).

e

p. PO...,(~fd ioawo^tOfIpati;r,J "aue umbera (p,S:3) X· Polartoordioal.frOflpalia.l.. aYfoumbm{p.SJ) Pol",eoor d iBat~fOl lp a.t ia.l ....aYf allmMtlj p.$3) 0: Polarcoordina.t~to,"!,atiaJ....a.Yf D" mMn l p.53) Joll): Ot l.tlluncl ioo ofO/du O lp·M)

6: Norm31iad~urfac~imp"d ance(p,li")

~All~n u a lio n rua ttion( p.M) P.: Numt tiuldilt311t~[p.liS).•

IIeigb t l3iap3ram~tu(p ,liS) cr

I

^I:: ComplcmeDtaTY UfOf'uDtt ion (p.Sli).-~

Z.:I Surr:u:~impedanceioo!lm5(p,li71.

zo: Imp~d .nct of.ffetspaee lp.57).

E,: Taaa:entialel~ttric,..:,.L-- _.Geld(p.57).

II ,: T~n,~ntialmagottichid (p.57).

•

K...: Aven ge modilled surleee impedanceof a roughocun ."rtace{poS'J~

F(p,q ): Kernel

;:o~~;o~D~rg:~:~:::i:;~~~;dn:%t:r:i:~:~a:(;.i~~gral ~

^a

"1

li'lp,q): OetaDlu,f:u: eh(gbtapectral denlit y(p..fi9).

An&lebttWhnradiopropar.ationpathaod willd direction(p.63).

Empi ricalO(t3no&raphiccO~(P .Sg).

9: .Accelerat ion duetor.f&vity(p.W)-.

V: \Yindlpeed.iDmIl(p.Sgt.

>.: Polar cOordinlteIOf orUnheilh tlpectnJdensity_(p.~)

~: Polarcooldillate'ro'OCuDbei&htlptetraldeuity(p.&I).

CI'!

FIH P, m:

h!...:

d,

t .

hr: hR:

;, D,

E. ,

~

fT(AT) ,

fR(A,

I'

II'~II), YR:

Yr:

t.: A( q.,t, ):

q.:

K:

/ .

-xi ...·

DIf~ctiorl2./ rornpOD rDlllrthe(I(tlllh.. i, htlpr c t nld u';ly10 polar(orm(1'.61)

Ampl it udrcomponeet

o r

the OC"tlllb~ishtlJ'l'ctraldruity in pol wIerm(p64l

Tu m.mit lot.nna,br isht(p.7~)

Rte t iH"n1l'Daahf'ir.ht(p,7-t) Approximatrra<!iulof t bturtb{ p.7S ) RadiowavtltDgth(p-7~J

Stparatioo di$tanccbet wcentrlll,mitandrtctive3.nltno;l.'l(p,76j

•

Cornpoot otoflb...leeuleorientedradially ltomlb.iflbni<:al 5urb ctof t b. wth(p. 76]

Sphtricaltartb he ig hl&"inruoction( p.78) Spbericalt a1lbhe igbtSail\ rUPdioo (1',78) Ail}F'undioo5(p.78).

Traosmit:&DltDlIlhtisbtga inpan m..tcr[p.78).

poltlort ht ruid uemru(p.78J Rt5idutIt ri., fupc:tioo[p.78).

Sutfactirni>edaocefUOCti OD(p.79) St t (p.7'i1).

Stt.(p.70).

Sf t(p.SO).

)'-q.", :

Modihdlurrac:r imped an ce 'uDctioo[p.SOl 6: Sur taerimpedeeee fuo ct ioa te Bremmrr' 1form (p.821 T,: Polesoftheresiduesrrie~iD Bre mmer'sform(p.82) T"O' Limit iolvaJu n folr.(p.ll4)

',,00: Limitinlv'l.! ucs 'OT1. (p.8 4) 1,'"': Themit ulilioo 00t,(p.81)

lip':

r

Ai(t):

Ai'(l):

oq ll' ' tB;'III'

n!l~ (l):

11'1' ,1'1' Lit}, Mill' FI,,,i, l,

Sphrriul Hank elfunct iolll(p.861

Airy funfliOlu ulinl,",lternaudrlllfi rion(p.86)

DeriV1l1ivr ofAiry(uortioDlulilli1111al trr ll a ~ de'llnitio n[p.86j.

Airy"runctiolliusiol alteruledelln itioll( p.86)

Deriu'tivfof 'AiTyrunc-ti,,os.usill,altc,03~de611itioD(p,86).

DerivativeofAiry functio ns(p.86).

runc-t ionror coDvcrl cDt l cricscxp allSioll ofAiry fUllCtiollS(p,87)

FUlletioD rOrCotlVctgeotseri clcxpusiollofAi ry fullctio llS(p .87) 1.serin'or asymptoticcxplllsioB

?'

Airy functions(p.88).

M leri« forasymptot icupal"iollorAiry functiODS(I',88).

RHidue .erits fUDct ion(p.90).

Eo:,Electri c6eld'~omaDcJrme bt~ryvrrlical eleetriedipo le ie Ireespaee(p,1l3). ' W.: ·Spbe ric-alellf t battro uatiollfuDetioD (p·,9j).

GT:.Tralismit t.lItr nnlaill(p . 94),

G_R: Ree~ivr antcllb.gaio (p.94), P,: Transmit ted power Ip.04).

P,: Ren iv edpower(p.~).

• xvi-

I

" _1-. '"

INTRODUCTION

1.0GENERALINTRODUCTION

In radio comm unicationsa prac tical qu estionwhich arisesis the maxim um usabl erangeofagivcDtransmitter. Amajorcomponentof suchapredictionis thenbilityto estimate th'e~lnngth^oftheelect roma g netic fiel ddist an tfromits

"0"'.Models10' tbe

'I"t,"~neti'

^{(EM)fi,ld;n' emplysp'"ar-e}relativ ely simp le;itis the problem of

detet~iDing

^themodi6cation tothis fielddue tothe presenee01the earth'ssurrac~which is not trivial.It is theproblemofestimat- ingthe eart h'seffectsonthepropagat iollofelectromagneticwavestowhichour att entionisdirected. Inpar ticularIitis the numerical evalu a t ion of"models for radi o propagationeve rtbe ea r th,inefforttoestimate thepow erlosseslI.5aIu nc- - lionofthepropagat io n distan ce whi chis of int erest.

Analyti calmodel s for groun d waverad iopropaga t ionba.ve been develo p ed

~

.

formany years,Arou ndthe timeof the deve lopmentofradio, at tbeturn of'tbis cen t u ry,physicis ts , matbcmaticilDs andeng ineen developed analyt ica l models whichpredicted.thebehaviour ofelectromagmtie 'fiel d s in the prese nceof tbe. eart h'ssu;ta.ce, Newtheori es aswell asrefinemen tstothe old havebe en dev elo ped in the subseq uent years sothat moreaeeu ratepred ictions for an ele e- trom nglfltiefieldinthelresen ceoftb~eert baiepossible,In t,his th esismode ls

for_E~propagation over the earth are consideredand compute rmodelsforrad io wave propagationlossesin th-e presen c e oftheocean-su rfacearepropo sed.Many significanttect o rewhichalTect thepro p agationofradiowaves,such as the electr- icalproperti~ofthesurface,the surfaceroug bnesa, say forexamplecaused b / : - Oceanwaves,aswell as the curvatureofthe earth'ssurfaceandthe dilTraction

losses associatedwiththecurv a tureareecnsidcred:Ofcoursethecha r n.cleristics ofthesource,such astheoperatingfrequency,arealsoincludedin the modelling elIort.

Asa firststep inthisinvestig atio n,asolu t ion tothe classicproblemof radio propagationovera,Oatsurface isdeveloped,by all.alter nateanatysis. Byusingn.

spat ialdecomposi tion method,expressions forthe electricfieldfromannrbi~rrLry sour~overaplanar su rfacewith arb itraryelectrica l parameters is dcriv ed

i^{Th is} expressionisin~beIor m of the spatia lFouriertransform ofthe-field. Anele- menta ry verticaldipolesourceas wellasa.high lycond uctive earthsu r face,such as th eoceansurfaceisassumed,andtheclassicalintegral solution to'the plane earthproblemis derived, Tb..:.resultsare notstar~ling,but significan tsince au alternat ea pproach~otheproblemhasbeenus ed. Aswelltheelect ric~eldfor :uiyfinue sou rceand. anarbitrarysurfaceimpedancecould be determ ined, pro- vidin g lheinversespa.t ial' Four ier tran sform ofthe elect,ric6cldcouldbedeter- mined.

Diverging slightlyfrom th isresult ,modelsforthesurface impedan ce,which representstheelectrica l 'properties oftheeurrece, for propagationove raroug h oceansurfaceareexamined. Theseresults willenabl ethe prediction ofro.dio wavetransmissi~nlossesfor "r.~pagationover arougb,aea.Assuminga rougb

.1

lion bet weenthe EMwaveand theoceansurfaceareimpleme.n t~in •c:o~puter program. Thenpressionlforthe modifiedsurf.c eimpedancearein terms ofthe oceanwaveb!ightsp!Ctra l density._Forthesurfaceimpedancecalculationsa sta ndu doceanogra.vhic: mod elfor;tewaveheigbtsped ral densityisassumed.

Thecalcu latedvaluesforthesurfacei~a~ce mayhe used innumer)caJ tragsmisaionJossmod ~sto enablet~epredictidb of transmission lossesin the oc ean environment.

T~e'

^{planeearth'modelfor EM}

·~ropagation

^{over theearth is}

s~ita.ble

^for relativelyshortdistances. For longer dlsten ceetheeffectsof diffractionaround the'ph,d,,1;.,1eee or·the

'~'lh

^become,1, eIR,.,,,'Analrtl, .1modelshave been deve lopedforplopaga tion over a homogeneousspJlerical eart h byother

inv~t;gll.tors:

Weproceedto develop a

c~~p~Ler

programwhich

i~plemeDh

a rC'Sidueseriessolut ion tothe sphericalearthmodel. Theachievedresultisa numerical modelwhichpred ictsthetransmission lossesforCToundwavepropaga- tionoverth!ocea n surface

indud~

^{tbeelJ';tsoftheear th'}s eurveture. The

, I

in ft uenc~ofOCUliwavtll on thepropag ationofEM waves~redetermi nedthrough theimplementaHoD ofmodifiedsur faceim~edance.e]{pressions. fot theoceansur- face.Typicalnumericalresultsfor thesetransmission lossesare presented in

gra phical.rorm. t: ,

1.1

LITERATU'~

^REVIEW

s'

Manytheoretical models fort~eprOpaj l!'t ion ofelectromagneticwaves.along theearth's surface,~3.vebeenproposed in tbeliterat ure throughoutthis century.

.l

Som merleld{I, ,,,,,;026,IO<OIpresenteda

"'I~ll~.

^lortheprop.,atio;

.:,~a

. <- . , .

planar earth transmissionI~s^versusdistance~raphically.

.. ...

plautsu rfacesepanling two hcmogeeeouebalfspa«:s of diflering eletlric:&Ipl(p.

perties. The

upp~r

hanspatewasc=hu acteriled as airand'the lower..,..dissipa·

livelfound.Thesourcewa,sesaumed10be..ytrtiul dipole locat edill theupper hanspac e.Somme rfeld'spbysiu.lexplanationW3.$theextsteneeofaspace wave andasu rfacewave,bothecmponeeubeinzrequiredtosdi d y theMaxwell', equationswit\.thespecifiedboundarycon d itions.

BasedODan'iougr.1formula tionoftheplan ar ectbproblembyVa ndel Pol andNiesscn (10301,No~tohpg3S,I036,IQ37)proposedaseriessolut ionfor- mula.Nortonproposedthat theelectric fieldcould bedivided,i ntothreecom- poncnls;thedirectray (directpa.t hbetw eensource Andobsc,;nl iin'points),the

f ..

reflectedray(dcpen.~ingonaFresnelreflectioncocfficlent ],andasiu fncewave.

Theseform ulafacilitatednumericalcompu tations,enllblingNorton to pr{'S('fltthe

/ Wait11054,19S71gaveNorlon'ssolu tio ntotheplaneearthproblemin"n alternate form. Utililingthe surfaceimpedanceconcept,Waitdeveloped tbe sameasymptoticandecnv ergenr seriessol~tion5^asNorton.In addition,Wai~

de velopeda.nother urm

~t~e

^{Nerice asymptoticseriesvalid whenthepba.seof}

the numericaldistecee,;,isr>#>0,whichgivesrisetSl~trapped lIutface wavepb~nomellondiscussedbyWait11070). ..

Solutionsforth~propa gationover asphericalu.ftbha vealsobeeninvet j.. gatedbymany aut hors.ThesemethodsareutensionsofWalson 's11018,10101 inv estigationsof thefield from a radially-oricuted dipoleinthepresenceof, hom~gencousdissipativesphere. The solu tionwas intheformofase~iesof sp h erical Hankelr~&nsandLegendrepol ynomials.This series was.~~!r~ti.

I, :1

! /

cal lo rpropagationproblems duetotheeno rmous Dumber ofterms01~bese ries req u iredforcoonrgente.IDd~thisseriessolutionwu ooly"Pplicab~toelec- tric fieldproblemswhenthewu elengtb wasa sip ific&ntr,adioDof therad ius01 thesphere. followiDKWalson'sapproach, the harmonicse~ie!wastn.~r.?rmfd intO an iDt~ral in the ecmptex plane. Van der PoIjand Brem mer II037,UI38,lg30jformulatedthistype ofcootou r integrallor thepropagation01 radio.waves alongt.heearth. Thesphelical. Ha~elIueeucnswereapproxim~ted byIbnkdfunctions oforder1/3 and'tbeLegendre polynomialsreplacedbyf be leadin gtermintheirasymp toticexpa nsion. Usingtheseapproximatio nsVandet, Pol andBremmer wrotearesidue se ries solutionlor tbeco ntour in tegral which wll8. 5uit:ililysimple (ornu~erieal.com.pu~a~ioDs. NonceP9il]used thisfcrrnu- lat ionUsingtogencntean indep endent analysis,

.

numericalresults.Foek(lgotSIoblaintdasimilar residue seriei solu tion. Fo ek usedan

approli~ation

^for

t~e

^Watson's

sp beri~a.1

^{Hani elIuue-} nonsin terms ortheAiryIoeeucns(Abramowit! apd Stegun,106S).It isthis approximationwhichtommonlyapp e an in theliterature,although WAit

1 19701

suggeststhatbothresultsachievesimilar resu lts.

In adifferentap proachtctbespheriulearthpro par;ali oD problem,Bremmer IIg401also usesthe geometrical)heory or dilfradionto determ ineano ther approximat esolution.Tbis saddle point apprOJimationisvalid on lywbee t~e sou rceandobscrvaJ ion pointsarewellabov ethebcrbce,that isror highreceive and transmiteetecn e elevations andshortseparationdislaD ~es.For these eit ue- tio naBremm ersuggeststhai.tberesidue seriesmaybepoorlyooflvergenl. For short digtan c esthere aretwoadditio nalapp roximation formulaercr thespherical I .

," -

r...-'

j-,

t

{'...

. .

.--/'.

"

eart h altenuationIuneticn. For&smaU"radius o((ui uture aodlowhequeDC=y• powerseri es..xpansioo maybeused,udevelopedby Wait110~,UI58I 'DdBrem- mer(1058). Atalargeradius(smalleurva ture) aDapan siollillterms ofthe plaD;~arth^(Norton)attenuation(unctionisliven byWaitIIO~I^{and Oremmtt'}

~~.

.

(lgS8f~ullsusin gbothth~methodsandforav:u;elyofmli('t'imprtb,ocrs haveb: ri-presented.

~y

^HillandWail

11~801.

The investigat ions ofthe elf{'Clsofsu rreee

.

roughness^" on tbepropagAtionof elec t roma gn eticwavesoverthe earth',surfa ceeommeeeed~tbFe.in bcrg's!HJ.t.I\

resul ts.Feinbergformulat ed the problem inIninll'lraJequatjon and g:\Vca resu ltfor sm allsurface height irregulariti es.TheresultdidncraccountIur

Jil l"

effedorfinilesurfaceconductivit y. RiceJIUSI],using'\pertu rbationa lnnalys is, treatedtheproblemofsr aUtring Iromslightlyrough

rando~

sllrfales . Wait 11057\derivesanexpreseicnfatthesurface impedanceofa.!lightlyeonu gnted butotherw iseperfedlyecnd uetingsurraee.Thisresult"oufortbtin d u9tivt'eon,- Lributionwhentheheightand periodofthecorrugationsaresmallcomparedto an electricalwuelenlth. Wait (IOSg,par t I)alsod~,esan elfectivtsurface impedancefor•perltctly conduc::tin g surfacebavi~g^{auniformdist ributionor} hemisp hericalbosses whoseelect rica lpara metenare arbitrary.Wai tpgSO,part

2f.

alsodiscu ssesthe etrtctofthe earth's cu rvature usingsucbarou.giisur~~ce ... model.

IBarrick[1071a,107l bjderived aresult IorthemodiOedsurface

impe~~~ce

^of

B;roughsea,u~ingRice',pert urbationmethod.This analysisASSumesa random pe riodic sur face which may bedescr ibedbytbe average beigh t

~'pectral

^{density of}

the surface.

Us~,oceanograpbic

^modelsfor the ocean ave rage wave

r

J'

-,

1.2SCO PE OFTIIESIS totberouSbllessofthesurface.

Byall alt e rnattepproeeb,Srivastava

\10841

derives&IIexpressionfor the

.

\

modifiedsurfaceimpedanceof a roushoceanas put ofhis aaalysisor the blCk-.

scattered radarcros,;-sect ionofthe ocea ns~rface. Thean ~lysis.basedODthe thoory ofgcnetnlired fund ions,isan extens ionof Wal$h's

119801

genua)., approachtoroughsurfacescatt er.The analysis

assum~

anelementary v.ertical electrjed"lpolesourcelocated near~asurfacedescr ibedby the avera geocea n wave height spectraldensity.

Tb~

^{surfa ce}impe d anceexpression

~blained

b; Srivashy a

as

wellas thalobulned byDn/rickboth reducetothatofFeinberg.in'th e'limit- ing ease.

.-

I .

Inthis thes issolut ionsforgroundwarepropas.tio noverahomogeneous ('Drtb[sph ericalandplanareart h.m~els)areexam ined.The prima ryobjectiv,eis to deyelOpcomputet'progralm whichwillpredielt6etraDsmissio~lossfor radio.

"'!avepropagat iono'er a planarorsphericaleart hmodel with orwithoutsurbee roushne5SatIlF(~30MHz.)andlow"~rradio frequencies.

Initiall y,grou ndwavepropagat ionoveraplanareart hwithar bit rary electri- aal parametersand an arbit rarysource-arestudied.Asol utio~forthisprobl emis . ~~iYedinthetwodimensional5p ~ti a.1Fouriertransror~ma!n. The analysis~

....oJed

ona tech.niqlledevelopedby Walsh[lggOJlor" general

rormulatio~

for

toughsu rface prop asat ion Ilndseeu enn g. ThemethodusesHeavisideIunctlons to spatia llydecompose tbeelectr icfield equation intot~reeequat ions: thefield' 'RboYetbe surface,belowthe sur-face andaneq uationIJoking thefie'ftft at'the

. ! ..

I

-I ^[

.: \ ^.

⁸

"

,boundary.Tbusitis. ch·l.c.,t ba tthe

mt'b~

supplin jUOWGboun4ary

·rondit~Ds.

By~umi DS:aoelemtlltaryvt'rti~eleetne...dipolesourcetbeiflle~aJtolu~ If lion derivedherein istbe same es tbat derivtd-b ySom~trrrid^{pOOO,102 61.For} a.hir;hlYronductiv"fSU, rartlbeintt"f;r&l solutionreducestoth..rdenvedby W"il_ (lQ70]. Folwwing

W. i,',

results,(hiseriessolutionsIcrthisinl(' gralhave been presented.

~

This-series solutionmaybeeasilyimplemen ted inacomputerptogr.amand resultspresentedingraphiul re-m. Forrelati velysmallsepara t iondisl noc7' betweensource andobservatio n points ,diffractiontrrc('tsaroun dtheaphencnl surfaceofthe eartharene g1i gi~le,so thatlh(>·pl1l.na.rear th mode lwillyicldll nlis.

' .

.

ractory.propagation los sresults..The-lim itofn.pptitoabililyof thcpll\n rncl\rt h solution.is

ge~crall,.

^{consideredtobe}

~

^{...60//'/1where}

~

^{is(hese; :t.; a}t ion.dill- raneein ~iles andI thera·di~_Irequene y in ml:'gahl'r h [JordenandOalm~in.

Ig68].Theobvio us adv80b .geforusing thissol utionforshortdistancesi!Ithe smallamo untofcompu ter resourcesrequiredto cakulalethepla nar eUl bseries

t·

^{solutio n. . -}

~

^{'. •}

t \) . _

^{Whenlargesl'pu}ation djstanees betweenthe 50Uru andobservationpoints are(()n sid~rtd:theadditionaleffec tsof diffracl ionarouod the cu rvat ure, ofthe eart h'beco me

_.

sig nificant.

-

Severa.lauthorshaveprestllled

~ion.'J

^1.0~^{the prob·}

tern

Q J

ground waveelechoma go eticpropag at ionover aspherical euth..Based onclassical te<:hn;uesF()(k..

IH J451

andBrem mer

I I~OJ

^have

presc~t cd

^residue

seriesapp rcximattonstothe contour iotegra l formulAtio n or'tbisprob lem,as givenby Walson Ilglg ]. Usingth~ereeulta,anemdentFor tra npro;ra mis developedwhichC"Yaluate: th:resid ueser iessolu0oll forthe groundwaveeleetne

- ,,",

..

,

field forafinit elyC911ducting sphe rical earth. A previouscomputer program, writtenbyBert)' and Chrisman 110661,also implementedtheresidue aeriesequa- tionsfor the"electr ic field.The program docum ent edinthisresear ch offers man y

iLdvllntag~s

^{overthe.Ber!y}

a~d Chris~an

implementation.In

parti~ular,

^{an alter-}

nat etechniqueis usedto eva luate the-poles of the residueseries. Berryand Chr ism anuseaseries expa nsio nfOTthe poles, asdevelopedbyBremmer11949].

Thenew prog ra musesaNewto n iteration tec h niqueon thepole defining equa- tion,to estima tethe poles ofthe residue series.Aswell,thenewprogramiswrit- 1('11

ig

modern Fortran-77 sourcecodeu~ingcomplexarit hme tic,pertnit. t iuga com pact , fastandeasyt~followprogr am. The meth odsusedby Derryand .1'-9 rismnn placedsignilica~tlimitat ion s onth e adaptability of theirprogram to

sm allercom pute rs.

To accountfortheeffectsofsurface roughness~Dthepropag ati onof radio wavesover8.sphericalorplanar earthmodel, expr ess ions for a modified surface impedance, Ior'a rough winddriven sea,lrave beenexaminedand implemented.

Themodified surfaceimpedancepresented ,by Barrick

/19711

bas'been imple- ,mented in acomp~t!r.^{program, usinga suitable}oeeenogrephicmodelfor the oceansurfaceheight spectraldensity . Thismodel is implemented in a Fortran subroutin~su bprogramofthe planar and spherical eart hpropagatio nprogram, anduses astand~rd.packageprogram(IMSL)to performtherequire~integration . AnII.ltern~teexpressionfor the modified sur fac eimp ed ance,developed by Srivas- tava110&41,bas also been examined,Theexpress ion developedbySrivastavahas ,beenimplemented,withsomesimplifying'assumptions.A Neumann-Pierson[N eu- ,mantiet ai,10S51 model-fortheocea n surfaceheight spectraldensityandcalee-:

/

\0

latelrabSm issioDkisses over arougb spherical earth usingthe SriVistavt.mod elis ass~med. Comparisonsoftheeeulre(rom the twosurfareimpedecee expressio ns arepresented. Calc ulationsfor the transmissionlossesusing theOatrick model areavailablerorcomparisonfromBa rrick 110701_Forlransource codeli3ling~of theroughsurfa.ct'.sphericalnrl hmodelisincludedintheappendix.

, ) .

I

./ . .

\

,, ' ^.

. ,-,,'

CHAPTE R 2

T HE PLAN E EARTH SOLUTION FOR T HE ELEC Tll-IC FI ELD

2.0GENERA L

Inthis sectionaclassic problem in electromagneticpropngatlontheoryis approachedbyanewformulation.The problem istbatof,propagationovera pl:ma rsurface

o r

finit e elect rica lproperties.Th isanalysisfollows them~lbods

o r

Walsh[tUBOI.originallydevel~p;d^{forroughsurface}propagatio llandscatter.It 'is.~otespeeted that theanalysiswit.1levu!anystartlingDewresulta;ratherit will yield . seto( gt'neralequationsfor the eleetrie fieldinthespatiaJFourier transform,do main. Intb;estequationsthe ebcieeof8.source.remainsubitra ry endno assumptionsare maderrgarding·theelectricalpropertieS.ofllicplaoar surface,Of00thebehaviour ofthefieldsOiltbenrfaet The,electricfieldfor'.

l:iVI'Dsource maybe evalualt'd,-nsu ming theinverse spatial..tiurier

transr~O}S

maybedetermined.

I~ ~he

^lasttwoseericesor tbischapter,theelectricfieldforelement ary Yert i- ul djpoleanleIUUlSl,derivedas"a speci ficcase. Thisresult, intbespat ial(x,y ) Fou rier transformdomain,iscquiv~lent^{to.the integral}equation derivedbySom- meefeld11000,10261.~well, a highly conductive eurface,such astbe oceansur- rar e,isassum edyn-Idioganequ iv~leotres~lttothat or.Wf Pg10!.

r

" . ..

Themeth od ofsolu t ion utilizes .. spat ial decom positi o n ofthe electric field.

tor co m pon ents in thehallspace; abo ve andbejewtbe ear t hint erlace. The med iu m abov e thesu rfaceisapproxim at ed by'free epa ee'andis assu me dto eo n- lainthesource.Themed ium beiow-thesurface iseba ract<'rizcdbyit"(,ledrit'll ! properties,nam ely;tbecond uc t ivity,thepir me a bilil y andthe perm ittivity.Fig - lUI"2.1illustratesthe geomet ryofthe problemassumedfor thrs annlys is.

A basic pnr t.ial diffl.'t l'nt ia ] equation,whichtheelectric fieldmust9I1ti~ry,is derivedusingtbe Ma xwell equations, the electtical prope rtics ofthecomplet e space and the spatia!dcco m po s ition ofthe fields,. ThepartialdilJerentinlcqun- Lionis itselfdecomposedinto.two waveequat io ns ,forthefields abovenndbelow tbe su r fac'e,,a nd~.thirdequa t io n whicht,hefieldsmust sat isfyattheboun dary (boundaryeOllditio~s).

l!-

may be notedthat no e:dernalbounda rycond itio nsnrc applied;theboundarycq~ation^{is aprodu ct,oftheanalyeis.}

Aset of two co upled,convolut iontype, integr alequationsarcthe n~cr ived using thefun da mentaj"solutio nstothewave eq uatio n.'Solvi ngj.betwointegr al

~ i · .

equationsyieldsafun c ti onal rela t ionship betweenlhe.sourJeelectr icfield and the.

electricfieldabovethe int erf a ce , in the spatialFourier trans form dom ain.The eleetrtc'field ,for aD\Ygiv en sou rce,maybe deter minedhomthes e equat ionspro- vide dtbeinversespa tialtrans formsmay be det e rmined.Forelemen taryvertic a l electricdipoleante n nas , tbe resul ting integralequation for tbe elect r ic. field is show n

.

to beeq u iv ale nt

.

totha t wh ichwas der ivedby Som.lJler feld\l000, IQ261.

Fin ally, ahighl y cond uc t ive surface isassu med , the re.sul 13 of wh ichare equivalent totheint egralsolve d byWai,t[IQ70),

/

FIG URE 2.1

p.o.E"o;

snowTHE SURFACE

G~om c tryofth ., Pla narEarth Propagation~lode'l

~,

z·oSURFACE

:~

..

2.1 INITIALASSUMP TIO NS

Theproblem ofdete rm ining8.model Ior the electric fteldabovean assumed planar eart h mod elbasbeen approached andsol vedby manyinvest igators, among theearliestbeingSommerfeld[IQW,HI26]whodeterminedexpeeselone for thespace wave andsurfi ce waveportio nsoftheelectricfield.Theplane eart h prob lem is described asthe propagationofelectromagneticfieldsthrough a medi um approxim a te ly &.cribed as'Ieee space'over a homo geneousplanar sur- facewitharbit ra ryclectricslprope rrica,Sommerfeldassume dII.vC]liulelectr ic dlPoleso~tce.Thiswork derivesthe completeelect ric fieldabo vea planarsurface for30arbitrary sourcet1..'Ia specialcase of the~lSh[lgaOIgeneraltrea tm ent of

propagationandscatteringfrom roughsurfaces. ,

The'analysisbeginsby derivingthebasicpart ialdifferent ialequat ionfor propagation ofelectromagneticfieldsov~raplanareart h.Thisisderived by an electricfielddecompositionap proachasdescribedby Walsh[10801.First,exprcs- sionsdescribingtheelectric-a.1 propertiesofthecomplete'spareart>derived.Inthe hall-space abovetheplana.rsurface theelect rical properties are describedby the following:

Po '"tbe peTmeability

to=.tbepermj~tjy.ity "

! .

~ 'OBductl¥ity .

^'.'

Similarly.

fnfbc '

halfspacebelow the pla.narsurfacewehave Po""tilepetmeability

fl- tbepermitt ivity

"l~tbeconducth ity

-..

A3';"'1'1I,it " essumedthat, - 0describestbelocationofthe planarsurrace separatingtbetwo'half-spaces .The eledriulpropertiesof thecomplet e'pace may be describedu, i'n,theHeaviside(undions,which aredefined&5

·1'1-

{~:

^{;; :.}

Using theHeaviside(uocl ioo..(zI. theelectrk elproperties of thecomplet e space m:l..Ybewrittenin terms of thetheelectricalconst antlprescribedfor.tberom- ple~espaceas

nod

(7"",,{I-h"1)171

"e=-,10' (') +fl(I-A(f l l

/l-Po

(U) (2.2)

(2.3)

(2.') (2.S) (2.6) The! terms contlining(1-"IIII are the electric!,,'propertieso~tbespacebelow the

.surfaceandtermscontainingon lyAI,I are electricalproperti es of.thespaceabov~

thesurface.Thus . setof three equa t ions,'(2.1),(2.2),and(2.31 describethe

eled.ricaJproperties.of tbe eornplete speee. The.MuwelltqUatioosin time-

h:lfm~cform,usingtheusualconventionsforsymbols,uegivenas

9?<£'--j ..,iJ , 'vxR- j ..,D . l

Q's_ot and

(2.7)

".r· Itisassiim edthal theMaxwellequations apply1.0thecompl ete space. Wealso'

assume tb&tthe mediabot habove and belowtbesurface are linearandisotro pic.

I "~

'\ ;

..,

I i

1",

. , .

With these assumptionswe alsohavethe Ja llowinlrelationships:

D_".R forall• • .D-lE- [r, ' (I)+r, (1-'(I)

J]

^E

J,-"'.

11-'(, lll

TheparameterI,isdefinedas thecondu ctioneutteat dl'Dsity.

2.2BASIC PARTIAL DIFF ER ENTIAL EQUAT IO N

(28 ) (201

(2.101

neticfieldRas(ollows:

Wenow proceed toderivethlbasicpart ial dilTercnliall'Cluation (ortheplane eart h

pro~m,

^{byusingtheMaxwell}equat ions and the assumpuons in the

prt~.

ous section.The curloftheelectr ic field,Vxl,iswritten in terms ofthemag-

r:

(2.111 By takingthecurl

o r

both"sidesofequa tion(2.111.Vx vr lisexpr~Ma.5

(2.121 We 5ubslitutethe expression for thecurlofRinteemsofthe displ3Cemcnt cUHenl.vector,D.and the curr enl density,

t.

from equation12.51intoourpO

expressionfor V x\Ix

t .

(Th isyields

v;

^V

x

^{£- -J101"0}

[J...

0'+.1] 12.13)

Tbe currentdensit.y1maybewritt en astwoseplI.ra,le ebmponents,one tortbe condtJ'e1ioncurrent density and a secondforthe ecarcecurrentdl.'nllity,Wewri te

(2.141

.\ ⁱ _{\. . .}

The paramet u1,i5thesourcecu rrentdensity aDdtheparamtt~I,istbeeon- dUC1~Qcurren tdensity.8yusin! thi,CODl'tOl,ioorOtt~currentdl'u it y.equa- uce(2. 13),rortbecurl orthecurloftheelectricfield,maybeexpa ndedto obtain

vxvxt- -i ...".[iwD+ ls+-J.] (2.15)

A usefuly~to,.idl"nt ity,whichmaybeappliedto eqeatioe12.15) to decem-

Theequation (2.15) maybedecomposed byusingthe abovevect or identi t y.We liseexpressions(2.1)and(2.2)lorpermittivityandcond uctivit y.The expa nded version of(2.IS) is

Q'

tv -

E.I-

v'

E-

-~ "'1'.

["J!I-A{,1lE+j...

[1 0

^"('1'"lill-'II) }]E+1,](2.16) Theappesrance ofequatioD(2.16)maybesimplified greatlyby 6t'l1Jtmakinr;tbe followi~gdefinitionsfortherelativepermittivity aedtherefractiveindex:

Byusing theabovedefinit ionsequation(2.16) is

~E +...

IPilIO[N :

(t -Als}}+ A]t-ilJPo1.+v.fv'£

J

(2.17) Theright handsideoftbisequationcootaios thelTadieo torthedivergence' or

t

wbichmaybeinterpretedbyus ior; oureprev joua-results.Ccmm euci ngwit b eq ua tio ns(2.Q)and (2.10),wema ynot ethe following:

"

I

r

\8

- [-, 1' - ' ;. 11 +

iv

I ~'l' )

^{+<,f1-.{ . ) )}

l] ^t

- iv[I,, +f;;II' -'I') )+,.I .) ] { .

12.18) The quantitiesiJ,and(',are defined asronows'

0,-0 +3.-

Iv ll'-(I+~_I v

The definitionsIcrii,andf'aare appliedto (2.18).wh i~hmaybeweiuenas 12 \0)

F.,quation.12.19!may beinY~ ,yiddinganexpressioe forE:intermsortJ••the HeavisideIunetions,andth~!lectrin.lpropertiesortheeomptetespace.Thi:!

expressionis

12.201

Thisreb ,lion.hiprna' be usedtoiDterpret the diYergeDceor the eledrk filld,v'

t.

Bytakingthe divergence orbothsides ofequation (2.20), wearri"'e Atasuitable relat ionship betweentbedivergence

0 '

th~eleetne field.tb~quan tit y D.aD,dthe requisiteeleetrleelproperties.Thedivergenceol~is

'V"£

_--';" v · O.o + (1'-:0

v-(1( 1) 0,) )2.21)

(, loll

'( The term

c i

AII) D,)•i'nthe above expressionmayalso be

i ntetJ)~C!d

^by

,expanding the derivatives as(ollo.,~"

f ;.

"

V'(l ll)D,)..'I,)(V'D,),+[v:l(, l)D,

- · ll l l v · jj. l ~ i · D" ~1 1

. where

0,+_}~r:.11, .

.

^;

i5thevalueoftbeqlJl.otitytJ,imm ed iat ely abo' tthe surface and6(M)isthe Oir;c*deltaIunetionan di is •unitvecto ralong the1u~. Equat ion J2.211lor thedivergcn ~ofEmybewriUenusing the aboveresults as

v·g'"'"

~ v · D,

⁺

~

^{[A(, l l v-D,I}+"0,+6(,)] {2.22}

(, l~ l • •

In ordertointerprd the divergenceofthequantit)· 11,1weret urnto equa-

.

/' (223) enableswritingVxflinthe following

,,-

Cor m:

vx

n

_1 +'jwD-ls+1,"+iIJtJ-Is+iwD.

.tion(2.5)rcrthe curlof

, n,

^and_expandit usingthedelinition lorlJ,.This

I '

Oyusing theident it yv·

f

V x OJ... 0, we obtainanexpressionforv·0,in

gee ee or both sides of~Uallo n(2.23)yields •

termsofthe souree turren l density,Isfrom equa tKlD(2.23).By taking the diver-

~

v·lv xRI-v·l,+ iwv·1J.-o .-.

Thediverge ncenr6,may'be obtAined'

fro~

^{the aboveas}

v ·iJ, - * v ·7 .

(2.24)

By assumption,tbe support orthesourcecurretltdensity·t, lies whollyin tbe halfspaceJ~o.Therefore,itisobvio us thatA(JJ (v'D,J may be deduced i,mmooiately-!romeqUAtion (2.24)as

. , ';

!. I :

20

.II ll v-O,l--~^{.IIl v·1, .}

The expressionlor v'£in equation(2.22)maybesimplified conside rablyby using theaboveresult lor.(1 )

tv

D,). Artersome algebrav·1:iswritten in

'/

122') whereD,'"istheu.lueofthequantity0,immediately above thesurbte.,;.))ioct'

0,+- t.,E· .

where E+ is

th~

^valueor

lh~

^{elcciriefiefd}immt'di:l.telYo.l>ovethe surtece,'we

.~IlY

writeequation (2.25)tOtV·E. interms

or

thesur(nce"field.For n'olat ioollleon- veniencewe usethesym bol

to n;'presentthe valueo(theelectric fieldimmediatelyabovethe plana rsurface in thepositive balrspace.Equatjon(2.2&) maybesimplilied usingtbenprlMiOb rcr0,+,and weDOWwritetbefollowing ab ousingourecreuontottbe eurtaee elect ric field;

. .

v ·l - ·i ~ t.,

^{'1'.1. +}

"~i ' [i .£,6(1 . ]

^12.26)

(.' Theaboveexpressioll forv-Eisallint er pretafio nilltermsofthesourcecurrent

~

density]"therefractive

index "~

^{and thesurfac eelectricfield}

e .

^ill

lhe~ositiv~

hair-sp ace.We may alsowriteasimilarexpressio nin termsortheSUrrllCeelce-

/.

:/

tnc

fieldinthe~ehalf-spac e.Returningto equati on (2.1Q) wewrite t -

[ l!:!fJl

_"

^+.'k!

₁₀

¹

^{D. -}

[ 1.

₁₀

^{- 1 1 -' 1 '11} ~

_loCI

1

D. ^12.27)

I

i

(2.20j"

,.

21

foll o wingthesame meth od used toderivett:jua tioa(2..261'Wema ywrit e

v'£

- ~

^v'D,

~ f:;l~

^{v:[(1-.1' 1)D.]}

_4

^j

~lO v:

I,-

I ~l~

^{[fl-lll))V·V,-i'D,-}

~I l ]

- -j

~ IO

^{V 1,+}

II:O~I:O [i

.D,-I(')O] (2.28) The expressionD.' isthevalueof.0,immediate lybelow thesurlace,andis definedby.

- -D.-: -

!~j}.

"Alsoitisapparent fromequanc u(2.lg)that, /J,--I,'£-,whereE'isthe electric

fieldimmediatelybelowthesurface inthenegat i,ehalr.space_

t - ,

isdefined as

£.-

!~r:.E ""' ~ .. ~

wherewenowuse thenotation£..COT the surfaceelectricfieldinthenegative.

halfspace. ByIIsingthese results,asecondequationforV.

r

mayb~writtenin- ter ms ofthe surfaceelectri cfield,t..,andthesourcecurrent densit y1_s. WeDOW

write

\

Bylaking thegradien.toft<Juatio n.s(2.27)and(2.20),twoequationsmay bewrit - tcn.r~r.v(v·t~Thetwo equationsareasrollo~

v lv v lv

I . ",I~^I

£)...

--- ,.w••

vtv 1,)+- .- 9(' £' /.('))'"

". t: \

l)--j

~ f,

V(V 1,)+{1I01-I19(^I.£.6(6))

(2.30)

(2.31) Bitheroftheseequationsmaybeusedinequation(2.17)toob tain thebasic:par-_' tilLldifferenti alequa t ionfor theelect ricfield.; However,sinceour present

"

.:I ~·

inte res t lies mainlyin deter rniniag theelectr.ic fieldinthe halfspaceabove the pla~ ar.surface, equati on (2.30) in termsofthesurfacefieldabovethe surfaceis most suitab le. By. using (2.30)inequ atio n I2J7),thefollowingexpre ssionis obratned:

V 2l +<'?Po'G

[ni

fl-~{~ ))⁺

hlzl]

£-jWI'GJ,.

i -;;;

^vlv

^{1 ,1}

+ :~/ Q'[ i · g, ~~I]

^(2,32)

Inorde r to simpliry the appea ranc eof equat ion(2.32),A'Source Current Density Operato r\operatin gon the'sourcecurren t densit y']$is dt'flnt'dbelow as

T"

[1,]-; :,) " 1;, 1>4

+t'

1 , J

Also,twoadditio naldefinit ions may be medewhic frepresent the theelec trical propert iesof thecompletespace asfollows:

Tbe precedin gde6ni~onsareapplied'to equation(2,321"and theresul tingeque- tioniswritt enas

(2.33 )

Wehaved:ri vedthebasicpartial _diff~~en;ialeq~at icin(2.33)whichtheelec- tete field

~

^{mustsatisfy. It}is obvious that

t~e

conditio n has been usedif!

deriving (2.33). Byasim ilarapproac h,

.

anexpression for the magn et icfield,

R ^.

could be achieved . However,our primaryinter estisagainLh~eleetriefieldso tha t weneglectthe details ofthis'd'erivationandpresent onlythe finalpartial

\ , ^.

(

I

difJert'ntial~ualioQ.The magnetic fieldmust satisfy thelollowiog equaltoo:

V

R

+'ft'

n - -

T!II

I1s

J-iw

h -

t.) (;x

RI

~I)

Eitbe! of-then equation, att' equally suit able ror,tbisualysi!lbut we ebcoee equatio n(2.33)in thefollowingsectio ns.

Wenowproceedtospat ia lly decomposethe eleetrie field.Thisdeecm pos i- tionwillresultin three separ ateequat ions.Thefirst twowillrepresen t theelee- triefieldsabG¥C and belowtheinte rface(respectively)separat ing the twomedia.

•The thirdequation willdefinea set of bound ary condi tio nswhichmust be satisfiedattheinterface.~nthismannerno externalboundarycond itionsneed ..-be appli ed.

2.3 ELECTRICFIELDDECOMPOSITION

.The.complete electricfieldmay be separatedioto fieldsaboveendbelowtbe pillnarsur face byosing tbfHeavisideIueet jcns.To effed this deecmposi tic n we first·writethe eledri.cfield£^L5

.£- ""1£+(1-""1)£

Thillexpression may be used to spatia lly decompose the~ave equat io n

1q2l

+ ",:

£)aswritteninequation (2.33).Therightbandsid eof (2.34)aboveis

subStitu,ted(or

t

intbeleft handside of(2.33),the baaie partial dilJerential equetl on for thef'!edri~field.From equation(2.340 ),we may prceeedwit hthe

.complet edeco~ poIIitioninatermby term manneras:follows:

V2£_V2(....

"'E)+V\[II_A"n E]

^(2.35)

Each-termofequation(2.35)~ay^heexamine~individu ally.Thefirst term onthe right hand sideof(2.3S).!Or theelect ric field in the upper(posit ive) half-spaceis

:,.'.

.. - , . .

2.

6rstdecomposedintoitsCart esian ecrnpc aeete . Wewrite vI~l l )r1-

9'!

~(JI£,IJ +v' 14(.)£,Ii ...v' I'(11 E,.JJ Cons ider, only,tbt;tum,""14(1)E,I.

The gradi(' olof~,. )E.maybeexpand ed

lL'

v1"{IIE.1-'(I)vE, E,'Q'A

L

^{-.(z ) V E iE..o!I,)}

...andby taITngthe divergence, we writ...

V'l li(I )E.J-V.<;}IAIIJ£,I-v',[AI' )v^{E,... i E..}

l4:~l

]

-A(, )V' 6,'tvE,v·,qz ) -tv·

[i

E..

$(i))

- ' I' I Vr E, +i ·I VE.) ~lirl"'V.· [ JE

..

6(II ]

(2.36) Simila rlyI...emaywrite expressio nsr&l-the y andI

«l~pon('nts

^u

v~l·h (l l tj l - "' (1 1 'l72 E; + i· I 'Q' £' J + 6(' I + V - [ IE" l(' I ] .

(2.37 ) 171\1(1 1£,1- "'{liVE,+i·(QE,) +lJ(1)+

v ·l1

^E..

^{6(,) ] .}

^12.381

whereE..,E..,E Narethe Cartes ianeompc neuts

or t,. t.

^isthesurf..ee elect ric fieldimmediatelyAbo~e^{the hi t trrace,}defi~~^as

We com bine theequatiq ns(2.36),(2.37)and (2,38) to obtain the espe eeionlor

Vl ~ (;rl £i..,

i v'l"(6 )

t

1-

~

^{III v'E+}

I ~ I

^{+6(6I +{v·{IE.,6(61) }J}

~.,...

. '

" "

I ,

I

J I

!

I !

~I('('trie^fieldbelowth~surrae~,^...nd~1J~et^{thesame type}01deecmpceu leeas above,via.,

'Q"il(I-A(l l l tl-'Q"

[ ll - ~ II )J

E,

J

f +'Q"

[ "1. ' (11 ) Ej. ]; ~

, 't'Q"

((1 -

A(I llE.]I

.p

^{Taking,rcreram}ple,only tbei eompoeenr,we expand'Q'

(11 0

.l(IIJE,]as

'Q'

(1 1-"/111

^E,]- 11-.l(' 11'Q'E,.-i.f;..6(1) • :lndbytak ingthedivergeee e weobtain

v' [I, ·. (, IIE . ] -I ' ·'( " lv' E. - [ ' _{I V E. I '] ~' I - V ·[ ' E·· ~' }l}

Omilti ng thedetBilsofthenl?ansioDslorthe; and; components,we may write

vll

^l-

.i' li e 1 - (, -' ( ,11 v'£ -I¥. I" ~,)

-I

^V·[IE_

_~'J 1 1 ' -I

^{V [I}

_{'E~ ~.)} J I;

-I v · flE • . ~'I 1 1 '

(2 '0)

10 theabove1':..,,f;. ,,E...are thecartes ia ncompo nents01

t. ,

whichisthesur- Iaeeelect ricfield immediatelybelowtheaurface.E.isdefine?as

E . -

lim

E

. - '

Inrquatio o5.(2.30)and(2.40),the symbols

I ~ )

^+end

t _~

J⁰denotethe Dormu derivatives01£immediately above andbelowthe surface. Forreter eeee

Byinser ti ngtbe spati al deecmpcsiucn ,equation(2.34.),into (2.33)aDdby

,

...

t:

.~

apply iogtbe expression sforV-,[.l, )E] ae dV[(I.l(Jȣ ]USbOWll illequations ('l39)and(2.401.itisobvious lhtthebasic:equatioo(2.33)illsatisfiedif the eleetriefieldsatisfies thefollowinr;equa tio ns:

[H I'))

( v

^f£,+

'l:t]-

0 (2.43)

[1 ~ (I ~ I}( ;I -I"[ i 'E .. - r.., I ~' I Il · ^, + I,, ·[• ^t

^E••

^r.••.' ~· I 1 1 ;-I" [• .

^{(.Eo -}

^{r. ••.} I t ^{1 1}

ⁱ

- ': .1 ' " [!

·E,

~'II

Thesym bolsj.iI,iartvheCa,leSlll.1lunit vect orsand1\'

_.t,,:.

The equetioes(2.42 )and (2 .43J arethegoverniDgeq uatio Dsfor the eleetrjc"

field.aboVeaedbelow "heiDter fatt; Thethird equation (2.41)It'pt~cnuthe bQund aryecndition whichthefieldmustsatis fyat tbelerer teee.

2;oCREDUCTION~OINTE G RAL EQUATIONS

The trme'equations(2.42•2.43,and2H)mayb~redueedtoeon tion type iolegra lequations.W1mak~lise of thefun d amentalso~ulionstothewave equat ioninthe rorm orGreen's(unctions,' \

~ \

KOl(~^{,r. I}1-uP

1:: *'! , )

(2.,,~)

I i

I

I

(

Kaz(;r" .I) - np(-j -h ,1

" 4"

In tbe abovewebave usedthelollowinr

"

r_ (1' .11'7+1')

a_w~ ,

lll - t' " I.= l: 'lf· - ~I -

Thesefunctions,K01'lPl~K~must sa l is!y tbefollowingequations:

qIK!IItl:IKo1co-6( z) '{r )6(I)•

(2.4&)

(2.4 7)

(2 '<8)

Twoidenti ties enablethe use01Green'sIuect jonstodetermin e expresaiona forthe ..Imriefield. Theidentit iesare

12.49 )

)

and

v'Il I -AII))tj.K.- I(1-A(;rllE I -v'KI'lI (2.50)

Theasterisk(.)has beenusedtodenote a threedimensionalspa tialeoavolu- tionwith respecttox,7.andI.It has beenassum edinequations(2.40)and (2.50 ) thatthesetonv~lulioDS-6ist. The..bove identities maybe use-dwith tbe deco:position sfottheetedric

6cld,

equa.lions (U 'iI)and(2.40),\0write convolu- tionequationsfor the decomposed electric field.Werepealtbe decom positionfor v'

[A( JIS ]

^{homequat ion(2:30)as}

v'I III" !:1-III, )v²£..

(~(

^{<1(,) +}

tv - ^{[ I E..} ~: ) )}

^{i I}

+1 ' · [JE''!')J I I+ ! ' · [ lE. ~') J I '

⁽²⁵¹⁾

Also,thedeecmpesuion(orVI(II-•I'II

t ]

^{Iromequation(2.40)isreputed L5}

\

+ ^l- E~ ~' I] I' -I

^{0 [ ,E.,}

~'I J I;

-1

^{0 [,E••}

~'I II'

^(2)'1

The identityequation(2.411I)and theexpressionrorQ'I1' ('1~I,~\I:\tioD (2.52),may beeomb iuedtoformequation(2.53)asfo llows:

'(' Io 'l ; K ., [1*" ~'I l ' ^{K . ,} [Iv [,

^{W ..])"}

(c'

[I ~' l

^E.

^{J )} HI" [, _{~' I}

^E..

II ' ].

^{K ..}

(2)31

Severalatthe termsinequat ion(2.&3)abovemayberegrouped.This..yield!

I'(' l l l' ['(.11v'l,.'t )]•K. - -[

1 * l' ~' l l "K '

-II

^v-[,

~.

JE..

J) , , I" 1

^{q .J}

E. II'

, I

^v:[,

~<1

^E..

II' ).

^{K.. (2.5<)}

Thetormatequation (2.54) maybesimplifiedby eumininr;ands.impl.ify inr;

severaloftbp terms.Theseterm!are

I

1 i

I

i

I

_I.

I

equationisgivenbelow:

Theaboveterms are combined toyield

II ~ [i ~d

^E"

J I•

^{+ ( "}

[I ~.)

^E.

J I

ⁱ⁺

I" [ q.,

^E"

^J I' I.

^K•

. -H, [

^E"

~.,

^{]j +}

f. [

^E"

~'I 1

^H

f. [

^E"

~' I

^].

I .

^K"

- If. [ E, ~" Jt' K"

Allbefor e,E.is thesurfacefiel d inthepositivehalf-spa ce .Thissimpli6ution maybeusedtorewriteequa tion (2.M ) intbefollowingIorm:

1'(" E

I + [ ^{'(- 11v'}

^E

+ ".

^E

^I I·

^K"-

- [I* 1+ ~.) j . K~ .

-[f. [ E· ~·)J l· K.I~")

Equation(2..12),thedeeom pcsedbasicparlia)dilfE'rtnti..lf'qua tion. maybe sub-. stituted into (2.55).Thisyieldst~l'followingtquatioo fortbeeleernefield I.00Vt thesurface:

An expressionforthe electric field below tbesurfacemay he obtainedby a' similardecomposition.Omitting the details ofthisdecomposition,tberesulting

\

" r.:

!

I 1-

"

!.".

! I

30

(U'I.

WeDOWretueetothe equatio n 10f_thebou n daryronditions,equation(24 -1) Thesame simpli6.tatio o appliedto rquation12.S4)lor tbe 6eld~ytthesurrul", may be usedtortheboundary conditionrqult ion.F.quahoD12.H)torthe~n·

daryconditionmaybewnue nu~in r;thesesimplifi('ltions as

(1¥o J'- 1 ~H1 ~·) ·H I ~, ^{. e : I} ~'I ^J

-

.~;

^I•[

.~, ~' Il

- I· t ·'1 . [. t, ~'.I] ^{1 2·"1}

Equaro as "(2.56) and(2.51 )decomposetheelect ricfield intotwocom- ponen ts,the electric fieldabovethe sur face..adthe electricfieldbelowthesur- (ace.Theseequat-ioDsalonr;with(2..>8)expressthefieldinterms ofthe(ollowing four functions:

..

Itistheproblemof determining thesefund kinsto whichOIIfaUenttonisIlOW•

directed.Totbisaim,we define theincident (orscuree}eleetriefieldin

. .

terms of the sou rceeurre ntoperator,operatingon the soeree current density. The incidentelectr ic8eldis

l,

~ - T,; r1,1• «; I

The~x p.ression^-Tn11$1beealready beendefined.BylIsing"ttttincidentelectric fieldnotationequatio n (2.56)iswrittenas

12.5')

i I

. \

It1/*1 abobe.~oti(edthat

sinceE,is afunctionof(I.,,)only..TbefunctionifI)isthederivati,ve01the Dirac della function.defineda.s

1(. ) -:'.5(0 1 Thispermits(2.M~1to be writtenas

(11(11£1-

J;.

+[R+{J.')6(1)-

t. 6 } ' I] . K Ill ( .'

withthe

run~ tion n+-(I,~

^{Idefined as}

~ { *] .

^{Byusingaproperl yofaconvalII-}

tion,

'~ I I ' K. _ ~I I ' 8 :,.' .

equation(2.S0)may nowbewrittenas

(2.60) Equa tion(2.60)representsthe elect ricfield ebovethesurfacein terms01the incidentorsource electr icfield£,^IthesurfaceelectricfieldE" endthe(undi0'l i+{I,, ).Thesame operatio ns areperformedODequatioa(2.57),the equation for the fieldbelow thesurface. Tbisyields

111-

'"n t I

~-R11,,),~,~.Kn+l•• ,I).Kft

--{Ul l,ld6(II-E.'" I}'KOf • wherethe Iunettce 1111,, )isdefinedas

I ,

i·

(2.611

-ic

_I

,