the ancient and modern history of em ground-wave propagation

7/23/2019 the Ancient and Modern History of EM Ground-wave Propagation

1/18

T Ann n Mon Hsy EMGnW Pogon

James R WaitUnivry of AizonaTucson, Arizona USA

Kwor: Gound wav propagaton

Abtt

Radio-wa ranmiion or h urfac of h Eh i abj of nquy gog t bging o nu. Inh vw an m md cbe t gn-wvemm a omni-present. We s ll ann lynalyta tibts f Zn n Smmfl ba on a-Ea modl T ubun nv ptll wregar o l of Znnk ufa wa oulind Fr

r opm y n sc a n Pol Fok,Brem, an Mngon, rw an atmp sma o p h n mod conx W ao how that hrapp sa w n b a ignian ono t h lgonwv wn n uc nv Mxah o an oning mod b Ry King dscid bry long w lua gaion u fwo- an tr-sin pas appn iliogpy nuefeences to r , uh a topr racti dopgrap nuncs.

ndu

h progon adi w bn b f inrs a nury Beginig wi te oigna insgions fHnic Hz [] t nun nrnng obuon n h wa qon o po d. In paricar Marconi isg wann of ng wn a wa lown ansmitin d iving annna

a con, u a b pe Nla l [3]w a t gnl w gd in m fion by t i-ony At ti, th pil n o inspw lso broached [4]. Bu vang a a h ie (i a u f nry) wa a mw n of wa a y acor o b kon wih Thi wa h bginnn o huj ha w now al "gronwa propgaion n fact suh

Eiors n: Jim Wait

Oo 1 198 his abeen eviwed an fo plicatio pr o hs dath AtP. Wai' eque n at hs da Davd H on oh mmndaon f vw an mad t nssane Jefy Young roo un M i o Po Wai o man o aci ppor o anonruion o h Magzine. n moria or ro Wai appawhr in hi i Hi man onrion o h S anri con of which ti t onxml a h i mot-ting mmol WRS]

h opi f h riw Of cor h ionopr pay a rott y t nrndng mnm of unwv ransmsson mpott in m aplatis.

n wha oow w wi amp o mpoy a conin noaion whi omim ma difr raiay fom a fon n ly l Of s w all al mo S rain l MKS) n an ni m ator xp+jmt) ml ougot.

3 Te Zee ue wve

op w h pp of ai thotica pysicists h ay w bl a a ua wa wh a low anaionnd hs wa t sujc f a mna papr y nnk [5] Hisml (i ou nttion) wa a coning lpc or 0,wi oniy an pi . air or inat frz 0 iiv Co ual a or a T wlgo aum t nnme, n n a neabiliy is am a f-spc valu or anmon nhe ireci folowing Znnk s amd a o 0,t ol magn ompnn n b pr y

a oponing o for z < 0, by

Hy=bexp(-+uz jgx),

wr bo an b a nn n wr

uog +y '

jJOW( jw

u= g +y ,

Coni wi Maw h angenal l nin o Euatins (1) an (2), r

Ox xp gx and

= +Kb x u (4)

IEE Antennas and Ppagaton Magazine, Vol 40, No5, October 045-23/$ .


2/18

wh Ko = u/JEoO) and K = uH jO ha th atu owa ipdacs

Bda cond a tat gnad nt aotnuo at h iac z = O Ca th a ha ad at a d

5

hu h did sluin fo th propagao of th Znek wavi

= Yy Y +Y 6Now t be angfu al pars of both Uo nd u shold bosi. Th a a conq th agi p of U is ngai whil th agna p o u s posii Thus i kpigwith th phsica cp f a gdd surfac wa th daitds atenaed th direcos away frm the inerfabt th pha vlocit ar dwnwad boh above and bow hnrfae a

A ca vw to cod a pa wav cdet

om aove whre vaiaion is xd accordig o exp- .In h ca h rcti cocnt a h racc = , ifd t Ko -K)/(Ko K, n ts f h imdance Koad h pe o t recio cocint s dtined prcy by Equato i the cpex plane ithe cae, hep veiy of h Zenek face wave he dieci is

(7)

whch is r tha th spd o t n a Alo h attenuaton in the dirct th real pa f n pr/m

Aothr asiy dmined propey of th nc sfacewave th wae tit of the lectic ld t n a at th ntrface i dee by

(8)

A smp xrcis hows tha

9

whch is a rekably ple rut h tip h ctrc ed pe ly traces ou a eps, a shw nielyb k [] aot a he coonding sfacimpedanc, not srb by Zk gv y

(10)

whch s aso rmaaby e bt a rsl ot foud in lasscaltts.

i ueu o not that i h postatd mod was a grazigp wav g would b pacd b Yo o jk. Th th wa iecs

= (IY 1-YY , 11

which rduce to uation \ 0 when 6y 1. he corpondng sufac impdanc f th gzngnciden wa

?]=fy -rolft h dea ha a Zncktye sfac wav cud b epoed toveigate the sfa r ayers ws pnted o bh

Zck hsl d b his clague Hack 6 A we ndatedab wav il d i ipc poaiaton is an dicatr the and he fo h hogene hapace Acualy ack[6] also endd c's oalism to a wolayer-ea mdln fact Gokpf and Vog itrretd heir adwait data n o th nka hy A w nownow th gzng laeralwav pola a me raisdscptor whe pacuarly at the hgr unce jEOIis not arg cped with e. fo feqncis ab 1 Mve tpic god cductiviies of 13 m

o u h Zenck wav n esectiv w nd o deal wihth citaion mhanism; ha s u x ask

4 h S bl

h gudgwav canism o th ar-earh tracscmd to b a ladg condr to pan he ong-distanceramion o radi igna basd on th a h knowdga bgnng f h cntu old Somrfd 8, in ariculr ook t lad re in an atempt to put the queon o a aaica as H lasicl analy deal n a ful gos fashon wh th prm of deng h d erc ad magntc dioles ocat i h insuatng ha-space ova condcing hal-pace fac ao cnsdrd t dpols b locaed the paar er but hs foaon smd oha d to om uedd mateatca cpis r wju da with the dipoles bg aboe h nrfa, bt thei

heght ay bcm as mall as dd. Th sution is otliedve bry here

h rrc Fg 1 a va o cn lnt I i located a v h coductng alpac f cductivy d pitivi Th gotr, ashwn ad e cylndica crdiates wl b u oatr fnc Of cus in gu for h rtical ctrcdpol excitatn zutha ymmtr vais ad u 88 0 ommd's pocd rcapitlad by man is r

ge 1. ec eec poe cen eent oceve ne s ce

8 es d oto g N 5 8


3/18

the eevat eld cmpets i tems f a Herz vectr, whicheed have ly cmpets deted by Uo r > 0 ad y Ufr z < 0 Thu fr z 0

(1)

(14

15

z 0

(6)

()

()

Nw w w f baic thry that th prmy potential

ug, i the t o the p f1 8

where R = 2+ h?t he key t the slt is t emlyth eqalt frm

U S4 J ul exulz h Jggdg 0)

where Jg t B co o od ro ad ametg. ?)

(

21/ .

ga w ve U g + =

q

alld th Sld Ir Idity 8 wic i atdsrpt

h sdary ptetial i the reo mst be addedt U ad t resltant ptetia r > 0 s the

= J{e I

+Rexp[u h]}ggdg 1

where R is a eecti-ecet ctio. Fr the reiz < 0, a sabe f fr the tetal s a smla teral wherthe terad hee nles ly the fut eJ hodv stio mplcitly cotaed i Smmereld's909 ad 96 paers 8 ead t the reut

22

where dd r K u!jO d K u! + Cer the terd qo 1 has a p t = asive b t 6. s hee ae bah pt atg = g w N ded bew Eltxpio or t tl a my obtd by o

i the eatis idicated by Equts (3), (4 nd (5whch dy d be writ ot

We t s en feable t emply al thds rth it teal disussd he t is alwy drable tamie asmttc sluts piculaly whe k s learamr ad/ whe ad h are mal cared wth erewe wll jut ute me thee results.

I the rst ae let us l at the eressi f the eicalelectric el, i th ar at + ad fr a sue dil hehth = . Als we fllw Serfeld [8 nd Nr [0] dassume a least iitially that ad als that I 0l 1 eiec hr ad later we dee the mple efrat id i low hal-pae by N d ts

(

wh l tt a vetalldeed he retrti here that N2 ees t have beeid by eary all wrers beii with Smerfeld ad uutl the rt era.

t te be rss

2

where E is a refeee ld ive by

E = 0 l expk) 5b th ectr d d tn or cty codcti t ah Wth thi coit ormiztio t eect te rund dtty s aunted fr by the "atteatn

cto" F s t s my desrbed. ccod with

Smerfeld 8 ad N 1 it dd by

where the meal dtace is e y

t i mptan t t tht t tn mpleet Eqat i deed y

cerfc{= Jez2dz

28

where thc ctur he le plae is m 2 ia astrah le t the ri ad the al the eal ai t + Aswe c o he SmmefeldNt prble rnes

fm t adians the hae ale ranes 0 to/4 rd Wit the iit llo tt o I 1 lead asmtti appiat fr Equati i

F( 1 9)

w cdi tr y 2 - I i cIEEE Antnnas and ropagaon Magaz Vol0 No5 e


4/18

ck whch has he xpeced atral wave beavor o lw cotemporay noenclature [1]

In iew of he abv, w mgt ak dost te ZSW(th ek ufa ave eerge this lmt? Mathematcally,he aswr tha he S pole, v quaton (6), mproper and occus on he oer Rmn seet of he cope pl. Pyscal, th xpanaton is tat t ciaon of th SW

by he highly ocazed oce weak u ndeed, at hot disaces whee pi s elativey sma, t ZSW does pay a ro inspt of s beg o e "wrog ema het. idt bhe rss evp of Equaion (26) ive y

F I P = -) t P }f P +. (3 )Th second rm w ulipd b o ads to r!ampltude dpndnc ut caclations of IFp ch as dlayed y Noton 10 for te ne 0 I < ad !2 ar< 0, demosae tt Fp neer exceed oe.hs in ect, hi r term s bein wame y t oters ecept we Ipi is smal.

T et cople of e asi ee evn for tshy ide


5/18

naltl sudies o he bsic ntegras were cred out by Rce[7 and Wise [8] Bot validated the Sommered-Nortonfor for F

In spite o the appren closure of te antcal deate theontroverses woud no de Th late Kene Noton n eminentrdo engneer n e US ecnged numerous eter wthSommeel An emple [1] fm ommerfel o oon waswrten while te ormer ws on hoday in te Austrian rol.

Somfld ne ag n r iu ad v be ade.Tw lter etters from oron [, 1 ndicted s views on teseparaton of ae wa and sc wve. ommerfldkowlge N' ats, and sggese e compe his reslts wit those of van der Pol and Nssen [16 Ofcorse, tere consstency here ecause oton ] basd spaers on n der Po and Niessen

t ag eah n e USA w a the BelLboratories, ws Ches uows. H carefuy mesured theld streng a MHz for disnce range fom one to00 m o deep lm lake ner Senec n upe ew Yorkstate [22 He owed usng daa on te elecrica properties o teae water pendey bae) ha th obsrvd eldrnt anc wa n ll onom wh Noton 0, bu

dffered rom olf [2.

6

o simplfy he bove discusion of pysic pnciple w O F ey al pcta a e bh heigts z and h ae noneo. There s been gret dea of discussion in t litratur for te pst 5 years or more on the for teteoy hould tak fo h enrl a o rased enals Tepenent geomety is gan sown n Fgure 1 cul vn derPol nd iessen [6] ad consdered ths a. t t was the ireles [10] wo evelope n mproved fom fo he ertotental for atray eghs (.e., z / ed ot sacompared wi on) s wrir and aticully his forer

cllae Ry J. Kn [4] posed vios fos at ad tenfo roper of reducing eacty to geomerica optcs fo any t igher elevon ages and unde the condion tt 1H R [++t as indcated n Figure 1 e form1 he su adop he and h rater th r equal ero is

wee

P_.( / !/ k e r,c e ,

C=(z/R' ,

= rjR'

38

(39)

40)

(41)

f course, in tems of he eal nge shown in Figre n ltve sall heghs S s nerone Suc s oen ssumed, ele n wrting uatons ()nd (0) witout pror explnation. Also, in such cases C s, ineffec repaced by

aletve fom oP Eqon 3) is(4

whee

4)

h same functin a Fp ded b ton 26sng aton 4) t is now ver instrucive to eite

uaton 7) in the decomposed fom

(4)

wee

45

and

46

nd were D 4 en, n ung uton (9), we seett

(47)

48)

In Euaton 47 C/C cn e dentied s he exact f e n cen a e ncdentn h a- mx nx uI + De-jkR/ can be releled fsw to indicae tat it s heNorton Surfce Wve. oosey sekng t is te correcton togeomercal optics to ccount for te wve dyics low elevaton gles t s useul to note tt for te graing limit e or 90 te sce wve vses Te

(49)

wh

=j/, (5)

5)

IEEE Antnnas a Ppagaon Magazn Vol 40 No.5, October


6/18

Thn

)

whch is n cco with Eation 3.

Smeimes the es assocated h US are dentie satera waves nee, such s o nappropae bu the stcequivlence s relly ony vad for lage numerca dsaes Ourpont s that he NSW Nn Srfce Wave) is to be dened ste dece betwee e oal wave eld d he eailycom-pued space wve r gemetrcaloptical ed. [Founte, such sconsisen wth the most recent recommentons f the IEEWae Propgtn Sandar Cmmite; see h ppend frentons of vrous wves]

At ths stage perhas s sel t say smehng aginbout the concept of srfce mpedance n he presen contex .The asetion (or w o a b rd) i o sei , hesurfce f the halspce

5)

whe, wth a ein amoun f hindsigh, sutabe fom for hesuace mpedace Z is selee It's an nresing exerse shw h luio f U o po o mo s ngre cn btne by yg tis impence cndiion,pried h w Z 10J where is given b Eua-tin (1. Acuay Ray Kng 4 hs ls pd th srce o ed -ao oo oobain esseniy he sae result as given he by Equaions )or 4)

The releanc to yeed modes

he beu in the yes of th beholde!) o he srface-impedane tehnie is tha etensons ayereeah mdels rerealy caied ot, whot havig to dea expcy wih suinswh he layers. As an eamle consider te twolaer srcueshown in igure 2 Whle s a saghtrd bt eis eercise to sove his parcua problem ab into here s a grea savg bo d et ang ecognize tat th rfceimpeance in the specra plane is 2

)

Hee ZAg, s given by Euaon () is exat bu be in mndit is uci f respondng epessions o any nmber ofayers are givcn esewhe 12 2. he qivn crcui fr hepresen twolayer case s shown n Figure 3. In this case Zsg isthe npu impednce a o a section f ansmisson lne ofngth wih ropagaion cnstant u" ad chracestic imped-ace K . he line is terminaed by n impedance 2

z

I

Figue 2 A veal ee poe loae oe a w-aeonie hafae; ohew, he gemer he ae as

n igue 1

I

-

Fgue 3 A equale ansissiole a wlae one halsae.

ow folows ha he esre soluion r Uo fo z > isgie eact y Eqn () bt now w rele he expresionin Eqaio 2 for he reecon ncton by

Z g K Zs

(55)

Z Zg) gven y Etin 4) heprblem psed th suin is eac s in he case or the hmo-eneos halfspe

hen ge tha he su gen b Eons ) or() is still vali in ppoximtc sense if he s nwrepae by the effecive nmied surace medancegiven by

5

a Eq ()g

ev lce b was beore

his seeming hc pceue cetainly eds the coect estor the space w p of Uo n Eion () B inU62 shoud no ngr be cal th ton Srface Wave or a eral wve. As we shl see below, the physics e quie deren

IEEE as ad agato Maga, Vl 0 o 5 Ote 1998


7/18

8 The apped suace wave

To ustt h fo o th layerd-arh soution, l again ke h = 0 hh men a th s wa vanishs.Th rsutnt soltion g gn b Eon (24), whr dd by Eqaon (26 bu w a disa y

Now it n be h nded n b osi frarios ayr paamrs lrt this i h cox o hod sow Fgr 2 l s asm tha h frq s uiy ow hat CIW n z cw, ad s aso gltan agnc cnrs . I = = - ' Tn wh

r ( . )12 don n JW au lPW)2 i oow adi om Eqaio (5 ha

wh

IZ jk OW I

-=

YI Q [YI + ah yh ] t

Ylh

]

'

Th fola fo h ma dista is ow g y

wh

(8

(59

60

(1

6

A idid o h mod h ox dmsossfution Q a ma o ad b alid nd pha rs ar how lotd a a io

t paet (UPW) h o io as of th ti

yi/y (/U12 n pir, i nted h ao s ssthn on for a raty h-ondin bsam hha a of Q n h h phase ag o a posiiSh wod o o a omogo ah o

Aoh sial isa a s whe th owr a p ndin ie = Thn aodng o Eqo ( wiho h aoxao

(3

th o osids ha h ayer is intl thi h atiu w s ha

64

u

:

Y

.

5

4 -

4

3 -

30

2.5

0

-

=

:

00 10IW 00

Fgure 4a. e amude of he coeco aco Q o a wo

layer odg halae, a shw Fgure

00

0

0

'40506

': o

c =

_

O

Fgue 4b The hae of he oreto ator Q fr a wo-ayecondtig half-sp, as sow gur

Th agai g PI Po P

(6

whr y?jpW(IW Now w kow tha i O7 0 d 0 wh E E o h a o in qaio ( w wl aroximad k rhof 1 0 rsndn o a thin die yr

(66

his rst is idia o Colin's [6] sson or samda or gazg id of a TM a wa oo sh

Anna ad Popaao Maan. Vl. 5 Ocbr 3


8/18

srfce It s aso sef to noe that nifor peectl condctingsce o sml rogess also exhbts a sce mpedncewhch s hgly ndce [] Accordngy, n sch cases thepase of he ueca ditace apoche +/2 adns

A reevnt eercse s to show how the mgtde ()1res as fncton of I for various phase anges o. o allowor cases of hysca neest arg b c ge om /

+ rdans or rom -90 to +9. he elic xpressonor the atenton functon s witten

where

2 2Z) e z

(67

6

Th ou eofuci igl frm e ogi t ecomple Z n te ane, v stight line. We note here j!2=l1 I exp[(J+arg)/] pecifes the comle locationohe per mt he foowng asymptoc xpnsons [] srctlyald o I , re seful:

1 l3 lx5--- . or b < O 6 )

where b s he phase ngl of The aparnt dscontiitybetwee Eatons 69 nd (70) b 0 s of no conseence yy h mlr e-P ashes Ally theowr-es aion o () ge by Euato id

for ll comx ve of fo th ag fom J2 to / bt iis ony usel for smer values of In gee, nmerc inegration of quton 6 rodes the dd data n wh foows

The functon s shown poted in Fgure or Iyg om - o 50 fo b vng from 9 to +0 Not srisngy thee is considele eancemen of he ed forostie phae nges of hs is n conomty wh qo 70; the edig e, whe mlled y cn be idented as trped uface we t h an mpitde ratonaccoding to rep( Re , nd a phse eocity ess thn = E t t Zeck Wave which woldhave a phase veocity greae thn !

he mpoace of the ted sc wave T forgrond-wve rsmission ov strted o form rog eain hs no been adequaey steed n te pat t frtr dscssion cn fod n th rfrences [] s ntertgo ot ta Che 5 of [] ad i h del ihsimi layered modes but the traed sface waves do noappear

For e sake o concseness we hav retrctd attenion upo ths oi o at- ode f a vica-ctc-do

32-

;"

0

=

F al of oaao ao () o haao fo a a fo of h al (l l o o

excitation As ch, we e eally takng out vertcally orizedgrondwe trnsmson o distances whee eth ce ist mot. h tsons to mgetc dpoe excitato n d to

biy oenons of boh doe types, gn fo at eat redescibed elswhee [ 1 ] so he ecttion of heeec uface by an appopaty tard vetc crt cosid

9 Ea by o

We hae defeed the dscsson of earth cuvare in odeo clar some contentous issues tha have arsen n the distantnd ecent ps Bt o de wth the ct physc word wcly ed o dot a hecl modl Perhas the ey aerhere s y G N Wason 2 He bgn wth the clasic hamoncseie solution for a diole in the pesence of homogeeosshee of di nd wi eecical popes iv y [

ad he suondig sace had roperes nd wthro conductivi Whil the soluton ws mattcal oech sees resetation ovig shcal Bese fction ofneger order w otorously poor in conergence, becasek theshee rd n weenghs was enomos Wtson rst stepws to repesn th eres by a cotou nega tht eosed teel axs o the compex wae-nmber lane hen he waped thecontor arod mnageabe numbe of comple oes whichrovdcd hghly congn rsdue sees n h shdow egon e the eade o oher laces [e.g 44] or to Wtsonsorgn nlyis [2] fo deais.

o impemnt th sd s oluio t was ncessay toot the oe i the c w as here s to eal wi te souo of e egvale equatio

(72)

where h s he spherl Hkel fncion of complex order nd aument k In our leicon ' s the sufce mpednceof the phere which stcty pkng fncton of v s un our exposton, ' hs been noized by O! o 10he discree poe loctons, t v v deterine the ngul vario of th popgtion Styng away fo te atpode we cy that he ed of an idividal mode o order ves s

I a a Paai Maa Vol 40 o Ooe


9/18

si(dat exp(-vsd/, whered is te greatcrcle dstace tohe oberer. mpotant cnributios to determinng the complexp locatios V were mad e by Wwedensky [46 van der Pol andBremmer 7 8], and Eckersley and Mlligon 9 Recogizethat fr he imprtant mods (ie. thoe with ow attenuatio) both

vad ka ae lge parameers but their dierce Ivs -al ' is of order f 3 his sggests tat a ey arable s t enedy

2 13t = vka (7

as emplyed cosisteny y Fok 50 einerg 1 a othervets The ke asymptotic pproxmation amuts to representig the spherical Hanel cio y

w) is the Ar inegral, deed here by

1

Jxp

[

JdZ ,3

7

(75)

where the cntour C rs i the compex Z ple om ej/ tothe igin ad ten ln the eal axs + Atll uatin 7 r its eqiaent wa empled over a cetur ago bthe De dwig z 52 wh emploed cyndial Helfucts f orer 3 ad d cmplex argumens an der Plan emmr 7 in their xtese tdes also cosstentempl hese rather awward multaled ntions that areproe to misiterprea. his wrier al defrece to isetr Hedis Bemer pefers te more ceent Aryfunctin fms. The, te mode eqatin as given y Equatin72) taes he simper frm

w'tqw(t 0 6

whee

-a) A (77)

al usel o nte tt (t saes what is nwn as Ste'eqatin

w"(t- (t 0 7

vald or al complx vales f Primes here n quatis 76 ad7) indicate differenta with respect to

The ex nntriial chre is to eemne te roots of Equatin (6) wh are esgaed ts Tere are ny god reerecesto this parcul tak oaly [12 48, 49 53 54] he ealatioo he needed resdes of the omplex ple sragt fwadwher tpl mk use Stokes euatio or ts eualent

Leavig aside man detls we wll now write dwn theresuling rada (e, veically orete eectrc eld O i the

air at a greatirle istance d for the radial ie ecal eectrcdple source f momen d he reslt s given by the soalleesidue seres:"

where

=-d e nd

79

(80

is a reeree ed d Wxq e pagaio fato. Actally,quations 7 and 0 are analogos t utos 2) ad 25respecvely for te lat-e cse. In the preset case

x=kada is a malize range parameter, a q isdeed b Eut 7 en

where

8)

is hegt-i uci Here [ka f kh ndYb (ka)t kb were h a b are the eghs of the surcedipole d the seer rspetive aove he surfce f theeath. A umber of approximations are risic here These arethat h a hb are somewha less th he rge d whle e latteis lss tha h arh radis a. Acualy the pprximatiGsY I + k where h or is id or the domitmoes beig the same as fr a at eah.

he ms etese presentato f nmericl rests or is othcram [5 who als accuted or the modcaticased y nm moheric eraio. The wellow and sill

e usel paper by No 6J i ased esenially on the eadigte o Euato whih e augmened by e omplemetar aterh resls e trsi ewee the spherial dthe planar prictn i cginrd y a graphil preduhc is suprsngly acurae Ante ery d al ey seuldeepmet ws made y Buows nd Gry 7. her shaowfcors in a sense were a oncept later now as ck TheoryIn th cection tere exss beautiul expsitio i a kheedreport y eso Lga A followo paper by Loga nd Yee eal speccay wth he atteuatio a phase characteristis f the creepig waves on onvex cyliers

A iteresing questin n e psed. \ht happens to thereside-seres represetatio suh as gve here y quatin 8)whe the eh c rvte aishes ta is whena The onergene o he series wuld seem to be icreasigly bad Earliernuerca studes showed that he spheraleath prpagatn fac dd ed if 1 or mre terms in the resdue series were aken Bu the aalytica conetio was not at allobios ere, the genus f remmer 60 came to te for Usighis approh this wter 5 61J fu tt the fllowig series iniere powers f ka was aid

IEEE Atennas ad Popagaton Magae Vol. 0, No 5 Octob 15


10/18

xq FP)-(} 1 +2PFP]06{1 ;1 _P)- 2p+ r FP} + em n etc,

wee

V(q3)

j Tis ye of epsion wa s

in a region were e at-earh aaon Fp w becomingiaduae to peic te phase f e gun wave Of ourse atlrger ditces te resiue erie as a maeable conerenceFuheore n he ovrlap reg at intedat rang, h conisency f the eahcrvatecete frm a gven bEuaton (83) w rede sees was an exceen ceck onhe nmercal wor.

Thee a eoecal pont f vew whic help one(ie ths wrier reconcile the apparen isp f he olionfo bewen te aearh et a e reiue series a shoge ve te peal erh h orgnal Smmerld soluonfor he dipole ver the hmeneus hafspae invled an integal w a brnch cut a = k he corespong brac ite lower afspace of he omplex pane, le to a onnouspectn o ave B h w eall eqiet o e tliy fte esde erie, when te nnite n of these was oeren he o ero cuatre

related and il esiabe prope of the sphercalearth

- ouo i ha he radial wave funcon hae rthoonal over e range < r < he pyical pobem mat wit an mpcdanc bunda condon a r = [he jor f cor s tha e srae mdnce m notdepn on te mode number. In tis case, fr a ni e eh radis he moe spec entrey iree muc o he pie osome of my eteemed oleas].

Mixeh heThe neee extenin of te peal eah as foate

above to aw r lateral nnunforiy s now consere i ater lime cnext h claic eample i he iuon whthe pa beween ure an obsere pae from la to sea ofr ea t and. Apa from me early emprcal appraches topredic te edstegh-ersuitance vaon, te rs analtcal prposa wa pt f eorge lngton 62 Hege a p to te ase om a an-baetransmtin ntenna, behaed n he maner as for a ll frmor homogeneou pah Beyon e coaie propagain wasaume to beave od -a pa w aow onn e ec a tecoatine or ranmision in the revee recion, e employe

te ame argumen. Bt d behol, e fod hat he resantelds fr te ame ourcedpole curent moment were dffeentUndeteed he then took he geometr mea of te w predctos nd aeed at ch ws "go imte of he acalel. Indeed, recprcity is nw atise Geores een intitionid no fail hm, because aer aalyical developmnt as dcusse elow owe h c a e-geer folatina valiiy in a imited sene

Again it wa Brmer w wa nspre hen o poe anacal bass fr tis mixedpath probem is wter

receied early personal accont of tis wok in 95, mBremme follwing h presentaton at the 954 RS eneralAsembly n The Haue. Hs mltipecattein approac, whieah cmpcated dd a aymptoic mit repouceMillingtons geomerc-mean fmla

A smewa moe geneal e m-pteometes i based on the compensation heorem as frmuaeby G. D. Mnteat 64 J. I a he go fone to vst

Mony a t BBC Researc Lab n Tawor, ey, in May156. At the ime we scssed ow i fomulation cld beepoyed o e u an intega uaton fr propagan over lateral inomgene paths Bt to inroduce te topic ere e meolne t asc concepts.

Monteths cmpensan theorem i acal eated

renz' ecprca reaonhp fr ecor and els 6]In the preent ontex, it is appie o a mo sface o inerfaof srface mpedancc wch may be ncon o te srface cors The ufae is aenwhch ha a lateraly unfrm suface mpedance wat foows Z and ae refeed o a te ded ad nmode srfae impedes repetivel Now we locate wo sh ateasdenoed and , ovr h urac. hen te mtua mpednebeween the terinas f thee tena s enoted Z for temoded sface, d z for he modied rface. olowngMotea 4 w ave t legan-tdeepvely-mpeappear ret:

= Ua1hflJEx H b L 4

where d are te vetor elecric d manetic es oerte modied srface fr a uent nece te n anea b d E an H ar e vcto l or the died rfac fr a curent nced no e termina of nea A icate b he bsp in qatin (84) nrmlcmponent of te co pruct are taken an te interati is

o h S Z dffes fom Z I our duon tenentia ed safy e appropriae surfaceimpence bunay conition bt acualy ation 84) i vald n a mre general ense.

Mnea' engneerng reerh uaton (84) waapped to anea grun sstems at MF were he modie andnodied angntia d o de ls e mde angential e cou b eac b henmded eld t e moied nentia eleric el sn replcd b ' . h n pproach an be e el fr a nmber f problems in rder to gain initial sessment.B we wl go a sep fer an how hw one acco ore change o te ode angeta ed foe dong the du

ble integral Euation (84). simplify te icsion we cnier ieaze wo

ecion mixed pa as icate in Fgre 6, where the grea-circetanc btween aneas d d. e uface impedance aonta Z ecep fr e moid poion of on rfacempedace Z of lengh mesed from ntea ow,accorng qatin (84)

U(I -Z)t' t dS ]

1 6 IEEE Aes d ogo Mge, Vl. 40 N, October 998


11/18

Figure 6 Th gr go oc

hcl h h gr -rc dc d d

Fgr 7. h gr fr hr hrclrhh

whee ad are te tagetial mageic ds ove h surfce SI te ight of he junti btwee e tw egions hdouble ingal ca smld f w ay a aonapha

argmnt to rduc t to a ie itgral of egh This i jutied by ong a cotbions from oIh gracrc pat edto be selcacelg is asec of th roblem is disussed nsom dta lswh [31 6 67]. Also n quato 85 i ssefl no tha

ad

_

a h.JflOw

- W( ) - 2d xq

(8)

(87)

whe ' ad W ar th popagato facors for h moded ad

h umodied cases esectvel. Here, ad are he cv lgs of teas d b, esectvely We ca also xprssH ad b m of thes ropagato faors. Lang asideurth a we av a te folwg msoa itgauatio 68:

(Xqql ) (x/)l Z e}4(ql _ q)

lxfW(xX" , I)'(ql q)[(- r12d

whe, o be cnsst wh ai oaon

x =

ka1 3! ,. _ ka1 dl x = (ka)1/ 3 ,

2 X

-

2 a(k ( )13jq = JI I

10n

(88)

r h unknown atenuaio functo or popagaion facor s tob dind Vais ucal tod can b oug o ba

wthout making ay fuer appoximatio But in te cae ofradowave propagaon ovr th eath's surface aross ucionsh as lig oie, e refeins a y wak Thimans hat whn X i negaie o when dl < 0 n Figure W'(Xq ca be reacd by Wx or the am rasonW(x" in te integand of Eation 88 a be replacd forthe ase oiie or fo l > 0 by W l h n ffecw hae educd e inegra equao o a intgral ormula to alculate the eld vaiato as the obsever crosse coaln. smwht n m ver he whe w plac W h inegand of ao 88 y Wx" q whh would rsord trbao

e exesio of h aboe pocedure can be caied ou for amtscon pa. or eap in t as of a te-econ a,suh a how i igu te rsulant for for the ropagaionacor s gv by

l[ ]1/2 f [ ]+ U) q -q) ( - " q",q x(" x"

l?

[ ]/2 f [ -2+x/Un) (qz q) W( - )W'X,I Z) x"x-x") d(9)

whe

Uing Eqa 88 we ee h n 9)

WX",oq = Wx" z) 90 /

+[x"/(Jn)] q2 I W(x" ', QI WX' q2)[X"-' x't/Zdx

Th an (89) s xplici ormla o caclate ranmisionover the theecion pah caly, he iegraios ca b donemloing t rsidu-ris rresnaos o h W fncion

Fr 8 Mg' f d hr h c m f h h dr cu)

IEEE a a Paai Maai V 0 N 5 Ocr 7


12/18

but e ead o douby ad ry innite ee uay t ise to emoy nmerca interaion of thee convoution integals

A indcaed ae thee xed-pth omlations coisteny negect eections a the junctions between different hoogeneos egos l neeld s ae noed t sttcand induction elds e not incooraed into foruaon Fherdcuon ofth methodoogie uch a reviewed here nd extensions of he nayica tecnqes are gven n he references [96, 6 67, 7 70

Experimental sudies

Th expeenta conon of xe-path rogaiontheoi is aalale in ew lmited ce. The casic exmle ishown in Fe 8 whee the transmssion o a BBC tsterin Englnd t 1 1 2 H i recorded a a nction o dtnce for css tl 80 om soue ou over eNoh Se o ange o 00 The dt oin re ten omilington 6] whie he ccuaton re bed on qution fo se conducivi of 4 ho/ and nd conducivt of

9 x nd a dect conta o 0 he dashed ce s hecaculated exension auming he th ws and The ceebated "recove eect i cetiny i eviece ee. One old

the ancient and modern history of em ground-wave propagation

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