Diffraction,
Refraction
and
Reflection
of
Radio Waves
V.A. Fock
Published for the
National Aeronautics and Space Administration
1975
Translation of "Osvoyenlye Kosmicheskogo
Prostranstva v SSSR, 1957-1967," Moscow,
"Nauka" Press, 1971,
TRANSLATED FROM RUSSIAN
Thirteen Papers


AFCRC-TN-SM02 A4TIA DOCUMENT NO. AOIWJ74
DIFFRACTION, REFRACTION, AND REFLECTION OF RADIO WAVES
THIRTEEN PAPERS BY V. A. FOCK
INTRODUCTION BY V. L SMIRNOV APPENDIX BY M. A. 1EONTOVICH
EOHOB, N. A. IMAM ASSOCIATE EDITOR, P. MACKSMITH. JR.
ANTENNA IABORATORY HICTRONICS RESEARCH DIRECTORATE




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EDITORS' NOR
The S osiet p h y sic ist V.A. Took 1* w ell known ty p h y sicist* fo r hi* work l a quanton nechaniee, p a rtic u la rly In eonnwtloB w ith the B artreoPoek thaory o f ealf-co o e lo ta n t f ie ld s . The purpoie o f th ia eo lleo tlo n i t to »equaint th e reader w ith Peak'* nore recent work on the p ro p o r tio n , r e fra c tio n , and d iffr* o tlo o of radlowavaa. Pock's e a r l/ papare on tM i subject (th e f l r e t f i r e papers in th is c o llec tio n ) appeared la B igllah alnoat a doeado ago. Boweror, a l l of h ie aore recant work haa been pabllehed in Russian and le r e la tiv e ly unknown ontelde the S o riet
The tren e la tio n a In th ia co llec tio n bare been baacl upon tra n s la tio n s obtained fro n e e reral aoureea. H r. Barman V. Cottony o f the Rational Bureau o f Standards and K iss A. P in ta il of th e Raral Reeseroh Laboratory, ree p eo tire ly , aada the o rig in a l tren e la tio n a o f Chapters VX and XX o f th ia e o llo etio n . The tr a n s la to r o f Chapter P IU ia unkn o n to tha e d lto re . The r m e l n il lT ig ahaptsrs ware tra n s la te d ty K orria D, Prlednan. Chapters T ill , XX, and X ware nade ty Itor r l e D. Prlednan, Xno., Kew tonrllla, Kassaohuaette. Chaptars XXX and XXIX wars aada in oooparation w ith Llnooln Laboratory.
According to tha L ibrary o f Congress aohonc f o r the tr a n s lite ra tio n o f the Buseian alphabet, Pook'e nans appears aa Pok. Howerer, because o f the aore general use In s c ie n tific lit e r a t u r e o f tho fo ra Pock th e ed ito rs have reta in e d th is fo n t In th is c o lle c tio n .


:3.IESiS
JHTKOXTICN: V.*. FOQC'S CCMEB'.'TTCKS TO DIFFRACTICS TKE3SI ( 7 Ia iiji.tr AIsitsa n d w ic h Fok. 17-32, P rin tin g Rouee o f the Acedesr Science, Katcow, 1956). 1
I . m METHODS IN DIFFRACTION THECST (Philosophical Haraslne 2 ° , 2AM.35, 1946). 1
n . THE DUTHIE7E OK CF CHWfllTS EIDUCBB BI A FLANZ WATE CB THE SURFACE CF A CONDUCTOR (Journal s i Thesis». USSR. 1,0, 130-136, 1946). 13
r u . m p m cT ioK of radio wire? around the earth' s surface (Jou rn al o f Physics. USSR, o , 2 5 W 6 6 , 1945). 31
1 . Stateaeu t of the Problas and i t s Solution lr. th e Fora of S e ries 32 2 . The S-jnoatlon Foraula 35 3 . The E rol'jaU ca of th e H erts Function f o r th e lliircia aied Region 41 4 . Asymptotic- E x p ressio n fo r the Henkel Function 46 5. The Expression? o f th e Herts Function Valid In th e Pemmbra Region 51 6 . Discussion or th e Expression f e r the Hertc Function 55
17. SOL'jTIW OF THE ROEi iM OF ISf-FACAT.'W OF ELECTROMAGNETIC WAVES AlOW.- THE KARTj"S S’JRrACE ffi THE METHOD OF PARABCUC EQUATION (sJc'^M k =£ 13-36, 1946).65
1 . Tr.« Csse of a Flar* E arth 2. The Cate of s Spaerioal Earth Table 1
7 . THE FIELD CF A PLANE WAVE HPtR THE SURFACE CP A CONDUCTING BODf (Jou rn al £ Physics. USSR. 39*-409, 1946).
1 . The Geometrical Aspect o f the Problem 2 . Sim plified Maxwell's Equations 3 . Sim plified Bcur.-jary Conditions 4 . D sterttinetloc of th e F ield Ccopanent
V OTIC
5S§s 3 sa s


C o n o rs (cant'd)
5, D etiralnation of tha Ccxsponool H. aad the Othor Mold Cooponents 116 6, The Plaid In the IU u a ln a tad Region 120 7, Conelualea 124
n PROPAGATION CP TUB DIRECT NAYS AROUND THE BARTH WITH DOB a c c o m fo r d iffra ctio n akd r e fra c tic h Ciffi. ( s jr . Q | . ) , & 81- 97, - 1948) . 127
1. D if fe re n tla l Equations and th a Boundary Conditions of tha P ro b lm 129 2. Tranafar to Dlaanslonleae Q uantities 135 3. Solution of Equations 139 4. Inv estig atio n of tha So lu tio n fo r tha Region of O lraot V is ib ility 146 5. Inv estig atio n of tha Solution fo r tha Raglan of th a Psntasbra (F ln lta X and I ) 1£L
m , THEOfll OP RADI0WA7E PROPAGATION I I AH UWCMOGBiBOCS ATMOSPHERE POB A RAISH) SOURCE (UK (Se£. & * .) , JA, 70-94, 1930) . 159
1. Pundasental Equations aad M a ttin g Conditions 160 2. Approximate Porn of th a Equations 164 3. Analogy with tha Unsteady P roblm of Q u an ta Xeehanlea 167 4. Trenefomatlon to N ondlmnslanal Q uantltloa 175 5. Properties of Partloular Solutions of ths D lffsra n tla l Equations 178
6. Construction of th s S olution as a Contour In te g ra l or S e ria l 184 7. A pplication of tha Saneral Theory to the Superrafraotlon Case (Schematic Kxanple) 191 8. Appradmata Fomulaa f o r la m s with low A ttenuation 198
7I U . THE FIELD FRCH A VSiTICAL AHD A HORIZONTAL DIPOLE, RAIS3 ) ABOVE THS EARTH'S SURFACE C z m . 916-929, 1949) . 30?
1. V ertic al H&lsed Olpola. Solution In S e ria l Pom 207 2. Appradaate Series Sum st ld o fo r tha Harts Function 209 3. The Attenuation Paotor 212 4. R aflaction Pomnla 215 5. H o rlw a ta l E la o trlo al D ipole. PrimaryP laid 221 6. S e ries f o r tha Total P la id 224 7. A ppradaate Expressions fo r th a P laid 228


Corral13 (oo n t'd ) H . FRESNO. DIFFRACTION FROt CONVEX BODIES
3 . Computation o f the I n te g r a l $ 4 . Evaluation of the Integral 5 . The A ttenuation Factor In th e Region of the Shadow-Cone
Z. FRESNEL REFLECTION LAWS AND DIFFRACTION LANS (SEES* & 30S-319, 194«).
1 . F resn el R eflection Lave 2. Cross-Section o f a Bean of Refleoted Bars 3 . Klectroiaagnstlo F ield o f the Refleoted Wave 4 . D lffreo tio n Laws In the Penuabra Region 5. Investigation of the Expressions for the F ield s In the Dobra and D ire c t-V isib ility Regions 6 . Comparison o f th e D lffreo tio n F o w l* w ith th e F resn el F o w l s f o r the Lino-of-S ig h t Region
U . QENERALIZATIGN OF THE REFLECTION FORMULAS TO THE CASE OP REFLECTION OF AN ARBIIRAKI NAVE FROM A SURFACE OF ARBITRARI FORK (ZSTF. 2£, 961-978, 1950).
1 . Fresn el Formula* 2 . D iffe re n tia l Oecnetiy of th e Reflecting Surfaeo 3 . Croes-Seotlon of th e Bundle of Reflected Rays 4 . C alcu latio n of th e Determinant 5 . D iffe re n tia l Geometry of th e Nave Surface 6. Reflection F o w l* 7 . R e flec tio n o f th e Spherical Hava from the Surface of a Sphere
H I . APPROXIMATE FORMULA FOR DISTANCE OF THE HORIZON IN THE PRESENCE OF SUPERREFRACTICB (BadloteRh. 1 Elafctr. i , 560-574, 1956).
1. Introduotlon 2. In itia l Fow l** 3 . H o w l Refraction Case 4 . Asymptotic In teg ra tio n o f a D if fe re n tia l Equation with a C o efficien t Having a Mini 5. Investigation of the Attenuation Footer 6 . F o w l* fo r the Distance
(UFR. *1, 587-599, 1951).
1, Fonmlaa for the Attenuation Factor 2 . Reformulation of the A ttoiuatlIoonn FFaacctor
233
d
S S b S S 8 2 g * 8 2 ? 4$ g 5 S £ m


CONTEXTS (c o a t'd )
n n . OR RAITOWAVB propagation near the horizcx with SUFQQUTRACTIOH (R adlofkh. 1 B * t £ . i , 575-592, 1956).
1. Introduction 2 . On tb* Horlooa Conoa£t l a tb« Prosene* of a Tropospheric Wsreguid* Hoar th« Earth 3 . Fundamental Fom alaa 4 . R eflection Formula 5. Nuaarloal Result* l a Nondlaanalonal Coordinate! 6 . Attenuation Factor l a Deep Shad*. Raaldua Sariaa 7. A a a rlo a l Reault* f o r a Concrete Caaa
APPEDDIZl APPR0X1XATE BOUNDAKT CCNDITICHS FOR S I KUCTRCKAGNSTIC FIELD CM THE SURFACE CP A GOOD CCNDOCTGR
(InvMjl|a&}0g | {ft BadlowaT* Proaa«atlon. P a rt I I . 5-1 2 ^P riatin g b a a * o t tha Aoadsqf o /S cia n ca a,
Abbreviations f o r Soviet Journal*
IAS (Ser. F i t. ) . Isra a tu a Akadcall Rank S3SR 3 * rrla Plilc h ask a la (B o lla tla o f th e Aeada
S a rrlaT falch o ak al* (B o lla tla o f th e Acadsay of Solaneaa of OSSR - Physics Sa riaa ).
Zfauraal E kiparlm entalcol 1 Theoreticheskol Ltlki (Journal o f Experimental and Theoretioal Physios).
Radiotekh. £ K lsk tr. . ( R a d lo an g ln e ar la g
Radlotalchnlka 1 Elolctronlk* and E lectronics)
**m \


7 . A, FOCI'S COHTRIBOTICSS TO DIFFBACTiai THBORT
1 . DITBODOCTICa
7.A. Foek became in te re ste d In d iffra c tio n problem* cim peratively rec en tly . Within a abort tin e he succeeded in obtaining nueeroue re su lts whioh e re very la p o rten t both In th e o re tic a l end In p ra c tic a l aspects. By fo re ca stin g the paths of fu rth e r In v estig atio n s In th is f ie ld , they undoubtedly a re epochal In d iffra o tlo n theory. The solutio n of the problems of aleotronagnetlc w*ve d if f ra c tio n c onsists of fin d in g so lu tio n s of th e Xaxwell equations su b ject to specific i n i t i a l and boundary conditions on th e d iffra c tin g surface and rad ia tio n c onditions a t I n f i n ity . The I n i t i a l conditions a re often re placed by the requlrenent th a t the solution be sinusoidal in tin s . Foek devoted h la e e lf to an an aly sis of problems of the la s t k ind. P rio r to the Foek In v estig atio n s In th s theory o f electromagnetic wave d if fra c tio n , only so lu tio n s f o r a small masher o f prob i n s f o r o b stacles of a sp e cific shape were know , such se t th e In fin ite wedge, cylinders c irc u la r, e ll i p t l e and parabolic - and a lso fo r the sphere. In addi tio n , the problem of d iffra o tlo n from a paraboloid of rev o lu tio n , solved by Foek b ia se lf in 1944, should bo added to th s above l i s t .


The previous so lutions o f th s problems mentioned aboys, which wars represented by s e rie s o r by I n te g ra ls, ware not eery u sefu l In th s Important p ra c tic a l cass tdian the wavelength Is small In comparison to the dimensions o f th e o b sta cle, and thay should be considered as only the f i r s t step In solving the problTM , The next step must bo ths de riva tion o f form ulas fr<a which q u a lita tiv e physical consequences can be obtained and which a re . In ad d itio n , su itab le fo r p ra c tic a l computational Hence, one of th s p ossible d irec tio n s o f work in d if f ra ctio n theory was th e development of a method of is o la tin g th s p rin c ip al pa rts out o f th e complex form ulas idiiofa c o n stitu te the exact solution of th e probing. The Pock In v estig atio n s ware mads In th is d ire c tio n whan solving th s problems of d if f ra e tlo n from a conducting sphere a s w sll as from a paraboloid o f rev o lu tio n , n a tu ra lly , the method c ite d la a p p licab le only in those few oases idian an axact solu tio n can be constructed su c cessfu lly . Consequently, an u rgent need existed f o r th e c re atio n o f an approximate method of solving d if f ra c tio n problems tA ieh, w hile being g e n eral, would lead to r e la tiv e ly simple form ulae. The fundamental works of Pock on d iffra o tio n a re devoted to the construction of auoh an approximate method and to th e so lu tio n of a rnanber of p r a c tic a l important problems by using th is method. Pock developed and used th e parabolic equation method propoeod by Lacntovioh. Thla permitted him to give not only new sim p lified d e riv a tio n s of re su lts he had obtained e a r lie r by o th er
11
but also to goneralleo


than 1a various d irec tio n s (to take the f in ite conductivity o f the tody in to account) to dotem in* th e f ie l d oloee to th e aurface aa v e il as on th e surface I ts e lf j to take ataospherle ln h c ao g w e ltle s Into aooount in the p r o b l« of d iffra c tio n of ra d io w res around the e arth ’s surface). As i s every approximate msthod of solving boundary value prob i n e , the Fock method Is based on the smallness of c e r ta in parameters en countered in th e problem, th e q u a n titie s which a re u su ally small In the problems of radiowave d iffra c tio n a re i -jA. and -^r-, th e re t; m £ * 1 i s th e aduplex d le le o trlo constant of th s d lffra o tln g body) Jl. i s th s wavelength of th o in c id en t wave) 1 i s a q u an tity of tho order o f th s rad iu s of curvature o f th s surfaoo o f tho body. I f |7^j ■ « (p erfe o t conductor), than th e f ie l d w ithin the oonduoto r i s te ro , l . e . . I t i s known In advance. T his c lrc u n ta n c * permits the d if f ra c tio n problem to bo form ulated only f o r th e space outalde th e body, which lead s to su b sta n tia l sim p lifica tio n . The situ atio n In th e Imperfect conductor oate Is sim ilar I f the In eq u a lities | y/1 ■>> 1 and
In th is eaae, the f ie l d w ithin the conductor appears to be van ish in g ly small everywhere except In a surface la y er o f thickness of order where th e lnfluenoe o f th is la y e r can bo taken in to account by using boundary conditions f o r th s ex tern al f ia ld
"V7 - »A> ’ V1*‘ n.*V >ett
in
(i)


idler* Jg, Jy, *re the components of th e ourra n t d en sity ; n^, n^, a re tbe u n it v ector components normal to the body aurfaoe. le ad . N.A, Lsontovioh f i r s t suggested tbe aforementioned conditions In a ra th e r different fora. Consequently, tbe approxim ate f em u latio n o f the d iffra c tio n problem I s thereby reduced to a problem Involving the f ie ld s e x terio r to th e body. A fu rth e r e s s e n tia l sim p lifica tio n in problene of rad io wave d lf f ra e tlo n fro * bodies of a rb itra ry shape r e s u lts fro * the p rin c ip le of tb e f ie ld being lo c a l in tbe half-shadow region. I f tbe eleotrcnagnetlo f ie l d n ear th e surface o f a conducting body were to be determined su c cessfu lly , and, th e re fo re , th e current d istrib u tio n in the surface lay er, than the solution of the diffraction problocs would be a tta in e d bjr simple well-known formulae f o r the vector p o te n tia l. Tbe f ie l d in th e illum inated region near th e body i s subject, with a high degree of accuracy, to the Fresnel lava of r e f le c tio n , and, th e re fo re , can be determined e a s ily ; the f ie l d decreases rap id ly to sero In the shadow region. Consequently, th e u n attain ab le lin k In th e approximate so lu tio n of th e d if f ra c tio n problems I s the tr a n s itio n region (half-shadow) located near th e geometric shadow boundary and with the shape of a band o f width
la the rad iu s o f curvature of a normal eaotloc o f the body In the Inoldent plans. Pock succeeded in showing th a t th e electromagnetic f ie l d in the half-shadow region i s , to the accuracy of qu an tities of the order of It


of lo c al c h arac ter, 1 ,* ,, i t depemfeonly on the values
of the incident w n f ie l d In tho neighborhood o f the g i f t s p o in t, on tho geometric thtpo of th # body near th is p o in t, and on the a la c trlo properties of tilt conductor. After tho p rin cip le o f tho lo c a l f ie l d had bean estab lish ed , there* remained only to find th e so lu tio n o f th e d if f ra c tio n problem fo r a convex body o f su ffic ie n tly general shape, and to d e rlfe the approxi mate formulae f o r the f ie l d on I ts su rfa ce . I t la convenient to take the paraboloid of rev o lu tio n aa euoh a body, l a solving the problem of plane wave d lffre o tlo n from a paraboloid, V.A, Pock usod separation of varlablaa In parabolic coo rd ln atae, Ha constructed th e exact so lu tio n In the font o f in te g rals and performed th a approximate calc u latio n of theae In tegrale under tho assumption th a t ka » 1 , where k la the wave number and a la a parameter o f th e paraboloid of rev olution: x2 ♦ y2 - 2a» - a2 - 0 . Tha c h arac terla tlc d ire o tlo u o f tho woric on d iffra etlo i. explained above le su ffic ie n t to In d ica te the Important p rln eip lao of th a methods developed. B asically, th e se methods reduce to th e follow ingi Pock Indicated an a ffe c tiv e method o f approxim ately evaluating in ’ln lt e aerlea and In teg ra ls (containing a la rg e parameter) which rep re sent the exact solutions o f c ertain problems o f electromagnetic wave d iffra c tio n . This method perm itted him to develop, f o r example, e rigorous theory on radlowavo d lffre o tlo n around th e e a r th 's surface surrounded by e homogeneous atmosphere '('D iffra c tio n of Badlowavee


Around the Earth*» Surface", 1946)1. Ha w i a lso th a f l r a t to a atab llah tha vary Important p rln cip la of tha lo c a l charactar of tha electro®*gn atlo f ie l d l a tha h a lfshadow reg io n ,, u tir^ widely th a Laontovich condition*2 la th a approximate form ulatlon o f radlo-«av* d iffra c tio n problem*. This work afforded him tha opportunity to conatruet an approxim ate, but y e t a u ff lc la n tly accurate fo r p ra c tic a l needs, theory o f radlowave d if f ra c tio n from conductor* of a rb itra ry abape aa m i l aa * theory of radlowave propagation around the e arth taking lahcmogeneltiee of tha atmosphere In to account. Tha explanation of th la theory la given In "Theory of Radlowave Propagation In an Xnhcnogeneous Atmosphere f o r a Raised Source", (1950)3 . These works on d if f ra c tio n have played a very im portant p a rt in the h is to iy of th la question and, a t tha present tin e e ra among th a c le a re st attainm ent* In d iffre o tlo n theory and i t a a p plication*. Let ua tu rn to a mere d e ta ile d explanation of some of those works. The problem o f radlowave d if f ra c tio n In a vacuum r e la tiv e to a oor>dueting sphere la solved in "D iffreo tlo n of Radiowaves Around th e E a rth 's Surface"^, Let the sphere be of rad iu s a and be c haracterized by th e d i e le c tr ic constant C , th a conductivity <T end the magnetic perm eability u n ity , Let th e sp h e rica l coordinate* ( r , 6 , f ) be introduced end l e t t v e rtic a l e le c tr ic d ipole be pieced a t th a point r - b , 0 • 0, where b > a . Tha e leotronagnetie f ie ld ax o lted by suoh a d ip o le oan be ex pressed by mean* of the Hart* funotlon 0 ( r ,9 ,f ) which s a tis f ie s the equation vl


A u♦ A - o
Bane*, 1® o ri« r to determine the value of the f ie l d on th o sp h e re 'i surface, I t la w ffle lo n t to W w tho quantitiaa:
(3) 0 . - U (» ,e ,f) and o ; - J M L I . vr I
In 190ft, Mia obtalnod an a n a ly tic a l roproaontatlon f o r th e function 0 as an I n fin ite a eries o f apherloal fu n ctio n s, The a x tr n e ly poor convergence of the a a rlss prevented q u a lita tiv e physical consequences from being obtained and prevented p ra c tic a l use o f the aforaaentloned exact solution o f th e problmt. A major step toward a p ra c tic a l use of these s e rie s was made by Watson In 191S. But th e transfom ed fo ra of th e solution was s t i l l u n sa tisfac to ry , both because of i t s complexity and because i t was only applicable in th e geometric shadow region ( l . e . , f a r from the h o rizo n ). Only In 1945 d id Foek suoeeed in obtain ing an expression fo r th e Herts funotlon su itab le fo r a l l cases. Foek transforms th e se rie s f o r Ua and O' in to ocoplex I n te g ra ls, But, In c ontrast to the preceding authors who tended to reduce the In teg ra ls to a sum of resid u es, Foek Iso la ted from the In te g ra ls a p rin cip al term which y ie ld s s u f fic ie n tly exact v alues fo r th e functions Investigated. I t was shown in th is work th a t I f waves passing through the th ic k ness of the earth and waves olroumsorlblng the e a rth because of d if fra c tio n a re negleoted beoause of th e ir n a lln a s s , then th e value of Da be r e p r e s s e d by the follow ing In te g ra l v ii
(2)


wke<“o
ft) UA
(5) 9TM
C ^ a y - i Xr . ^ ) f r J K d .) '
(*) X n ^ = J ^ i2 L ; * « I t O - i * th* w tv.lnjgth) *,<P> ' *
(9) . T)*e*£*-■£.}
a) 1* th* hrp*rgean*trlo function; tb« oontour C 1* * lin * ln ta r* a cting th* p o sltlr* p *rt o f th* r**l axlo going downward (to th* l o f t of th* pol*a o f {f(r))» A a ln ila r In teg ra l i s obtained fo r 0^. Tb* * * s* n tlal fo ata ro of th l* aothod of approach 1* th a t th* Integral* obtain*d oan bo oalomlatod • t a l l y and w ith gro at aoouraoy f o r a iy rain * o f 9 . Till


lb s o h a iae tn rletlo pnraantnr o f th« aforaom tlonad in te g ra ls 1* tii* qua ntity p oos y where y 1* tbo angl* between th a v e rtica l a t tbo observation point tad tbo Mors* dlreotlon, I f p>> 1 wA tbo oboorror la In tbo lin o of sight region (nor* aoeoratelyi I f kb cosT> > 1 , i t e r s h la th # h eig h t o f tbo I sure a above tbo e arth ), than tbo a - 'o a tlo n of tbo in te g r a l a loads to tb o wall-known "roflootlo n forn u la", This evaluation o f tbo In teg ra ls loads to tbo Wofl-^ran dor Pol fonnila v a lid f o r p o in ts a t la rg o dlotanoos fro * tbo oouro* but s t i l l w ell w ith in Um lin * - o f -e ig h t. th e half-shadow r*glsn ( i t e r s p « l ) , fo r whloh appraxlnato values of tb o f l* ld worn not know ) i s o f g rootost ln to r o r t. A nstbod Is lndioatod In t h is aork of evaluating tbo Intogrolo fo r th is oaoo and th s following fo ro n li la obtained
In which w^(t) I s tbo complex Alrey funotlon re la te d to the Hanke l function of one th ir d order ty tbo r e la tio n
The contour J 1 goes fro n l o to 0 and fro a 0 to |
(10)
lx


The formula f o r tho half-shadow region I s tho main r e s u lt of th is work. I t 1* applicable In a l l cases of p r a c tic a l in ta re a t. I t tran s forms Into tbe Weyl-ran dor Pol formula f a r from geometric shadow In th e lln e -o f-» ig h t roglon. T his fo n aila can b# reduced to a rap id ly converging s e rie s whan the tr a n s itio n I s nade In to th e shadow region idlers ( - p)> > 1 . I n the work "Solution of th e Problwa o f Propagation o f K le ctrcwagnatle Wave* Along the E a rth 's Surface h r th e Method of Parabolic Equations" (w ritten jo in tly w ith M.A. L eontorlch)*, a prohlma i s analysad which le alad la r to th o problma In th e paper mentioned abore but the method la e sse n tia lly d if f e r e n t. The Influence o f tho o a rth 'a surface le taken In to account by the Leontorlch approximate boundary conditions and ta rn s In th e f ie l d equa tio n s a re neglected which a re m a ll and a re o f th e order o f o ^ and . As a r e s u lt, the "approxlnate" form ulation of tho problem f o r
th e sp h e rica l a arth ease la sim p lified s u b sta n tia lly and la reduced to th e problem o f solving the parabolic equation
In the region e x terio r to tho earth and subject to the additional eondltlona


J t la d iffic u lt to estimate th« error Introduced ty discarding the • j ^ l l " terms vhsa using th le method. To do th la the vell-knom Pte«M l " re fle c tio n 0 formula n u t a lso be considered. The e sse n tia l advantage of the parabollo equation nethod i s l t a g rea t sim plicity as s e l l as the p o s s ib ility of solving oor* cocgilsx problems (fo r example, save d iffra c tio n from bodies of a rb itra ry shape). In th la work th e f i r s t oase considered I s th a t in which the earth la assumed to bo p la n ar. Than th e apherloal earth case Is con sidered and the sane formulas are obtained ty using th e parabolic equation nethod as bad been previously obtained ty approximately n m ln g the se rie s idiloh y ie ld the exact so lutio n of th e proMsm. The agreement between r e s u lts obtained ty these two methods provides a Ju s tific a tio n f o r th e use o f the parabolic equation method In problsns of radiowave d if f ra c tio n from good conductors. Pock used th is msthod widely In l a t e r works on d if f ra c tio n . In the work "Propagation of the D irec t Wave Around the Earth with Due Account fo r D iffractio n and R ofraotion",^ the problom Is solved undsr the assumption th a t the surface o f th e e arth i s homogeneous as wall aa th at the d ie lec tric oonstant of the a ir la a function £ ^(h) only of the height b ■ r - a of p otato above th e h o tiaan . A v e rtic a l dipole performing harmonic o o c illa tlo iis defined ty the fa c to r e”i4>* la placad on the surface of th e e arth a t th a p oint r - a , 0 ■ 0 . A rap id ly varying f a c to r la Iso la te d from the H erts function 0 ■ad a nsw "slowly* varying fu nction U2 I s Introduced ty moans o f th e


fo rm ula
■*%
^0 (h )r f a i n 6
tdiors a • a9 la the l tf g th o f are on th e te r r e s t r i a l sphere from the p o in t idler* th a dipole la to the p o in t above tha earth a t iddeh tha
obaeirar le eitoated,
/ 2V
the author neglecte quantities o f order r r ^ j In the equation obtained f o r Bj. A fter In troduetlon o f th e noodlnenalonal v aria b le s x and f ty naans o f the formula*
(
16)
vhor* “ ^ j ' ( o ) 1* th e equivalent radlue of th e T * IH o J
e a r th ’s su rface, and a fte r Introducing the new function ty means of tho formula
a
H' ~ W '
th e p re h lw i s reduoed to determining th e fu netion w^Cz*r) from th a aquation ,
(W ) ♦ 1 ♦ 7 (1 ♦ «)wx - 0 (y > 0) 3r
tinder th* condition*


end the n a tu ra l ra d ia tio n eoodltlon f o r h >> 1 . Tha q u a n tltla a q • a t g, e ntering l a tha fonsulae raduead abore, hare tha following T*lU8»
( » ) q - g - _ ^ « qW - . 4 < 2 ) . ^ {o) ] .
Zmrastlgatlon of th e aquation fo r *j_ above th a t I f g - 0 and I f tha radlna a la replaced ty tha equivalent radlue of tha earth a*, than tha n a th ao a tlc al prdhlan la reduced to exactly th a aana fo ra aa whan tha atmosphere la absent. In th a general case, g can be eoneldered aa a fu n ctio n o f tha product f i y , Wiere /3 ■ — la
a small parameter. Tha so lu tio n of tha probloa I s su c cessfu lly repre sented fay the contour In teg ra l:
*d»ere f ( y , t ) la aa e n tire transcendental function w ith a d e fin ite be havior a t in f in ity and sa tisfy in g th a aquations
(22) [ r - t ♦ y g ( £ y )J f - 0 |
s ill
A


The contour r 1* in f in it e and encloses th s f i r s t quadrant of the t plans. Investig atio n o f the so lu tio n of th e p ro b lai constructed shows th a t the lavs o f geometric o p tic s are co rrec t In th e lln e -o f-e ig h t region f a r from th e h o rlio n . Ths following in e q u ality i s th e condition
for this y2 kh2 (23) ----- ---------» 1 . 2s
The solution transform s In to th s K eyl-ran d sr Pol forsrula f o r a s l l values of x and y end f o r la rg e values of p ■ cos J - ,
Ths Investigatio n o f th s so lu tio n In the half-shadow region permits th e conclusion th a t the wave reaches th e horizon w ith an amplitude uid phase corresponding to th s laws o f geometric o p tlo s f o r an unbounded Bsdlun and undergoes d if f ra c tio n according to th e law of the fo ca l H eld in th s half-shadow region a t th e horizon. This r e s u lt agrees completely with the id eas o f L .I . H andel'shtan th at the properties of the s o il are essen tial not along the W ide ray tra je c to ry In ndiowave propagation along the e a r th 's surface but only In th a t region where the tran s m itte r o r rec eiv e rs are lo cated . Let us tu rn to the work In which th e problem d if f ra c tio n from an a rb itra ry convex surface i s analyzsd. An electromagnetic wave In cid en t on a conductor e x cite s surface currents Wilch, in tu rn , a re sources o f sc a ttered waves. Consequently,
xiv


ib e sse n tia l step in th e solution of the plane wave d lf f re e tlc a probfrom a eonductor o f a rb itra ry shape Is to fin d th e ourrents excited OO i t s surface. In th e work, "The D istrib u tio n o f Currents Induced by a Plans Ware M the Surface of a Conductor", the cu rren t d istrib u tio n ex cited by a plane wave on the surface o f a convex, p e rfec tly conducting, s u f f i c iw itly smooth body o f a rb itra ry shape i s analyzed under th e c o ndition th a t the length o f th e aleotrcnegnetlc wavs i s very n a i l in comparison with the body dIran alone and the r a d ii of curvature of i t s surface. A fundamental r e s u lt o f th e work i s the proof th a t the f ie ld has lo e a l character near th e g ecnatrlo shadow boundaries. I t ia show in th e work th a t when the in cid en t wave la polarized with the e le c tr ic v e cto r in the plane of incidence the c u rren t d i s t r i bution nasr the boundaries c ited i s expressed through a u n iv ersal (Id e n tic al f o r a l l bodies) function 0 ( f) of the argument f “ - j - , where / ia the d istan ce fro a th e g scaetric shadow boundaries meas ured in th e Incident plane and d I s the width of the half-shadow region. An an aly tlo expression i s derived fo r tho fu n ctio n 0 ( f ) and detailed tables are given. The solution o f th e problaa of th e cu rran t d is trib u tio n i s based e sse n tia lly on the study o f th e so lu tio n of th e In te g ra l equation fo r the c u rrent d en sity ? on th e surface of the p e rfe c t conductor. I f th e aonochrmeatle electromagnetic wave H ■ H** f a l l s on the eonductor and I f th e following n o tatio n is introduced


(24) ' - (1 - T - x H « ]
then the follow ing In teg ra l equation le obtained fo r th e su rfa ce cur rent density
(25)
idlers le the u n it v e cto r nonssl to the conductor su rface; "? and r*7a re radius v ecto rs o f fixed p o in ts o f the surface and of p o in ts with the surface element d37 and R “ | ? - "?'] . As an In v estig atio n of th e In te g ra l equation In the case o f very large values of k ( i . e . , sn a il wavelengths A ) shows. I t can be con sidered, with enough accuracy, th a t X “ 2J** on th e illum inated p a rt o f the surface (tdilch corresponds to Fresnel r e fle c tio n theory and J • 0 In the shadow p a rt. In tha neighborhood o f the geom etrical shadow boundaries, th e In te g ra l equation shows th a t In a bandwidth o f
(26) d - R* ,
sdiere Bg i s th e rad iu s o f curvature o f a sectio n of th e body surface by the Inciden t plane, the c u rren t d en sity and, th e re fo re , th e f ie l d has an approxim ate value dependent only on th e value o f th e ex tern al f ie l d H** In the p o in t under In v estig atio n , the geometric character^ is t lo s of th e eurfaoe element and on the e le c tr ic p ro p ertie s of the conductor. Such a r e s u lt means th a t u n iv e rsal fo n m las f o r th e cu rren t xvl


density on the surface o f • p e rfec t aonduotor in the half-shadow r e fl«B can bo obtained from the so lu tio n of th e d iffre c tio n problen f o r ^ p a rticu la r o im of • oonveac surfaoe, The u n iv ersal formulas h b tlenod oro obUlnod by considering tho probloo of piano wave d if f ra c tion fro* a paraboloid of rorelution. Tho roo u lt la
___ »iC f aK t
(27) t - r w ) - ^ w ~ ) « ,
rfioro * ( t) la tho oonplax Alrey fimotion and T la a contour In tho o o ^ lex plana going fro * in f in ity to Mro along tho lin o arg » y IT
and from taro to in f in ity along tho p o sitiv e p a rt of tho r a a l a x is. An investigation of tho asynptotle values of G(f) fo r largo p o sitiv e and Dogatlvo values o f £ shows th a t tho c u rran t d e n sity 3* transforms continuously whan tho tra n s itio n i s made from the half-shadow in to tho lin e -o f-slg h t or in to the shadow regions, in to tho values £3SX and J - 0, respectively. Detailed tables ara constructed for tho function
0(f). Tho r e s u lt of tho preceding work i s generalized in "Field of a Plane Wave Hoar the Surfaoe of a Conducting Body" in th a t, f i r s t , the fie ld i s determined not only on the body surfaoe i t s e l f but also in a certain surface la y e r with thickness —i l l in comparison with th e r a d ii of curvaturej second, the body is considered to bo not a p o rfa et, b at ® ly a good conductor in tho sense th a t th e )M « teentovioh conditions hold f o r tho ta n g en tial f ia l d component* on i t s su rface. Furtheim ore, x rii


the p o la risa tio n o f the Inc id ant wave nay be such th a t the e le c tr ic ▼actor Ile a In or la perp* d lo u lar to the plana of Incidence. Let ua dlecuaa the Fock work, "Fresnel D iffrac tio n f rtn Convex Bodies", (1951)7. Considered In th is work la the d if f re e tio n from a sphere, h e r e in re fra o tlo n of the etnoaphere la n o t taken Into account. I t la con sidered th a t the source and the observer a re above th e surfaoe o f th e e arth , where h^ la the source height and h j la the height of th e observation p o in t. The f i e l d I s expressed through th e two so lu tio n s IT and w of the equation A u ♦ k^V - 0. The follow ing notations a re Introduced In a ddition to those used previously:
(aa) r x - ( - ^ ) ' Tich1 , y 2 - 1*2 ,
(29) q - ( - I f f (? ♦ U * l j qt - (?) - l / 1 .
The following fornulas bold noar th e su rfsea of th e sphere:
#lka0 (30) D * T y r m v (l< 7 l' 72*q ) *
•
and the atten u a tio n fa c to r 7 Is expressed b j a c e rta in contour in te g ra l containing two Airey fu n ctio n s. A ll these r e s u lts a re contained In the work " F ield fro n a V ertic al and H orizontal Dipole, Raised S lig h tly Above mu


th t Bartli1* Surface*, (1949)® and In th a 1951 work, an approxiante «*preoalon la ji^ n i fo r T In th a region af th a ehadow oona. Henoe, i t la ooneid r red th a t tha paraneter defined >7 th a fo rm ic
y^ v7
0 .)
la la rg e and tha q u a n tity £ ■ x - ~ la f in ite or m a ll. Two funetione are lntrodaoad
(33)
(34)
1
f(aO - a - 1* 2- 1 ^ ^ j ,K ^ dee |
2^,-1
ppraxlmate expres aIon T(x^y^f y ^ .q ) la th a follow ing fo r tha ahadow oona
- - • UA p H ft.Q - g(f) ♦ —i - g '( ^ ) l ,
W* L J
Me do not c ite tha axprwaalon f o r 4>o . Tha p rin c ip a l ta n s la /* f(£ )> proportional to tha freenel Integral. I t la indepandmt of tha n o ta ria l of tha d if f ra c tin g body. Superlnpoeed 00 th e d iffra c tio n ploture (Praanal d if f ra c tio n ) datamdned h f th ia to m la th e background depandmt on tha function g ( () varleo »lowly in comparison with tha p rin cip al ta rn . Thia background deponda on tha s a te r i a l o f the d if fra c tin g body.


UPBBESCS
1 . A tra n s la tio n o f T.A. Pock, ■D iffrac tio n o f RadloiAToa around tha B arth 'a Sorfaea*, la a v alla U o th r o n g Morria D. Priadman, In c.] M ovtooriilo, Kaaaaotraaatt*. 2 . So# Appendix t o tb le e o lleo tlcn o f Pock paper*. 3 . See Chapter VQ o f t h is c o lle c tio n . k . So* Chapter I? o f th is c o lle c tio n . 5. See Chapter ? I o f tbit c o llec tio n . 6. See Chapter I I o f th is o o lle ctlo n . 7. See Chapter IX o f th la c o llec tio n . 8. See Chapter m i o f th la oo lle ctio o .


I . NEW METHODS IN DIFFRACTION THEORY
V. A. Foclc
The g e n e ra l problem o f th e th e o ry o f d i f f r a c t i o n o f e l e c t r o magnetic waves o o n a lata In f in d in g a s o lu tio n o f M axw ell'a equations, having p resc rib ed s in g u la r itie s ( fie ld so u rces) and s a tis f y in g p re s c rib e d boundary c o n d itio n s and c o n d itio n s at In fin ity . The s o lu tio n o f th i s problem p r e s e n ts s e rio u s m athe m atical d i f f i c u l t i e s , which a r is e c h ie f ly from th e n e c e s s ity o f talcing In to account th e g e o m etric al shape o f th e o b s ta c le s on w hich th e wave i s f a l l i n g . The p ro b lem I s somewhat s im p lifie d I f only m onochrom atic waves o f g iv e n frequency are considered, b u t the d if f ic u lt ie s are s t i l l so g re a t, th a t th e p roblem h a s n o t y e t b e e n s o lv e d , e x c e p t i n c a s e s when th e o b s ta c l e I s o f a p a r t i c u l a r l y sim p le fo rm . The b e s t known o f these are the cases of a p e rfe c tly re fle c tin g h a lf-p la n e o r a wedge, th e c ase s o f a sp h ere and a c irc u la r c y lin d e r. The c a se s o f an e l l i p t i c and a p a ra b o lic c y lin d e r have • I s o been c o n s id e re d , and th e f i e l d o f a p la n e wave In c id e n t on a p e rfe c tly r e f le c tin g p a ra b o lo id of re v o lu tio n (oblique Incidence) has re c e n tly been o b tain ed by the a u th o r. In the f *w c a a c s e n u m e ra te d a r i g o r o u s s o l u t i o n o f th e p ro b le m I n th e /°* » of an In fin ite s e rie s of In te g ra ls has been o b tained.
{'-)


2
The aim of a theory la to give a p ic tu re reproducing a l l th e q u a lita tiv e and q u a n tita tiv e fe a tu re e o f th a phenomenon c o n sid e re d . This aim i s not a tta in e d u n t i l the s o lu tio n obtained is of a su ffic ie n tly simple form. I f the rigorous so lu tio n has a com plicated a n a ly tic a l form, i t c o n s titu te s only th e f i r s t s te p ; a second s te p must be made - the d e riv a tio n o f form ulas s u ita b le fo r numerical c alcu latio n s. T his second s te p may be a s d i f f i c u l t as th e f i r s t one. To giv e an example, we may mention th a t th e problem o f d if f r a c tio n o f ele c tro -m a g n e tic waves around a sphere was BOlved rig o ro u sly some 40 y e ars ago (M ie). This problem in clu d es th a t o f the propagation of radio-waves along the surface of the e a rth . Owing to th e slow convergence o f the s e r ie s Involved, th e g en eral s o lu tio n could, however, not be ap p lied to the l a t t e r problem u n t i l 1918, when a tra n sfo rm a tio n o f the o r ig in a l s e r ie s in to an o th e r r a p id ly converging s e r ie s was found (W atson). But th e improved form o f tho s o lu tio n was B t i l l u n s a tis fa c to ry In sornt- re s p e c ts , being very com plicated and a p p lic a b le only In th e re g io n o f the g eom etrical shadow ( f a r beyond th e lin e of h o r iz o n ) . A f a r more s a tis f a c to r y form o f the s o lu tio n , ap plicable in a ll cases of p ra c tic a l importance, has been re c e n tly found by th e a u th o r .1 <mus, the way from the rig o ro u s th e o re tic a l so lu tio n to the approximate p ra c tic a l one took about 40 y e a rs o f re s e a rc h .
(2)


5
To fin d f i r s t a rig o ro u s s o lu tio n o f a d if f r a c ti o n problem and th e n to tran sfo rm I t in to a n o th e r form s u ita b le f o r num erical c a lc u la tio n s - th is s tra ig h tfo rw a rd method I s , how ever, o f a very lim ite d a p p lic a tio n . I t can only be a p p lie d to th e few problem s, a d m itting a rig o ro u s s o lu tio n In form o f s e rie s of I n te g r a ls . In o th e r o a se s ( e s p e c i a lly when th e d i f f r a c t i n g o b s ta c le l a o f a r b i t r a r y s h a p e ) a tte m p ts h a v e b e en made t o re d u c e th e problem to In te g ra l e q u a tio n s . These attem p ts have proved
su c ce ssfu l from th e th e o re tic a l p o in t o f view ; b u t w ith th e p
e x ce p tio n o f a p a p e r by th e a u th o r, no use h as been made o f the In te g ra l eq u atio n s fo r th e p ra c tic a l so lu tio n o f the problem , th e g en eral th e o ry o f In te g ra l e q u atio n s b ein g q u ite useless fo r purposes of num erical c a lc u la tio n . An a p p ro x im a te m e th o d , s u f f i c i e n t l y g e n e r a l and le a d in g to s u f fic ie n tly sim ple form ulas i s thus u rg en tly needed. In th e f o llo w in g we s h a l l o u t l i n e th e p r i n c i p a l I d e a s o f su c h a m ethod, proposed and developed by th e a u th o r. E very a p p ro x im a te m ethod I s b a s e d o n th e s m a lln e s s o f some p a ra m e te r s I n v o lv e d I n th e p ro b le m . Ne h a v e t o c o n s id e r w hloh o f th e p a ra m e te r s o f o u r p ro b le m may be r e g a rd e d a s s m a ll. We a re u s u a l l y c o n c e rn e d w ith th e p r o p a g a tio n o f w aves I n a i r , l . e . , I n a medium w ith p r o p e r t i e s w id e ly d i f f e r e n t fro m those o f th e s c a tte r in g b o d ie s ( o b s ta c le s ) . The e le c tr ic a l p r o p e rtie s o f th e se b o d ie s a re c h a ra c te riz e d by means o f th e complex d ie le c tr ic p e rm e a b ility
(3)


(1 )
4
(6 denote# a# usu al th e d i e l e c t r i c c o n s ta n t, -r - th e co n d u ctiv ity o f th e medium, <o - the freq u en cy }. Now i t 1b e s s e n tia l th a t in moat c&Bes |ij) » 1 . Thus we may choose as one o f th e email
Next, the w ave-length X in vacuo i s u su a lly very much smaller than the ra d ii of curvature of the scatterin g bodies. We thus have an o th er sm all p aram eter - th e q u o tie n t >.:R, where R is the radius of curvature of the o bstacle. I t is convenient to take .netead the quantity
In a d d itio n to the two sm all param eters defined above, th e re may be o th e rs , depending on the p o s itio n of th e p o in t of observation. For instance, in the problem of the propaga tio n of ra d io waves along the e a r th s u rfa c e th e angle of In c lin a tio n o f th e ray to th e h o rizo n may be regarded as sm all. Let us co n sid e r the consequences o f the fa c t th a t the p aram eters 1 :. 4 h and l:m are s m a ll. In th e lim itin g case | i ) |« a o (perfect conductor) a great sim p lificatio n a rises from th e f a c t th a t th e f i e l d i s known beforehand in s id e the conductor ( t h i s f i e l d being equal to z e r o ). Ve can confine ourselves to the space outside th e conductor by prescrib in g
p aram eters of the problem tu e in v erse v a lu e o f |q | o r the qu an tity 1:
(2)
(4)


5
proper boundary conditions to the fie ld In a ir (the tangential components o f the e l e c t r i c a l v e c to r should vanish a t the su r face). A sim ilar situ a tio n a flse s If - T u T Is very large. The f i e l d In sid e the body Is In t h i s case very small except In a th in surface la y e r (sk in -e ffe c t), and the Influence of th is la y e r may be accounted f o r by s ta tin g boundary conditions f o r th e e x te rn a l f i e l d . These are o f the form
? Jx “ ^ < Ex * nxEn> = V z ' nzHy ' e t c - <’ )
where ( J x , Jy , J z ) I s th e su rfa c e c u rre n t d e n s ity v e c to r, (nx» riy, n^) th e u n it v e c to r of th e normal to the su rface, Ejj th e normal component o f th e e l e c t r i c f i e l d , th e meaning o f th e o th e r symbols being e v id e n t. These co n d itio n s, f i r s t
e q u a lity sig n ifie s th a t the thickness o f the skin layer should be sm all as compared w ith th e ra d iu s of cu rv atu re of the o b s ta c le . C onditions (3) may be e a s ily g en eralized fo r a r b i t r a r y v alues o f the m agnetic permea b i l i t y m. C onsequently th e sm allness o f 1: J h i perm its us to confin e our a t te n tio n to the f i e l d o u tsid e and on the body, which c o n s titu te s an Im portant s im p lif ic a tio n o f the problem. We now proceed to examine the Influence o f the sm allness
s ta te d by Leontovlch^ In a somewhat d iffe re n t form, apply I f The l a t t e r ln 
(5)
of the wave-length.


6
As w ell known. In th e lim itin g case o f sm all w ave-lengths th e laws of g eom etrical o p tic s become v a lid . P a r tic u la r ly , the boundary of the shadow on th e su rfa c e o f th e body becomes sharp and well d e fin e d . On th e onp sid e of th e boundary — In the illum inated region — the fie ld obeys very nearly F renenel's laws o f r e f le c tio n , and on th e dark sid e the f i e l d ra p id ly decreases to zero. The approxim ation given by the geo m etrical o p tic s I s , however, not s u f f i c i e n t f o r o u r purp o ses. The p o in t o f I n te r e s t f o r us I s the d if f r a c t i o n phenomenon In I t s s t r i c t sense, l . e . , th e bending o f the ra y around th e o b s ta c le . This phenomenon cannot be tr e a te d by th e means of g eo m etrical o p tic s , end to g ive a th eo ry o f th i s phenomenon a more a c c u ra te s o lu tio n o f the fie ld equations Is required. The au th o r succeeded In fin d in g t h i s s o lu tio n by means o f a new p r in c ip le which may be c a lle d "The P rin c ip le o f the Local F ie ld In th e Penumbra Region". This p r in c ip le c o n s is ts In the fo llow ing: - The tr a n s itio n from li g h t to shadow on th e s u rfa c e o f the body tak es p lace In a narrow s t r i p along the boundary o f the g eo m etrical shadow. The w idth o f t h i s s t r i p I s o f th e o rd e r
where R0 I s th e ra d iu s o f cu rv a tu re o f th e normal s e c tio n of th e body by th e plane o f In c id e n c e . . I t may be proved th a t.
(6)


7
with neglect of email q u a n titie s of the order fie ld In th le s tr ip has a local, character: I t depends only on th e value o f the f i e l d of the In c id e n t wave In the neighbor hood o f th e p o in t c o n sid ered , on th e geom etrical shape o f the body near th e p o in t and on th e e l e c t r i c a l p ro p e rtie s o f the m a terial of th e body. The f i e l d n e a r a given po in t on the s t r i p does not depend cn I t s v alu es a t d is ta n t p o in ts and can be c a lc u la te d s e p a ra te ly . To e s ta b lis h th e p r in c ip le o f th e lo cal f i e l d and to deriv e e x p lic it form ulas f o r t h i s f i e l d we have used two d if f e r e n t methods. One o f th e se (2) a p p lie s to th e case of an ab so lu te con d u cto r and g iv es the v alues o f the f i e l d on I t s su rfa c e . We s ta r t with the In teg ral equation fo r the surface current density J. This i s o f th e form
The v e c to r Jex (e x te rn a l c u rre n t d e n s ity ) Is defined by the ex p ressio n ( 3 ), where H i s rep la c e d by Hex, the magnetic vector of the external fie ld j z is the r a d iu B vector of the point of observation, z ' th at of the point of Integration;
(5) where f - (1 - lKR)e1KB ( 6)
R o | z - z ' | Is the le n g th o f th e chord between z and z ';
(7)


I s th e valu e o f th e u n it v e c to r of the normal, a t z. A q u a lita tiv e study of the In te g ra l equation perm its us to e sta b lish the p rin c ip le of the lo c a l f ie ld . This p rin c ip le once e sta b lish e d , we have to fin d a s o lu tio n o f th e d if f r a c t io n problem f o r a con vex body o f a p a r t i c u l a r shape and to d e riv e approximate formulas fo r th e f i e l d on I t s s u rfa c e . In v ir tu e o f th e p r in c ip le o f the lo cal f ie ld , these formulas hold fo r any o th er convex body having a t th e p o in t consid ered th e same v alu es o f the p r in c ip a l r a d ii of c u rv a tu re . (The p a r t i c u l a r body must o f course be s u f f ic ie n tly general to possess points with any prescribed values of p rin cip al r a d ii of curv atu re ; a c tu a lly a paraboloid of re volution has been u s e d ). Proceeding In t h i s way we a r r iv e a t a gen eral formula fo r th e s u rfa c e valu es o f th e ta n g e n tia l components of the m agnetic f i e l d o r, which amounts to th e same, fo r the su rfa c e c u rre n t d e n s ity v e c to r. T his formula I s o f the form
I being th e d is ta n c e f r a n th e boundary o f the geom etrical shadow, measured along the ray ( l . e . , along the lin e of In te rse c tio n of the plane o f Incidence with the surface of the body) and taken p o s itiv e In th e d ir e c tio n o f th e shadow and n eg ativ e in th e o p p o site d i r e c t i o n . The fu n c tio n 0 (« , 0) Is d efin ed by th e In te g ra l
J - Jex a (e , 0) (7)
where th e argument ■ In G d enotes the q u a n tity
(8)
(8 )


(9)
a ( t ’ 0) ^ V ( c r ’
where C i s a contour in th e complex t-p la n e running from I n f in ity to zero along th e l in e are t • and from zero to in fin ity along the p o sitiv e real a x is. The fu n ctio n <o(t) may be c a lle d th e complex A iry’s ' fu n ctio n ; i t i s defin ed by th e d i f f e r e n tia l equation
u>"(t) - Uo(t) ( 10)
and by th e asym ptotic beh av io r f o r la rg e n egative values of t
*o(t) - e 1 * ( - t ) ‘ 1/4 • exp [ l | ( * t) 3/ 2] . ( 11)
The fu n c tio n 0 (« ,0 ) tends to the lim it 0 - 2 fo r larg e neg ativ e v alues of c , w hile I t s modulus decreases ex p o n en tially fo r la rg e p o s itiv e v alues o f «. Formula (7) reproduces thus the g radual d ecrease o f th e f i e l d am plitude when p assin g from l i g h t to shadow. The same r e s u l t s may be o btained by an other method** which allow s us to g e n e ra liz e them in two r e s p e c ts . F i r s t l y , th e body need no t be a p e r f e c t conduotor, but nay have a f in ite conductivity. I f only the boundary condltipns (7 ) are a p p lic a b le . Secondly, the f ie ld Is obtained not only on the surface of the body, but also near the aurfaoe ( a t distances th a t a re small a s compared w ith the r a d i i of c u rv a tu re ). The method c o n s is ts In s im p lify in g Maxwell’i'e q u a tio n s and boundary
(9)


10
c o n d itio n s by n e g le c tin g q u a n titie s of the o rd e r o f the square o f th e sm all param eters 1- J m and 1 : m. The wave equation f o r th e am plitude I s thereb y re p laced by a p ara b o lic equation of S c h ro d ln g e r's ty p e . The sim p lifie d e q u ations are v a lid In a lim ite d re g io n n ear a p o in t on the penumbra s t r i p . The s o lu tio n of th e se eq u atio n s may be performed by means o f th e s e p a ra tio n o f v a ria b le s and y ie ld s the f i e l d In the region co n sid ered and e s p e c ia lly In th e penumbra s t r i p on th e body. In tro d u cin g the complex q u a n tity
*A~
(th e modulus I q f i s thus th e q u o tie n t o f the two small m e te rs ), we may w rite In ste a d o f (7 )
(12)
para
j ex 0 (6 , q) , (13)
where
0 ( 6. q) - e f i - r c t r - v f t ) * (14)
th e c o n to u r C being th e same as In (9 ) . These form ulas give th u s th e d i s t r i b u t i o n o f c u rre n ts on th e penumbra s t r i p on the ' body and g e n e ra liz e our prev io u s form ulas (7) and (9$. The form ulas f o r th e f i e l d n ear the s u rfa c e are more complicated and w ill not be w ritte n h ere. I t I s to be noted th a t In th e outward p o rtio n o f the s t r i p , where th e Illu m in a te d re g io n b e g in s, approxim ate expressions can be derived from our formulas th a t coincide w ith expressions fo r
( 10)


11
the f ie ld obtained by superposing the In cid en t and the re fle c te d wave and u sing F re s n e l's c o e f f ic ie n ts o f r e f le c tio n . On the other hand, In the opposite portion of the s trip the fie ld Is p r a c tic a lly zero. Thus our form ulas c o n s titu te the m issing lin k Join in g the two regions where th e laws o f geom etrical o p tic s may be a p p lie d . Together w ith F r e s n e l's form ulas they allow us to compute th e f ie ld near and on the whole su rface of the d iffra c tin g body. • In some problems t h i s I s a l l th a t 1b re q u ire d . In the problem of p ropagation of waves around the e a r t h 's su rfa c e , fo r In sta n c e , we a re only concerned w ith th e f i e l d on h eig h ts not exceeding ten k ilo m e te rs—a q u a n tity th a t Is sm all as com pared w ith the e a r t h 's ra d iu s ( 6380km .). In th is Instance our form ulas, I f m odified so as to Include the case when the source I s near o r on the s u rfa c e , give the req u ired so lu tio n . In o th er problems, however, the fie ld a t large distances from the s c a tte r in g body I s needed. In s p ite of the f a c t th a t our formulas are v alid only in the region near the surface, they provide a means to c a lc u la te th e f i e l d a t la rg e d ista n c e s a ls o . Indeed, the f i e l d of th e s c a tte r e d wave Is generated by the c u rre n ts Induced on the su rface (In the sk in -la y e r) by the In c id e n t wave. These c u rre n ts a re given by our form ulas. Thus, by applying well-known theorems on th e v ecto r p o te n tia l due to a given c u rre n t d i s t r i b u t i o n , we may, in p r in c ip le , c a lc u la te th e f ie ld fo r a r b itr a r y d is ta n c e s from the r e f l e c t in g body.
(11)


12
The p r in c ip le o f th e lo c a l f i e l d In th e penumbra region provided thus a b a s is f o r the approxim ate so lu tio n of the problem o f d if f r a c ti o n In th e g eneral case o f a convex body o f a r b itr a r y shape.
1. V. POCK, Jo u rn al o f P h y sics. ix ;.2 5 5 , 1945. 2 . V. POCK, Jo u rn al o f P h y sics, x :130, 1946.
3. M. LEONTOVITCH, B u ll. Academy S cien ces. U .S .S.R ., s e r l e physique, l x ; l 6, 1944, (In R u ssian ), alBO H. LEONTOVITCH and V. POCK, Jo u rn al of P hysics. x:13, 1946. 4. V. POCK, B u ll. Academy S cien ces. U .S .S .R ., e e rie physique, &:171f 1946, (In R u ssian ).
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13
TT THE DISTRIBUTION OP CURRENTS INDUCED BY A PLANE WAVE
ON THE SURFACE OF A CONDUCTOR
V. Fock
The d is tr ib u tio n of c u r r e n ts . Induced on the
su rface o f an p e r f e c tly conducting body by an In cid en t plane wave Is con sid ered . The body Is supposed to be
convex and to have a contin u o u sly varying c u rv atu re.
The wave length X o f th e I n c id e n t wave la supposed to be small as compared w ith th e dimensions o f the body and w ith the r a d ii o f c u rv a tu re o f I t s s u rfa c e . I t
is shown th a t the c u rre n t d i s t r i b u t io n In the v ic in ity
o f the geom etrical shadow I s e x p re ssib le In terms of
an u n iv e rsa l fu n c tio n 0(%) (th e same fo r a l l b o d ie s),
depending on the argument £ » I / d , where I I s th e d istan c e from the boundary o f the g eom etrical Bhadow, measured In the plane o f Incidence, and d Is the width
o f th e penumbra region (d 16 the rad iu s of
c u rv atu re of the normal s e c tio n o f the body by the plane of IncidenceJ . For the function OU) an an a ly tic a l e x p ressio n Is derived and ta b le s a re computed.
Let us co n sid er a p e r f e c tly conducting body on the su rface o f which a plane electrom agnetic wave I s In c id e n t. The su rface o f th e conductor Is supposed to be convex, w ith a continuously vary in g c u rv a tu re . The In c id e n t wave Induces on the conductor e l e c t r i c a l c u rre n ts , which In t h e i r tu rn become a source o f the s c a tte r e d wave. I f the c u rre n t d i s t r i b u tio n on the conductor Is determined, then the c a lcu latio n of the fie ld of the sc a tte r e d wave may be performed by apply in g th e well-known formulas f o r the v e c to r - p o te n tia l. Hence th e e s s e n tia l ste p In so lv in g th e problem of d if f r a c tio n o f a p lan e wave by a p e rfe c t con ductor Is to find the c u rren ts Induced on I ts su rface.
(1)


1*
The p re se n t paper la a p relim in ary re p o rt on our work ooncam lng the approxim ate s o lu tio n o f th ia problem,
1. Let ub denote by J th e su rfa c e c u rre n t d e n s ity on the con d u cto r. The v eo to r J la d efin ed f o r every p o in t on the s u r fa c e and la d ire c te d alo n g th e tan g en t to th e au rfao e. I t ie com pletely determ ined by I t s two ta n g e n tia l components, th e th i r d component (normal to th e s u rfa c e ) being equal to a ero . I t may be shown t h a t th e v e c to r j s a t i s f i e s the follow ing In teg ral equationt
j - 2J« +J d8, (1<01)
with f - (1 - lkR)elkR . ( 1 . 02)
In t h i s equ atio n R I s th ele n g th of th echord Joining the two p o in ts of th e s u r f a c tt th e fix e d p o in t r ( x ,y , z ) , fo r which the In te g r a l I s e v alu ated , and the v a ria b le po in t r 'f x '. y 'j Z 1) , whose co o rd in a te s are fu n c tio n s o f th e I n te g ra tio n v a ria b le s , n is a u n it vector of the normal to the surface a t the point r , dS1 la th e eurfao e elem ent a t r* and k la the ab so lu te value o f th e wave v e c to r. The q u a n tity Jex i s an " e x te rn a l" c u rre n t d en sity defined by th e formula
["*«“ ] • <!•»>
where H** i s the value o f th e magnetio f ie ld o f the In cid en t wave on the aurfao e (" e x te rn a l" f i e l d ) .
I f the dependence o f th e e x te rn a l f i e l d upon the co o rd in ates i s giv en by the f a o to r
(a)


,lk(ax+ 0y+-y2 )
15
(1 .01)
then the c u rre n t d e n s ity may tje sought In the form o f a product 0f a sim ila r fa c to r w ith a slow ly v arying fu n c tio n o f coordina t e s . The I n te g r a l ( l .O l ) a f t e r d iv id in g by (1.04) tak es the form
i » J elk [R ^ (* , - x ) ^ ( » ,-y)+7 (* , -*)j gag. , (1.05)
where $ I s a slow ly vary in g fu n c tio n . I f the wave len g th la s u f f ic ie n tly sm all as compared w ith th e dimensions o f the body, the value o f th e in te g r a l w ill be approxim ately
I * ^ « , (1 . 06)
:< cos 0
where th e p o in t x 1 y ' z ' Is connected with the p o in t x y 2 as i t is shown In P ig s. 1 and 2, and 0 I s the angle of Incidence of the ray.
Pig. 1 Fig. 2
The a n a ly tic a l con n ectio n between the p o in ts x 1 y ' 2 ' and x y 2 Is given by th e fo llo w in g form ulas. Let n ’ denote the u n it v ecto r o f the normal a t the p oint x ' y ’ z ' and l e t
(3 )


16
a + 2 n ' cob 6
0 + 2n^ cos fl
y + Zn'x coe 6
where
(1.07)
cob ■ (<m'x + 0n^ + yn^ ) . ( l -06)
The q u a n titie s o ', 0 ', y 1 are th e d ir e c tio n eoalnea o f th e ray reflected a t the point x 1 y 1 z '. With th e s e n o ta tio n s , we have e i t h e r :
or
x - x1 ft
z
R = y (1 . 09)
RRR
th e form ulas ( 1 . 09) being v a lid , i f th e p o in t x ' y ' z ' is s itu a te d on th e illu m in a te d p a r t o f th e su rfa c e (F ig. 1 ), w hile (1 .1 0 ) a re v a lid , i f t h i s p o in t i s s itu a te d on the shadow p a r t o f th e s u rfa c e . In th e l a t t e r case th e " re fle c te d " ray is fictitio u s. With th e same degree o f approxim ation as in form ula (1.06) the In teg ral equation (1,01) allows the follow ing so lu tio n :
i = 2Jex on th e illu m in a te d p a r t , J = a 0 on the shadow p a r t .
Near th e boundary o f th e geo m etrical shadow (where cos I t f O ) , form ula ( 1 . 06) c e ases to be v a lid and ex p ressio n ( l .11) does no t give a grad u al t r a n s i t i o n from l i g h t to shadow.
(4)


17
2. In order to obtain for the currents an expression valid in the tra n sitio n region also , i t is necessary to use a sore exact solu tio n . I t is rath er d iffic u lt to derive it d ir e c tly from the i n te g r a l e q u a tio n , but we have succeeded to ob tain 1* in an i n d ir e c t way, on th e b a sis o f the follow ing c o n s id e ra tio n s . F ir s t o f a l l , i t is seen from F ig s. 1 and 2 th a t i f the p o in t x y z l i e s n ear th e geom etrical boundary o f the shadow, th e p o in t x 1 y ' z ' l i e s a ls o n ear th i s boundary and near the point x y z. T herefore, the value of the in te g ra l (1.01) is determ ined by the v alues o f th e in te g ra n d in th e neighborhood of the p o in t f o r which th e i n te g r a l i s ev alu ated . Thus, in th e region of the penumbra (n e a r th e geom etrical boundary of the shadow) th e f i e l d has a lo c a l c h a ra c te r. Secondly, the investigation of the in te g ra l equation (carried out under the assumption t h a t the chord can be re p la c ed by i t s p ro jectio n on the tangent p lan e) shows t h a t th e width o f the penumbra region ia of the order of
where RQ i s th e r a d iu s o f cu rv a tu re o f th e se c tio n o f the body su rfa ce by the plane o f in c id e n c e . But in a region of width d and in a c e r ta in more extended reg io n the nucleus of the in te g r a l e q uatio n depends e s s e n t i a l l y only on the curvature o f th e su rfa c e in the neighborhood o f a given p o in t ( l . e . on th e second but n o t on th e h ig h e r d e r iv a tiv e s o f th e surface equation with respect to co o rd in ates). Hence i t fo llo w s, th a t a l l b odies w ith a smoothly vary ing c u rv atu re have th e same c u rre n t d i s tr ib u tio n in the penumbra re g io n , i f only th e c u rv a tu re s and th e in c id e n t wave are the same n ear th e p o in t under c o n s id e ra tio n .
(2 .01)
(5)


18
The r e s u lt s s ta te d perm it us to I n f e r t h a t . I f we solve | th e problem f o r any p a r t i c u l a r c a se , we oan o b ta in u n iv ersal t formulas fo r the fie ld on the surface of a p e rfe c t conductor. I These formulas Immediately apply to the region of the penumbra, j but th e f i e l d may be co nsidered a s known everywhere on the su r- ! f a c e , sin ce f o r th e Illu m in ated re g io n and fo r th e remote shaded re g io n th e e x p ressio n s (X. 11) a re v a lid . j 3 . The d e riv a tio n o f th e s e u n iv e rs a l form ulas Is too complicated to be given In any d e ta ile d fo ra In a sh o rt paper. We c o n fin e o u rse lv e s to some I n d ic a tio n s as to th e method, and to th e statem ent o f the r e s u l t , which may be done In q u ite a sim ple way. The co n sid e ra tio n s developed above show, th a t fo r the d e riv a tio n o f th e gen eral form ulas we can s t a r t from an exact s o lu tio n o f th e problem o f d i f f r a c t i o n o f a plane wave by some convex body w ith a smoothly v ary in g c u rv a tu re . The s u r face o f the body m ust, o f co u rse , be s u f f i c i e n t l y g e n e ra l, l . e . must p o ssess p o in ts w ith given v a lu es o f th e p rin c ip a l r a d ii of curvature. There are two cases In which e x a c t s o lu tio n s o f th e problem a re known, namely, th e case o f a sp here and the ease o f a c ir c u l a r c y lin d e r (In th e l a s t case th e Incid ence o f the wave Is supposed to be norm al). These bodies a re , however, not s u ffic ie n tly g e n e ra lt f o r a sp here the two r a d i i o f c u rv a tu re a re e q u al, and f o r a c y lin d e r one o f th e r a d i i I s I n f i n i t e . The sim p le st of the bodies having a rb itra ry values o f th e curvature r a d ii are: the e llip s o id and the paraboloid o f re v o lu tio n . For these bodies only th e gen eral form of the s o lu tio n o f th e s c a la r wave equation I s known; th e complete s o lu tio n o f M axw ell's eq u atio n fo r the given p h y sic a l problem appears to be unknow ■>.
In o u r work we have obtain ed th e re q u ire d s o lu tio n f o r th e paraboloid of revolution (p a rtic u la rly the values of the tangential
(6 )


19
components o f th e magnetic f i e l d on I t s surface) and have used th is solution to derive the approximate formulas. Let the e q u atio n o f th e p arab o lo id have the form
x2 + y2 - 2az - a2 - 0 . ( J . 01)
If the parabolic coordinates;
u - k ( r ♦ z) ; « « k ( r - a) ; (3.04) ♦ - arc t* JC
(7)


(3.05)
20
with
are Intro d u ced , th e eq u atio n of th e p arab o lo id becomes
« * oQ - ka . (3 .06)
For th e g e n e ra liz e d (c o v a ria n t) ta n g e n tia l components o f the e x te rn a l m agnetic f i e l d we have th e ex p ressio n st
21u H*x + H*x = ^ _Juo e in * , (3.07)
- 21u H®x + H*x «= J u o e ltl ' 10 . (5 . 08) 9k
In the new co o rd in a te s th e ex p ressio n f o r (1 hue the form
0 = i (u - S') cos 6 + J uo s in 4 cos d . (3.09)
For the same componentso f the t o t a l f i e l d ex p ressions In form o f Fourier series with respect to the angle t are obtained. The c o e f f ic ie n ts o f s in s$ and cos s$ In th e se s e r ie s are d e fin ite In teg rals with respect to the parameter t. Involving some com plicated fu n c tio n s o f u, o, 4, s , t . These s e rie s and I n te g r a ls can be transform ed In to double In te g ra ls of th e form
21u H + H = - ° u * 2ir k s in (3 . 10) where the function g ( s ,t ) la defined In the follow ing way. D 5 ( o ,a ,t) be an In teg ral of the d iffe re n tia l equation
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21
0 4 + * i + ( 1 . • ! « .1\ c . de dp ^ 4 ' 4p 2y
having a t tj-»ooan asym ptotic expressio n
JL t \ s± l « i ^ I t , o
Tf t -1 ir -* + y 1 j
C (» ,s ,t) • . 01 - s - I t
where F2Q I s an asym ptotic s e r ie s o f the form
1 + a - It . l \ 2 ' "V ip to tlc s e r ie s o f the
(Vs. 0 .1 +f&2 +i«2^_&i6±Ui5+...
N (s,t) = | e C‘ ( t / B , - t - l ) + * ( s 2+ t Z) C2 ( 0 , 8 , - t + l ) (3.14) where p I s con sid ered to be th e q u a n tity (3 -06).
g ( s , t ) « e 2 ? (u ,s + l,t) ? ( c , s - l , t ) ( s - lt) N ( s ,t ) . (3 -15)
With g ( s ,t ) having t h i s v a lu e , th e expression (3.10) I s v a lid , I f - it/2 < $ < w/2. In the cases tt/2 < $ < jir/2 and - 3ir/ 2 < ♦ < -w/2 we have to take fo r g ( s ,t ) a somewhat d if f e r e n t e x p re ssio n , which we s h a ll not (9)


22
w rite down h e re . The In te g r a tio n In (3 .1C) w ith re sp e c t to the v a ria b le t I s to be made along th e r e a l a x is from - oo to + oo and w ith r e s p e o t to s a’.ong th e Im aginary a x is from - 1 ooto + 1 oo. The valu e o f -21u I s o btained from (3 .1 0 ), I f we re p la c e d by -d. The double In te g ra l can be e v alu ated approxim ately under th e assum ption, th a t th e value o f u « lea I s very la rg e . Let us Introduce the quantity Ju o sin A cos d - o cos 0 > '( » « ) ] w (.In . ) W ' (5-l6)
T t is easy to v e r if y t h a t on th e g eom etrical boundary of the shadow t « 0 j bu t In g en eral I w ill be la r g e , o f th e o rd e r of o1^ . T h erefo re, when e v a lu a tin g th e I n te g ra ls we s h a ll con s id e r o to be very la rg e and { to be a r b itr a r y (in g e n e ra l, f i n i t e ) . I t can be shown, th a t under th ese assum ptions the following approximate e x p lo sio n s fo r the in te g ra ls are valid with a re la tiv e e rro r of the order of
Hu + H^ * ^ - T " eUl + 10 0 ( 4) * (?-17)
- 2iu H + H. » — r S . “ - W 0 ({) , (3 . 18) vk
where
0(4) - e1 3 —1= J e l t ---dI (3.19)
4 * J w' (t) rl th e symbol denoting a oontour running from in f i n i t y to the
o rig in along th e ray arc z - 2 /3 it and from th e o r ig in to I n f i n i t y along the ray arc * » 0 (the p o sitive real a x is). (1 0 )


23
The f u n c ti o n w(t ) whose d e r i v a t i v e i s in v o lv e d in th e in te g r a n d h a s 'b e e n s t u d i e d i n o u r p r e v io u s p a p e r * . w( t ) s a tis f ie s the d if f e r e n tia l equation
w" ( t ) « tw (1 ) , ( 3 .2 0 )
and can be w ritte n in th e form o f an in te g r a l
where th e co n to u r denoted by r g runs from I n f i n ity to th e o r i g i n a lo n g th e a rc z « - 2 /3 if and from th e o r ig in to in f in ity along th e p o sitiv e re a l a x is . Com parison o f (3 .1 7 ) and (3 .1 8 ) w ith (3 .0 7 ) and (3 .0 8 ) g iv e s
% “ Ht g 0 ( i ) • (3 -2 2 )
Thus th e ta n g e n tia l components o f th e t o t a l m agnetic f i e l d a re eq u al to th e ta n g e n tia l components o f th e e x te rn a l f ie l d m u ltip lie d by a c e r ta in complex fu n c tio n o f a s in g le v a ria b le £. A B lm lla r r e la tio n e x is ts between th e to t a l and th e " e x te rn a l" c u rr e n t d e n s ity , namely
J = JCX 0 ( t) . (3.23)
Let us examine th e g eom etrical meaning o f th e v a ria b le ( in more d e t a i l . C onsider th e se c tio n o f th e paraboloid su rface by the plane of incidence passing through the given point ,(F lg . 4 ) . We denote by f the d ista n c e o f th e given p o in t from
* Jo u m . o f P hys., 1945.
(1 1 )


24
th e geom etrical boundary o f the shadow, considered p o s itiv e In th e d ir e c tio n o f th e shadow and n e g a tiv e In th e d ire c tio n o f the l i g h t . The d is ta n c e I I s measured In the plan e o f In cidence. Let R be th e ra d iu s o f c u rv a tu re o f th e su rface s e c tio n and k « & r /\ th e ab so lu te value o f th e wave v e c to r. Then th e q u a n tity
£ where d I s th e w idth (2 . 01) o f th e penumbra re g io n J la e a s ily seen to coincide with the quantity (3 . 16) defined fo r a paraboloid o f r e v o lu tio n . Since we know beforehand th a t formulae (3-22) and (3 .23) a re q u ite g e n e ra l, we conclude th a t th ey a re v a lid fo r a l l bodies with a given cu rv atu re. I f $ la given by (3.2 4 ).
(12)


25
These formulae give the transition from the shadow to the
light
Por large positive values of % the function 0(£) is approxi mately equal to
1 (*~ + bl
dU ) = ce \ 3 / e-b* , (3.25)
where a , b, c a re known numbers; namely
a - 0.5094 ; b « 0.8823 ; c « 1.8325 . (3.26)
Owing to the factor e'b^ the function 0 ( 0 decreased rapidly. This corresponds to the decrease of the amplitude In the Bhadow region.
For large negative values of 6 the function G (0 admits an asym ptotic expansion of th e form
0 ( 0 =■ 2 + + • • • (3 -2 7 ) 20
and tends to a lim it which I s equal to 2. This lim itin g value corresponds to form ulas (1.11) f o r the Illu m in ated reg io n . The d isco n tin u o u s fu n ctio n (1 . 11) i s thus replaced In our more e x act s o lu tio n by th e continuous fu n c tio n (3 .2 3 ). This en ables us to c a lc u la te the d is tr ib u tio n o f c u rre n ts on the su rface o f a con d u ctin g body w ith s u f f i c i e n t accuracy. In th e Appendix are given ta b le s o f th e fu n c tio n 0 defined by (3 .19) snd of the function g re la te d to 0 by the equation
0 <«) * • ’ «<x) ( 3 . 88)
and e x p re s s ib le in form o f th e In te g ra l
(1 3 )


(3.29)
26
8(x)=q r
The fu n c tio n O(x) I s ta b u la te d f o r valu es of x from x - 4.5 to x =1 w ith I n te r v a l 0 . 1, and th e fu n c tio n g(x) Is ta b u la te d fo r a range o f values o f x from x» - 1 to x = 4.5 w ith the same In te rv a l, Por v alues o f x le s s than x » - 4 .5 e x p ression (3.27) may be used, and f o r v alues o f x g r e a te r than x - 4 . 5 form ula (3 . 25) becomes applicable.
APPENDIX
Table o f the function d(x). g(x)
X Re 0 Im a 1 ° 1 arc G
- 4.5 1.'998 -0.0055 1.9990 9 -30" - 4.4 1.9997 -0.0059 1-9997 10' 10" * 1.9997 -0.0063 1.9997 10' 50"
- 4.2 1.9996 -0.0067 1.9997 11'40" - 4.1 1.9996 -0.0073 1.9996 12' 30" - 4.0 1.9995 -0.0070 1.9995 1 3 '20" - 3 .9 1.9994 -0.0084 1.9995 1 4 '3C" - 3 .0 1.9994 - 0.0090 1.9994 15'30" - 3.7 1.9992 -0.0090 1.9993 16’50" - 3 .6 1.9991 - 0.0106 1.9991 10 ' 10" - 3 .5 1.9990 -0.0115 ■ 1.9990 19140" - 3.4 1.999 - 0.012 1.999 21’ - 3.3 1.999 -0.014 1.999 23’
(14)


27
X Re 0 Im a Id arc 0
- 3-2 1-998 -0.015 1.998 2 6 
. 3.1 1.998 - 0.016 ' 1.998 281 - 3.0 1.998 - 0.018 1.998 31' - 2.9 1.997 - 0.020 1.997 34- 2.8 1.996 - 0.022 ■ 1.996 37' - 2.7 1.996 -0.024 1.996 41' - 2.6 1.995 - 0.026 1.995 46' - 2.5 1.993 -0.029 1.994 51' - 2.4 1.992 - 0. 03? 1.992 56' - 2.3 1.990 -0.036 1.990 - l°0 3 ' - 2 .2 1.988 -0.040 1.988 - 1° 10' - 2.1 1.985 -0.045 1.985 - l ° l 8 ' - 2 .0 1.981 -0.050 1.982 - l° 2 7 ' - 1.9 1.977 -0.056 1.977 - l°3 7 ' - 1.8 1.971 - 0.062 1.972 - 101*7 . - 1.7 1.965 - 0.068 1.966 - 1° 58' - 1.6 1.956 -0.075 1.958 - 2° 11' - 1.5 1.946 - 0.082 1.948 - 2°25' - 1.4 1.933 - 0.090 1.936 - 2°40' - 1 .3 1.919 -O.098 1.921 - 2°55' - 1,2 1.901 -0.105 1.904 - 3° 10' - 1.1 1.880 -0.113 1.884 - 3°27' - 1.0 1.857 -0.119 1.861 - 3°40' - 0.9 1.829 -0.123 1.833 - 3°51' - 0 .8 1.798 - 0.126 1.802 - 4°00’ - 0.7 1.762 - 0.126 1.766 - 4°05’ - 0 .6 1.722 - 0.122 1.726 - 4003" - 0.5 1.678 -0.115 1.682 - 3°54' - 0.4 1.630 - 0.103 1.633 - 3°36' - - .3 1.578 - 0.086 1.580 - 3°o6 ' 0 .2 1.522 -0.063 1.523 - 2° 22' - 0.1 1.462 -0.034 1.463 - 1°21'
(15)


28
X Re 0 Im 0 M arc 0
0 1.399 0 1.399 0° 00' 0.1 1.333 0.040 1.334 1°44* 0.2 1.263 0.086 1.266 3°55' 0.3 1.189 0.137 1.197 6°35' 0.4 1.111 0.193 1.128 9°51' 0.5 1.029 0.252. 1.059 13°45' 0 .6 0.941 0.312 0.991 18° 2f 0.7 0.846 0.373 0.924 23°47' 0 .8 0.744 0.432 0.860 30° 08' 0.9 0.634 0.484 0.798 37°221 1 .0 0.515 0.529 0.738 45°44'
Table of th e fu n ctio n g(x)
T?
O(x)
X Re g In. g 1*1 arc g
- 1.0 1.794 0.495 1.861 15°26' - 0.9 1.805 # 0.320 1.833 10°041 - 0.8 1.793 • 0.181 1.802 5° 47' - 0.7 1.765 1.766 2° 28' - 0.6 1.726 1.726 0°04' - 0.5 1.681 - 0.045 1.682 - 1°31' - 0.4 1.632 - 0.068 1.633 - 2°23' - 0.3 1.578 - 0.071 1.580 - 2°35' - 0.2 1.522 - 0.059 1.523 - 2°13' - 0.1 1.462 - 0.034 1.463 - 1° 20' 0 1.399 0 1.399 0° 00'
(16)


29
x Re e Im g 1*1 arc g
1.333 0.040 1.334 1°430.2 1.263 0.083 1.266 3°450.3 1.190 0.127 1.197 6°04' 0.4 1.115 0.169 1.128 8°37’ 0.5 1.038 0.209 ■ 1.059 11° 21' 0.6 O.961 0.244 0.991 1 4 °l4' 0.7 0.883 0.274 0.924 17°14' 0.8 0.806 0.299 0.660 20° 190.9 0.732 0.317 0.798 23°271 1.0 0.660 0.331 0.738 26° 38' 1.1 0.591 0-339 0.682 29°50' 1.2 0.527 0.343 O.628 33°02' 1.3 0.467 0.342 0.578 36°13' 1.4 0.411 0.338 0.532 39°25' 1.5 0.360 0.330 0.488 42°34' 1.6 0.313 0.320 0.448 45°421 1.7 0.270 0.309 0.410 48°48' 1.6 0.232 2.960 0.376 51°53' 1.9 0.197 0.281 0.343 54°56' 2.0 0.167 0.267 0.315 57°591 2.1 0.140 0.252 0.289 6l ° 00' 2.2 0.116 0.237 0.264 64°00' 2 .3 0.095 0.222 0.242 66° 5B' 2 .4 0.076 0.208 0.221 69°56' 2 .5 0.0596 0.1936 0.2025 72°54' 2.6 0.0453 0.1797 0.1853 75°51' 2 .7 0.0330 0.1664 O.I696 78°47' 2.8 0.0224 0.1536 0.1552 8l°43' 2.9 0.0133 . 0.1414 0.1421 84°391 3.0 - 0.0055 0.1299 0.1300 87°3v 3.1 - 0.0010 0.1190 0.1190 90°30' 3.2 - 0.0065 0.1088 0.1089 93°25'
(17)


30
* Re g Im g M a rc g
3 .3 - 0.0110 0.0991 0.0997 96°20* 3.* - 0.0147 0.0901 0.0913 99°15' 3.5 - 0.0176 0.0817 0.0836 102° 10' 3.6 - 0.0199 0.0739 0.0765 105°05' 3.7 - 0.0216 0.0666 0.0700 108° 00' 3 .8 - 0.0229 0.0599 0.0O41 110°55' 3.9 - 0.0237 0.0537 0.0587 H 3 °5 0 ’ 4.0 - 0.0242 0.0480 0.0537 116°45' 4.1 - 0.0244 0.0428 0.0492 119°40' 4.2 - 0.0243 0.0380 0.0451 122°35' 4.3 - 0.0240 0.0336 0.0413 125°30' 4.4 - 0.0235 0.0296 0.0378 128°25' 4.5 - 0.0228 0.0260 0.0346 131°20'
(18)


31
j j L DIFFRACTION OF RADIO WAVES'AROUND THE EARTH'S SURFACE V. Fock
The problem o f th e p ro p a g a tio n o f r a d io waves around th e homogeneous su rfa ce o f th e e a rth is In v es tig a te d . The d if f r a c ti o n e f f e c ts a re co n sid ere d but th e in flu e n c e o f th e Io n o sp h ere Is n e g le c te d . The aim o f th e p a p e r I s t o d e r iv e f o rm u la s f o r th e wave a m p li tude as a fu n c tio n o f th e e le v a tio n o f th e so u rce , I t s d ista n c e from th e p o in t o f o b se rv a tio n ( s itu a te d on th e s u rfa c e o f th e e a r th ) , o f th e wave le n g th and o f e l e c t r i c a l p r o p e r tie s o f th e s o i l . The main r e s u l t is the d e riv a tio n o f an ex p ressio n fo r the a tte n u a tio n fa c to r in form o f an I n te g r a l. T his e x p re ssio n i s v a lid f o r a l l th e v a lu es o f p a ra m eters which a re of p ra c tic a l I n te r e s t. In the lim itin g cases the w ellknown fo rm u la s a r e o b ta in e d : t h e W eyl—v an d e r P o l form ula f o r ill u m in a te d re g io n and th e form ula which c o rr e s p o n d s to th e f i r s t te rm i n W atB on's s e r i e s f o r the shaded region (th e la tte r in a s lig h tly c o rrected f o rm ). E s s e n t i a l l y new I s th e I n v e s t i g a t i o n o f th e reg io n o f th e penumbra (near th e ll.ie o f h o riz o n ). Form ulas a re o b ta in ed which g iv e a co n tin u o u s t r a n s i tio n from th e illu m in a te d re g io n to th e shaded one. Methods f o r n u m e rica l c a lc u la tio n s o f sums and I n te g ra ls Involved in th e problem a re e la b o ra te d .
INTRODUCTION *
T h ere a r e many p a p e r s d e v o te d t o th e p ro b lem o f th e d i f f ra c tio n o f ra d io waves around th e su rfa ce o f th e e a r th . A r e v ie w o f more r e c e n t i n v e s t i g a t i o n s may be fo u n d i n a p a p e r by B. V vedensky.^ The in te r e s t in th is problem i s J u s ti f i e d by th e f a c t, th a t a t sm all d lo ta n c e s , o f th e o rd e r o f a few hundreds o f
*A s h o r t . a c c o u n t o f th e r e s u l t s o f t h i s p a p e r i s g iv e n in our note.


52
k ilo m e tre s , th e r e f r a c tio n o f ra d io waves In the Ionized lay ers o f th e atmosphere may be n eg lected and the d e c isiv e ro le In the pro p ag atio n o f ra d io waves is played oy the d if f r a c tio n . In sp ite of the fact th a t a rigorous so lu tio n of the pro blem o f d if f r a c t i o n by th e sphere had been alread y o b tained some decades e a r l i e r , no p r a c t i c a l l y s u ita b le approximate s o lu t io n has been proposed up t o now. In t h i s paper we In tend to f i l l up th is gap.
1. STATEMENT OP THE PROBLEM AMD ITS SOLUTION IN THE FORM OP SERIES
We denote by r , 6 , $ sp h e ric a l co o rd in ates w ith o r ig in a t the center of the earth globe. The e q u atio n o f the e a r t h 's s u rfa c e (considered as smooth) I s r ■ a , where a Is th e ra d iu s of th e e a r th . Let us suppose th a t a v e rtic a l e le c tric dipole la located a t the point r - b, 6 * 0 (where b> a). S uppressing the tim e-dependent f a c to r e~lu)t In the f i e l d components, we can express th ese components by means o f the Hertz fu n c tio n U which depends on r and 0 o n ly . De n o tin g by k th e ab so lu te v alue o f th e wave v e c to r we o b ta in fo r the f ie ld In the a ir;
(2 )


33
the o th e r components b ein g equal to zero. Sim ilar equations hold for the fie ld In the e a rth . The fu n c tio n U s a tlB f le ^ fo r r > a the equation LV + k2U - 0 , ( 1. 02) and the r a d ia tio n co n d itio n a t i n f i n i t y
11m - lk r U ) (1.03)
I f b > a, 1. e. I f the source (dipole) Is located over the e a r t h 's su rfa c e and no t on th e su rface I t s e l f , U must have
a sin g u la rity a t the point r ■ b, 8 * 0, such th a t
_lkR
U * S jp - + u f (1.04)
and U* remains f i n i t e I f kR ■» 0. In th is formula
R = Vr2 + b2 - 2rb cos S (1 . 05) is the d ista n c e IVora the d ip o le . On the e a r t h 's su rfao e the Hertz fu n c tio n U has to s a t i s f y th e boundary conditions which ensure the c o n tin u ity of th e ta n g e n tia l components Eg and H^. I f we denote th e Hei^tl fu n ctio n w ith in the e a rth by Uj th ese boundary c o n d itio n s w ill have th e form:
k2U = k2 U2 ; £ (rU) « (rU2 ) f o r r - a . (1 . 06)
For O ^ r ^ g (w ith in the e a r th ) the fu n ctio n U2 has to s a tis f y an equation sim ila r to (1 . 02) and to remain f i n i t e . The q u a n tity k2 In form ula (1 .0 6 ) and In subsequent form ulas I s determ ined by the equation i,2 = „„2 * , „ (1.07)


3*
and by the c o n d itio n Im(kg) > 0. I t Is u se fu l to Introduce In ste ad of th e co n d u c tiv ity of' the e a r t h 0, a length I which c h a ra c te riz e s the s p e c if ic re s is ta n c e o f th e e a r th . We put
f « c/4sfl. (1.08)
For sea w ater the v alues o f I vary from 0.05 cm (very s a lty w ater) to 0 .5 cm (d careely s a l t y w a te r ) . For th e s o il th is length Is hundreds or thousands times g re a te r. Introducing the complex lndu o tlv e c a p a c ity of. th e e a rth
1 H I 0*09)
, fcg '■ k. J T ( i.io ) The s o lu tio n o f our problem In th e form of s e rie s Is well known. We w rite , down the n e cessary ..form ulas, w ithout giving th eir derivation
*n <x> *V"5 "Jn+* W.».
Cn tx) (*>.■■
( 1 . 11)
where J y(x) Is th e Beeeel fu n c tio n and H ^ ( x ) I s the Hankel fu n c tio n of th e f i r s t kin d . These fu n c tio n s are; connected by the re la tio n ♦n (*) C^(x) - ^ ( * ) ' C n (x) ■ 'l . ‘ (1.18)
We Introduce a s p e c ia l n o ta tio n f o r the logarith m ic d e riv a tiv e of th e fu n c tio n Fn(* )i *n<x >
. »A(«) *n ( x ) <*)
(1.13)


35
A8 s e e n rro m ( 1 . 0 1 ) , th e f i e l d on th e e a r t h 's s u r f a c e aay be expressed by the q u a n titie s
° a ” D| r ■ a 5 * 5F (rU> |r - a* ( * • » > For th ese q u a n titie s th e follow ing s e rie s In Legendre p o ly n o m ia ls may be o b ta in e d :
„ - - r V Y (2n + 1) <n(kb)
Pn(cos e),
Ua kab Z.!^(ka) - ^ Xn (k2a) Cn(ka)
D. = . k- F] (2n + 1 ) Cn(kb) x„(k2a)
r((-'os e). 1 ^ xn ( V > <n<ka>
(1 .1 5 )
(1 .16)
Our ta s k I s to p erform an approxim ate summation o f th e se se rie s .
2. THE SUMMATION FORMULA
The sums we have to c a lc u la te are o f the form
S - ^ v * ( v ) P y_^(cos 6), (2.01)
where the summation Is tak en over h a lf In te g ra l values o f v. I n th e sum ( 1 .1 5 ) th e f u n c ti o n $ ( v ) ( d i s r e g a r d i n g a c o n sta n t fac to r) is equal to
*(v) .. ----------------- j-i- 2 --------------------------- . (2 .( ‘ X y-i(k2a)
I n th e sura ( 1 .1 6 ) t h i s f u n c t i o n d i f f e r s fro m ( 2 .0 2 ) by th e fa c to r xv_
(5)


36
For the d i r e c t com putation o f th e eun I t would be necessary to take th e number o f th e tarm s approxim ately equal to 2ka, 1. e. to double the number o f th e waves which may be p ut around the e a r th circum ference. Since t h i s number la enormous. I t la e v i d e n t, th a t such a d i r e c t summation Is Im possible. For the c a l c u la tio n o f the sum S I t I s n ecessary to make use of th e fa c t t h a t d(v) i s an a n a l y tic a l fu n c tio n and to transform th is sum in to an I n te g r a l, which Is to be ev a lu a ted by some approximate method. Such a tra n s fo rm a tio n was f i r s t l y proposed by Watson^ In 1918 and was th e n used by v ario u s a u th o rs. But a l l th ese a u th o rs aimed to b rin g th e ex p re ssio n o b tain ed by th is tr a n s fo rm ation to the form o f a sum o f re s id u e s , while our aim Is to s e p a ra te out a main term which I s e a s ie r to In v e s tig a te and to estim a te the magnitude o f th e rem ainder. The method of com p u ta tio n of th e main term I s not p redeterm ined th ereb y . When perform ing our tra n sfo rm a tio n we have to b ear in mind th e follo w in g g e n e ra l p ro p e r tie s o f th e fu n c tio n d (v ). I t Is an a n a ly tic a l fu n c tio n o f v meromorphlc In th e r ig h t h a lr -plane. I t has poles only In the f i r s t quadrant and Is holomorphlc In the fo u rth quadrant. I t decreases a t In fin ity In such a way t h a t a l l th e I n te g r a ls co n sid ered converge. The Legendre fu n c tio n s t h a t e n te r (2 .0 1 ) can be expressed by means o f th e fu n c tio n
°v ■ £ {Y r { M t } p ( * ' * ' v + r a r * ) (2 ‘03)
( 6)


37
where F denotes the hypergeom etrlcal fu n c tio n . Denoting by Gf and by P*_^ the ex p ressio n s which are obtained from G and from Pymj = Py_ j(co s ®) by re p la c in g 0 by 71 - 0 we g e t:
P , ■ — 7= = f e iv 0 ' 1T 0* + e ' lv ®+1? o 1 . (2.0*.) v' s 11V2 s i., 0 I v v J
It. i s seen from (2.03) th a t I f the values of v l i e o u t sid e of a c e r ta in s e c to r , which Includes the n egative re a l a x le , and I f | v s in 0 | I s la rg e , then the fu n ctio n Gy (and a ls o G*) I s approxim ately equal to
Ov ~ t ^ A . (2.05)
S u b s titu tin g (2.05) In (2 .p 4 ) we g et the w ell known asym ptotic ex p ressio n f o r I f we denote by B(v) the f i r s t term in formula (2.C1.) :
B(v) = — -1--------- e lv ®-1¥ G* (2.06) 7i V2 s in 0
the follo w in g r e la tio n may be proved
P*_j ■ e 1^ - ^ " Pv. j + 21 cos vti B (v). (2.07)
We s h a ll use t h i s r e l a t i o n l a t e r on. We note th a t B(v) Is holomorphic In the rig h t half-p lan e. Let us c o n sid er In th e plane o f th e .complex v a ria b le v th re e c o n to u rs: 1) th e loop which s t a r t s a t i n f in ity on the p o s itiv e r e a l a x is , runs above the r e a l a x is , e n c irc le s the7
(7)


38
o rig in counter-clockw ise and retu rn s to the s ta rtin g point a t I n f i n i t y running below the r e a l a x is ; 2) th e broken lin e Cg, which c o n tain s th e f i r s t quadrant and la describ ed (in i t s h o riz o n ta l p a r t drawn s l i g h t l y o v er the r e a l ax le ) from the l e f t to th e r ig h t a id e ; 3) the s t r a i g h t lin e which crosses th e o r ig in and i s in c lin e d a t a sm all angle to the Imaginary a x is. This lin e is described from the top to the bottom and lie 3 in the second and fourth quadrants. He can w rite th e sum S in th e form
S • j J" v$(v) sec vii dv, (2.08)
sin c e th e I n te g r a l on the rig h t-h a n d sid e reduces to the sum o f th e re s id u e s in th e p o in ts v * n + 4. The fu n ctio n d(v) being hdlomorphlc in the fo u rth q u a d ra n t, we may rep la c e the contour Cj by th e contours C2 and and w rite
5■' * J C2
»♦(») ■
(2.09)
This tra n sfo rm a tio n o f the sum corresponds to the usual one; the In teg ral along the contour is neglected because of th e sm alln ess o f the odd p a r t o f $(v) (an estim ate o f i t s magni tude w ill be given below ), and th e in te g r a l along Cg is reduced


59
t o the sum o f re s id u e s . But we s h a ll go a ste p fu r th e r and d iv id e the I n te g r a l along Cg in to two p a r ts : the main term >nd the c o rre c tio n term . I n s e r tin g in the in te g r a l th e ex p re ssio n (2 .0 7 ) f o r P*_£ we s h a ll have
S = Sx + S2 + Sy (2 .1 0 ) '
where
S1 ’ ] v*(v) B(v) dv. (2.11) C
S9 ■ - $ \ v*(v) sec vn e lvfl Pv . dv (2.12) J C2
S3 = | j v*(v) sec vn P j_ j dv. (2.13)
C5
The in teg ran d in S1 has no poles on the re a l ax is (and a ls o in th e fo u rth q u a d ra n t). T herefo re, th ere i s no d iffe re n c e , whether we ev a lu a te the in te g r a l along Cg or along Cj We have denoted by C any contour, which i s e q u iv alen t to Cg o r C y The re p re s e n ta tio n o f S as a sum o f th ree I n te g ra ls (2.10) is e x a c t—th e re was made no n e g le c tio n in our d e riv a tio n . But th e e stim a tio n o f the magnitude o f Sg and shows th a t these in te g r a ls a re n e g lig ib ly sm all a s compared to S1 . In f a c t , i f we ev a lu a te the I n te g r a l Sg as a sum of r e s i dues a t th e p o le s of d(v) we s h a ll see th a t i t s r a t i o to S1 is of the order
(9)


(2 .I t)
40
| #2 iv i (* -« )|
where v1 Is th e pole o f d(v) n e a re s t to the r e a l a x is . The Imaginary p a r t o f Vj I s p o s itiv e and f o r la rg e values o f ka w ill be Im(vj) - c(ka)1/5 , (2.15)
where c I s a pure number o f th e o rd e r of u n ity ( f o r the p e r fe c t conductor c ■ 0 .7 0 ). Since ka Is very la rg e ,o f the order o f a m illio n ( f o r X • 40 m, ka • 10^), I t Is c le a r , th a t the q u a n tity (2 .1 5 ) w ill be la rg e ( f o r In stan ce, equal to 70) and the q u a n tity (2.14) w ill be n e g lig ib ly sm all. (In our problem 6 cannot reach the v alue s sin c e In th is case we have to take Into account the Influence of Ionized layers of the atmosphere and our form ulas cease to be v a l i d .) The v alue of th e I n te g r a l S^ Is determined by the odd p a r t o f $ ( v ). But the odd p a r t o f t h i s fu n ctio n w ill be o f the order | e 21k2a | . (2.16)
S ince th e im aginary p a r t o f kga I s a p o sitiv e and very la rg e , th e v alue o f (2 .1 6 ) w ill be Inconceivably Bmall. The fo llo w in g p h y sic a l p :c .tu r e g jv e s a n o tio n of th e sm alln ess o f the I n te g r a ls S2 and S^. The In te g ra l Sg Is the am plitude o f a wave which tr a v e lle d once o r s e v e ra l times around th e globe w ithout r e f r a c t i o n (by means o f d if f r a c tio n o n ly ) . The in te g r a l S j la the am plitude o f a wave which
(1 0 )


*»1
traversed a path equal to the diam eter of the globe with the abso rp tio n which ta k e s p la c e w ith in the e a rth . I t Is c le a r th a t both the I n te g r a ls a re nefellglbly small as compared w ith the am plitude o f the wave which reached the o b server through the a i r by the n e a re s t way. T herefore w ith the whole p e rm issib le accuracy ( l . e . with an e r r o r which I s n e g lig ib ly sm all as compared w ith the e rro rs Involved In th e p o s itio n o f our p h y sic al problem) the otu.i S defined by (2.01) may be pu t equal to the In te g ra l Sj alo n e. This In te g r a l may be w ritte n In th e form
= ^ v * (v ) e lv0 0 * d v , (2.17)
which follow s from (2.11) when the e x p ression (2 . 06) fo r I Is Inserted.
I f 0(v) I s the fu n c tio n (2 .0 2 ), then the re la tio n between urn S and the q u a n tity Ufl Is
ua ' ' TEafr S> (3.01) approximate ex p ression fo r U& may be
Therefore, w ritten
2e = \ v * { ,
V
elvO > d v .
us s (3 .02)


42
The p o s itio n of the main p a r t of the In te g ra tio n p ath In (3.02) depends on the p o in t fo r which the In te g ra l Is evaluated. In g e n e ra l the main p a r t I s In the v ic i n ity o f the p o in t v = v0 , where (5.03)
The q u a n tity h la th e len g th o f the p e rp en d icu lar dropped from th e e a r t h 's c e n te r on th e ray (1. e.} on the s tr a ig h t lin e which connects th e source and the p o in t o f o b se rv a tio n ). For the approximate evaluation o f the In te g ra l I t Is necessary to obtain the asymptotic expressions for the func tio n s 0* and d(v) v a lid on th e main p a r t o f the In te g ra tio n p a th . Since v0 and vq6 a re la rg e as compared w ith u n ity , we may pu t according to (2.05)
Q* = V w /v . ( 3 .0 4 )
For th e Hankel fu n c tio n s Involved In ip(v) one may te n ta tiv e l y use the Debye exp ressio n
Cw_4 (P)
V i - ( v2/ p2 )
(3-05)
where (3.06)
These expressions are v alid provided the condition
I p2 - v2 ! » PV 3
U2)
(3.07)


*»3
la s a tis f ie d - As to th e fu n c tio n i t s value near the p o in t v * vQ nay be re p re se n te d w ith a s u f f ic ie n t ap p ro x i mation by the e x p ressio n '
K - i <k2a > “ - W 1 - T 7 (3-03)
k2 a
In o rd e r to make c l e a r . In which cases the In e q u a lity (3.07) Is s a tis f ie d , le t us Introduce the parameter
(3-09)
where 7 la the angle between the v e r t i c a l d ire c tio n a t the observ atio n p o in t and the d ire c tio n from t h i s p o in t to the source. I t I s e a s ily seen th a t f o r v * vQ, p • ka th e In e q u a lity ( 3 . 07) Is eq u iv a le n t to the c o n d itio n th a t p should be larg e and p o s itiv e . Such values of p correspond to the Illum inated reg io n . The v alues o f p o f the o rd e r of. u n ity (p o sitiv e and negativ e ones) correspond to the reg io n o f penumbras the special value p ■ 0 gives the boundary of the geom etrical shadow (horizon l i n e ) . Large and neg ative values of p c o rre s pond to the shadow re g io n . In t h i s s e c tio n we s h a ll In v e s tig a te the case of a larg e p o sitiv e p (Illum inated region)j other cases w ill be in v e stig a ted in the next sections. We have seen thai, I f p » 1 the Debye expressions fo r the Hankel fu n c tio n s a re v a lid . I n s e r tin g th ese expressions In to
P cos 7,
(13)


44
(3 .0 2 ) and u sin g (3.04) and (3 .0 8 ) we g et
(3.10) e 1<BV 7 dv
where (3.11) lea
I f the condition kh cos y » 1 (3.12)
1b s a t i s f i e d , where h • b - a Is th e h e ig h t o f th e source above the e a r th , th e In te g ra l (3-10) can be c a lc u la te d by means of th e method o f th e B teepest d escent and the follow ing " r e f le c tio n formula" Is obtained:
In th is formula
la th e d istan c e from the source, and W la th e " a tte n u a tio n fu n c tio n " which In our case I s equal to
(3.14)
(14)
(3.15)


The q u a n tity d efin ed by th e s e rie s (1.16) d if f e r s (in our approxim ation) from Ua by a c o n stant fa c to r only. We have '__________
®; ■ - ^ \ f l - {£ s in 2 7 Ua . (3 .1 6 )
The l a s t form ula la tru e not only fo r the illu m in ated region, but also In other cases. I f condition (3-12) Is not s a tis fie d , the denominator In the In tegrand (3.10) cannot be considered as slowly v ary ing. I f In ste a d of (3.12) we suppose th a t the c o n d itio n s;
1 < < (ka)2 /3 , (3.17) n
1 « kR « a /h , (3.18)
are s a tis fie d (the Inequality p » 1, being a consequence of these c onditions), the In te g ra l (3.10) can be approximately c a lc u la te d by in tro d u c in g a new In te g ra tio n v a ria b le u, according to
» ' ^ ^ ■ (3>19)
For the function W In (3.13) the follow ing approximate expression Is obtained;
(15)
■ <3 -20 >


M-0 = h/R (3.21)
J*6
where
is th e in c lin a tio n o f the ray to the h o rizo n . The contour T i s a s t r a i g h t lin e which crosseB the p o in t u, * u0 p assin g th ere from th e fo u rth to th e second quadrant o f the plane o f n (or of u - u0 to be more e x a c t) . Hie I n te g r a l ( 3 . 20) can be c alcu lated w ithout any f u r th e r approxim ation and g ives the well-known Weyl—van d er Pol form ula. I f we put
<>•»>
we s h a ll have
W » 2 - i(de"((3+T)2 f e*2 1®
(3 .2 »
To o b ta in th e f i e l d components from our ex p ressio n s fo r Uft and we have to d i f f e r e n t i a t e th e se e x p ressio n s by 6 which Is e a s ily done, sin c e we may reg ard a l l f a c to rs in ( 3 . 13) ex cept e lkR, as co nstants.
it. ASYMPTOTIC EXPRESSIONS FOR THE HANKEL FUNCTIONS
In th e follow ing we have to c o n sid e r the case when the p o in t of observation is in the region o f penumbra. T his case is c h a ra c te riz e d by th e values o f th e param eter p (p o s itiv e s o r n e g a tiv e s) o f the o rd e r u n ity . Aa th e in e q u a lity
( 16)


U7
(» 01) is n o t s a t i s f i e d In th lB c a se , the hebye expressions 05) fo r th e Hankel fu n ctio n s are not v a lid on the main p a r t 0f the In te g ra tio n c o n to u r and must be replaced by some o th e rs, jhe new ex pressions f o r the Hankel fu n c tio n s s u ita b le fo r our purpose c»n be o b tain ed from th e asym ptotic expressions which
are given In our p”e v lo u s pap er^, o r from the form ulas given h
in the well-known W atson's t r e a t i s e , b ut i t Is more sim ple to deduce them Independently. Our aim Is to f i n d an approxim ate expression fo r the Hankel fu n c tio n in te rm s of th e fu n c tio n w (t), defined by the In teg ral
w (t) = Ije12-1/ 5*5 dz, (<(.01)
the contour T running frun in fin ity to the origin along the ray arc c » - 2u/3 and from th e o r ig in to In f in ity along the ray arc z ■ 0 (the p o s itiv e r e a l a x is ) . Tne fu n ctio n w (t) sa tis fie s the d iffe re n tia l equation
w "(t) - tw (t) (1. . 02)
with the i n i t i a l conditions:
"(0 ) = ■o / l = 1.0899290710 + 10.6292708b25, 32 /3 r (2/5)
w' (0) - e -i(it/6) . 0,791,5704238 - io.4587**5‘‘,'8 l. 3 / 5 r ( */ « (<<. 03)
(17)


48
w (t) ie an I n te g r a l tra n sc e n d e n ta l fu n c tio n , which can be ex panded In to a power a e rle a o f the form:
w (t) w(0) 1 1 + 2T^ + (2.5)h-«> ) + (5-5
; t “ t i t 10 i
(4.04) I f we s e p a ra te In w (t) th e r e a l and the Imaginary p a rts (for re a l values of t) putting
w (tj ■ u (t) + l v ( t) , (4.05)
then u ( t ) and v ( t ) w l l l ’be two Independent I n te g r a ls o f equa tio n ( 4 , 0£) connected by the r e l a t i o n
u ’ ( t ) v ( t ) - u ( t ) v ■( t ) - 1. (4.06)
The asym ptotic expressio n s of th ese fu n ctio n s fo r larg e negative values of t are obtained by separation of the re a l and Imaginary p a r ts In th e form ulas:
For large p o sitiv e values of t the asymptotic expressions u ( t ) , v (t) and th e ir d e riv a tiv e s are of the form
, i t 3/ 2 i ( t ) = t * 1/ " e7 ;
, t t 3/2
= t 1/* e 3 ;
(18)
(4.09)


49
2 ,.3/2 v ( t) ■ K l/U • ’ 5 .
. a t 3/2
v , (t) , . l t l / ^ e 3 k .
Prom the s e r ie s (4.04) th e follow ing re la tio n s deduced: w w (te ^ ) • 2elE v (-t).
w (te "5”) = e 5 [ u ( t ) - l v (t ) j .
These r e la tio n s d e sc rib e th e b ehavior, of w(t) In t l t-plane. Ve note th a t w (t) is e x p re s s ib le In terms of th fu n c tio n of the o rd e r 1/5 scco rd ln g to the formula
«(t) (-«)3/2) .
A fter having enumerated '.he main p ro p e rtie s of now proceed to deduce the asym ptotic e x pression fo r fu n c tio n H ^ ( p ) where v and p are la rg e and nearly that the ratio
remains bounded, while p tends to In fin ity . The Hankel fu n c tio n H ^ ( p ) adm its the in te g ra l tatlon
(19)
(*•10)
are easily
(« .ll)
(4.12)
e complex
e Hankel
(*.13)
w (t), we the Hankel eq u al, so
{*1.10
repreaen


(1.15)
50
< ‘ »w. 4 p
where the contour C c o n sists of a p a rt of the str a ig h t lin e Im(v) = - 7t d e sc rib e d from - n l - onto some p o in t v = v w ith He(v0) < 0 [ e . g . vo * ( - n /V T ) - lw}, a s tr a ig h t lin e Jo in ing vo to the o rig in and, fin a lly , the p o sitiv e re a l axis des c rib e d from th e o r ig in to I n f i n i t y . Let us express v through t , according to (4 .1 4 ), and In tro d u ce a new I n te g r a tio n v a ria b le
z * I T J /Z ’V (4.16)
C onsidering t and z as f i n i t e and p as la rg e , we can ex pand the lntergrand In (4.15) In a a e rie s of negative (fra c tio n a l) powers of p. Since th e re le v a n t p a r t of the tra n s formed contour C c o in cid es w ith contour T we can w rite
- A ( l ) ' 1/5 - & ( f f * * ...]« •
* (4.17)
and ev alu a te th e In te g r a l u sin g (4 .0 1 ). We thus o b tain
(4.18)
In v irtu e of the d iffe re n tia l equation (4.02) the f if th d eriv ativ e equals * ( 5 ) ( t ) = t 2 w' ( t ) + 4 tw ( t) . (4 . 19)
(2 0 )


51
In s e rtin g t h i s In (4 .1 8 ) and usin g ( l . l l ) we g et the follow ing ex p ressio n f o r the fu n c tio n ? v- l / 2 (p):
Cv- l / 2 ( p ) = - - ( f ) l / 6| " ( t ) - ^ { | ) * 2/ 3 [t2W ( t) + 4 tw ( tj+ .. | . (4.20)
D iffe re n tia tin g th is expression with respect to p (with account of the dependence o f t on p w ith v co n stan t) we get the following expression fo r the derivative:
t ^ - l / 2 (p)= l ( f ) " l / 6| w ' ( t ) - ^ ( f ) ‘ 2/I?f*+9 )^ (t)-iltw '(t)] + . . . } . ( * . 21)
These ex p ressio n s w ill be used In the next se c tio n . § 5 . The expressions o f th e Herz fu n c tio n v a lid In the penumbra reg io n . We re w rite th e ex p re ssio n (3.02) f o r the Herz fu n ctio n re p la c in g th e r e in the q u a n tity Oj by I t s approximate value V jt/v and th e q u a n tity s i n 6 b efore the In te g ra l by 9. We g et
f
U ( v ) e lve y r a v . (5 .01) 8 kab J
The contour C may be taken Id e n tic a l w ith contour C2 , which was d efin ed In ^ 2 , o r may be re p laced by some contour eq u iv a len t to C? . The main p a r t of th e In te g ra tio n p ath l i e s in our case ( l .e . fo r f in ite values of the parameter p) near
(21)


52
th e p o in t v = lea. Consequently, th e fu n ctio n _^{k2a) involved in (2 . 02) can be rep la c e d by th e value o f (3 . 08) fo r v = ka. I n t r o d u c e th is In d(v) we o b ta in :
*(v)
____________ t v.j(K b )____________ (5.02)
For ( v j and i t s d e r iv a tiv e we must uee expressions v a lid n ear the p o in t v ■ k a. Such e x p ressio n s were obtained in the p receedlng p arag rap h . R e tain in g In (<*.20) and (X.21) the p rin c lp a l term s only we g e t:
I v ^ O * ) - - i ( j * ) V 6 •<*), (5.03)
Cv.* (k a ) * l ( ^ ) 1/6 w '( t ) . ( 5 .0 0
where the v a ria b le t 1* connected w ith v by the r e la tio n
(5.05)
The num erator In (5 .0 2 ) la o b tain ed from (5 . 03) by rep lacin g a by b and t by t ', where
(5.06)
Equating (5 . 05) and (5 .0 6 ) we o b ta in th e connection between t and t 1. Since th e r a t i o h /a , where h • b - a , Is a n a ll £we s h a l l c o n sid e r I t o f rhe sane o rd e r aa (ka)"2^ j we must neg le c t I t as compared to u n ity . We may then put
(2 2 )


f = t - y,
where
kh ( 5 . 08) (k a /2 ) ly
Is a quantity proportional to the height of the source over the e a r t h 's s u rfa c e . We may c a ll y the reduced heig h t o f the sou rce. Hence, w ith n e g le c t of terras o f the o rd er h /a o r Oca) -2 /3 a
tv-i(kb) ‘ - 1(!r )1/6w(t • y)* (5.09)
where t la determ ined by (5 .0 5 ). (We have a lso rep laced b by a In the f a c to r b e fo re w.) S u b s titu tio n o f (5 .0 3 ), (5-04) and (5.09) In (5-02) giver, the desired approximate expression fo r ${v). I f we pu t f o r the sake o f b re v ity
' W iq \ (5 -10)
* (v) ■ - ( | r ) /? •
:mberlng formulas (1 . 09) and (1 .1 0 ), v Mty q
- _ , /naN1^ V e - 1 + 1(x/2ti/)
Q' 1W -eTilx/Lf)
(5 .n )
■may w rite fo r


w ith th e same accuracy
54
q ■ 1 ( ^ ) 1/3 - = = = ? = = = = = = (5.13)
VX / V € + 1 + l(X /2n/)
This form is s l i g h t l y more convenient fo r c a lc u la tio n s . We have now to s u b s titu te th e value o f $(v) from (5-11) in to (5-01) and Intro d u ce th e in te g ra tio n v a ria b le t . Making t h i s s u b s t i t u t i o n , we may re p la c e the q u a n tity V""v"in th e i n t e grand by th e c o n sta n t value V k a and a lso w rite b in ste a d a t a in the f a c to r b efo re the in te g r a l. The r e s u ltin g formula may be w r itte n in th e form:
Ua = e - ^ f e 1Xt I & ( t) d t, (5.14) c'
where x denotes th e q u a n tity
x = ( t ) 1/3®' (5.15)
which may be termed as th e reduced h o riz o n ta l d istan ce from the s o u rc e, while y and q have the values given by (5.08) and (5 .1 3 ). The conto u r C must be such t h a t a l l th e poles o f the in te g ran d are comprised w ith in the contour; as we s h a ll see la te r , they are a ll situ ated in the f ir s t quadrant of the t p la n e . Thus we can c a rry out th e in te g ra tio n in (5-14) from loo to 0 and from 0 to + oa In o rd e r to g e t a more c le a r idea on the r a t i o of the h o r iz o n ta l and th e v e r t i c a l sc a le in the v a ria b le s x and y , we w rite th e e x p ressio n f o r th e param eter p, as defined by (24)


55
■)), in te rn s o f x and y . Prom th e c o n s id e ra tio n o f the trian g le with v e rtic e s In the e a r th 's c en ter. In the source po in t and In the p o in t o f o b s e rv a tio n , th e follow ing approxlpate expression la ea sily deduced:
I t follows th a t the equation of the horizon lin e Is x = V 7 . Fu rth er we s h a ll need th e r e l a tio n between the d i s tance R from the source as measured along a s tr a ig h t lin e and the h o riz o n ta l d is ta n c e afl as measured along the arc o f a grea^t c i r c l e . Assuming a® >> h, 1. e . (k a)1^ x >> y , th l6 r e la tio n may be w ritte n
kR ■ ka6 + u)o , (5.17)
where
“ o = h + ¥ - h • (5 -18)
6. DISCUSSION OF THE EXPRESSION FOR THE HERTZ FUNCTION
The expression o b tained fo r th e Hertz fu n ctio n Is most conveniently w ritte n In the form:
lka$
Ua * S-g— V ( x ,y ,q ) , (6.01)
where
- . ' V ? " • (6 -02> c
(25)


56
The q u a n tity V may be c a lle d a tte n u a tio n fa c to r by analogy with th e q u a n tity W, which waa Introduced e a r l i e r ^see (3 .1 3 )]. Let us determ ine th e connection between V and W. Since In the d e nom inators of ex p ressio n s (3.13) and (6 .01) the q u a n titie s R and aO can be co nsidered as e q u al. I t follo w s from (5-17)
W » Ve-lu)o. (6.03)
We have now to I n v e s tig a te the ex p ressio n (6.02) fo r V. We s h a ll f i r s t co n sid e r the case o f larg e p o s itiv e values o f p(Illum inated reg io n ). This case has been already discussed by a n o th er methcd (f 3 ). B ut, as formula (6.02) w?i obtained f o r the case o f a f i n i t e p . I t seems to be o f I n te r e s t to v e r if y th a t i t 13 a ls o v a lid in th e case o f a larg e p . I f p >> 1, th e I n te g r a tio n p a th may be deformed so as to c ro ss the p o in t where - / T - P- I t s main p a r t w ill be s itu a te d In th e domain o f la rg e n e g a tiv e v alues of t , where expressions (it.07) and (<<.0c) i c r w and w are a p p lic a b le . Using them and applying the method o f th e s te e p e s t d e sc e n t, we o b tain
V ,1<uc 2
i - i(d /p ) (6.014)
and in v ir tu e o f ( o .03)
• ' i- - i h m ■ |6 -°5> The l a t t e r ex p re ssio n p r a c t i c a l l y coincided w ith (5 .15). We note t h a t in th e case when x i s of the order o f u n ity o r large the condition p » 1 is su ffic ie n t for the a p p lic a b ility
(2 6 )


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0f the method of s te e p e s t d e sc e n t. I f x Is :-mall, the fu rth e r condition y2 » 2x is n e c essary . I f th e l a t t e r co n d itio n Is not s a t i s f i e d bu t l.ie In e q u a lity
x « y « 1/x (6.06) *’
is s a t i s f i e d I n s te a d , the I n te g r a l can be c a lc u la te d by another method. F u rth er s im p lif ic a tio n s in th e asym ptotic expression f o r w(t - y) can be then made, and th e in te g ra l (6.02) reduces to the form
l£ IT
V =e Vs ] ----------- d t - (6.07) C
Taking V - t as in te g r a tio n v a r ia b le , we are led to an In te g ra l o f the form (5.20) £with V- t - (k a /2 )1^ u ]and we g et again the Weyl-van d er Pol form ula (3-23) w ith the follow ing values of 6 and t :
6 - ¥ q V7 , T - e 1 " — X— . (6.08)
2V x
These valu es p r a c tic a lly c o in cid e w ith (3 .2 2 ). Let us now in v e s tig a te tne most in te r e s tin g case when p is o f th e order o f u n ity ( p o s itiv e or n e g a tiv e ). Vfe know th a t t h i s Is th e region o f the penumbra, where th e d if f r a c tio n e f f e c ts p lay the dominant p a r t . I f the values of x and y are of the order of u n ity , the most e f f e c tiv e method o f e v a lu a tio n o f the in te g ra l (6.02) is th e re p re s e n ta tio n o f t h i s I n te g r a l in form o f a sum of resid u es
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taken a t the poles o f the Integrand. D enoting by t 8 * t g (q) th e ro o ts o f th e e q u atio n
**'(t) - qw (t) - 0 ( 6 . OS)
we o b ta in
(6 . 10)
The ro o ts t 9 (q) are fu n c tio n s o f the complex param eter q. For the value q ■ 0 they reduce to the roots t^ = t g(0) of the d e riv a tiv e w '( t ) and fo r q * oo they reduce to the ro o ts t ° ■ t s (a} o f the fu n ctio n w (t). The phases o f t ; and t ° are equal to n/3» so th a t
We g iv e h e re th e m o d u li o f th e f i r s t f i v e r o o t s t ^ a n d t ° :
s
For la rg e v a lu e s of s we have approxim ately
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(6 .12)


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To c a lc u la te th e ro o ts f o r f i n i t e values of q we may use the d iffe re n tia l equation
(6.13)
which can be e a s ily deriv ed from (4 .0 2 ). The root t 8 (q) Is determ ined e ith e r as th a t s o lu tio n o f (6.13) which a t q * 0 reduces to t^ o r as th a t s o lu tio n which a t q ■ coreduces to t®. Both d e fin itio n s a re e q u iv a le n t. S to rtin g from the f i r s t d e f in itio n , a s e r ie s in ascending powers o f q may be e a s ily co n stru cted fo r t B; th i s s e r i e s w ill converge fo r | q | <|)/"t^*|. S ta rtin g from th e second d e f in itio n we may co n stru ct a s e rie s in descending (negative) powers of q; th is w ill converge fo r M > 1 ^ 1 - These s e r ie s s h a l l not be w ritte n down here. I t may be n o ticed t h a t th e v alue o f t , which fo r larg e values of | q | Is clo se to qc , I s no t a r o o t o f equation (6 .0 9 ). I f th e co n d itio n y2 « » I V s l 13 s a t i s f i e d , we have the approximate re la tio n
^■w| t • ch(y sh(y y f t g). (6.14)
This r e la tio n perm its us to e stim a te the value o f remote terms In the se rie s (6.1 0 ). I f s Is 3 0 'larg e th at |q | « I V ^ I . we have approxim ately t g * t g (0) ■ t ^ . I t follow s from th is and from e x p ressio n (6 .1 4 ) t h a t the s e r ie s (6.10) Is always
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convergent. But I f x Is small or I f y Is la rg e , the s e rie s converges slo w ly , and to c a lc u la te I t s sum a larg e number of terms may be re q u ire d . In the shadow re g io n , where p Is la rg e and n e g a tiv e , the s e r ie s (6 .1 0 ) converges very ra p id ly and I t s sum approxim ately reduces to I ts f ir s t term. Our s e r ie s (6.10) corresponds to th a t o f Watson but has the advantage of sim p lic ity . The fundamental form ula (6.02) perm its us to In v e stig a te not only the lim iting cases (large p o sitive values of p -lllu m lr.ated region, large negative values of p-shadow region) but a ls o the In term ed iate c a se s, namely the region of the penumbra. While In the lim itin g cases our form ula leads to an Improvement o f form ulas p re v io u sly known (the r e f le c tio n formula and the Weyl-van der Pol formula f o r th e Illu m inated region and the Watson s e r ie s f o r the shadow r e g io n ). In the tra n s itio n a l penumbra re g io n I t y ie ld s e s s e n t i a l l y new r e s u lts . The case when x and y a re la rg e and p - f l n i t e (s h o rt waves, penumbra) Is o f s p e c ia l I n t e r e s t . This case has not been In v e s tig a te d b e fo re as th e known form ulas are not v a lid h ere. In what follow s we s h a ll d e riv e approximate form ulas, which allow a complete d iscussion of th is case. We In tro d u ce the q u a n tity
* -V 7 .
(?0)
2 = (6.15)


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which re p re s e n ts the reduced d ista n c e measured from the boun dary o f the geom etrical shadow (and no t from the so u rc e ). In the region o f geom etrical shadow we have z > 0, In the v lB lble region z < 0. Our p aram eter p , expressed In terms o f z and x, takes th e form
In our case x Is la rg e an z I s f i n i t e ; hence we have ap p ro x i mately p = - z. The main p a r t o f the I n te g r a tio n path In (6.02) c o rre s ponds now to v alues o f t o f th e o rd er o f u n ity ; but I f y Is larg e and t f i n i t e we may use fo r w (t-y) the asym ptotic ex p re ssio n (4 .0 7 ) which g ives
w(t - y) (6.17)
or approximately
1? -1 /4 i f y5^2- 1 V y t . (6.1ft) w(t - y) = e "y e 5
I n s e r tin g (6 .1 8 ) In to (6 .0 2 ) and rep lacin g In the fa c to r b efore the I n te g r a l the q u a n tity x^ y '^ by u n ity , we g et
yV2 V (x .y .q ) = e 5 V ^ x - V y . i l ) , (6.19)
where
f eizt
= J W*(t) - qw(t) d t ‘
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(6 .20)


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The terms n eg lected In (6 .1 9 ) are (fo r a f i n i t e z) o f the order of 1/V "y" (o r of 1 /x ). T herefo re, th e fu n c tio n V (x ,y ,q ) o f two arguments x ,y and o f the param eter q reduces In our ease to a fu n c tio n V1(z,q ) o f a s in g le argument z and of the same p aram eter q. The r e sulting sim plification Is quite e ssen tial. Let us now d e riv e the r e l a t i o n connecting the a tte n u a tio n fu n c tio n Wwith th e fu n c tio n V j. We have the I d e n tity
(6 .21)
where u> has th e v alue (5 .1 6 ). O m itting In (6.21) the la s t term we o b ta in from (6.03) and (6 . 19)
iz ? W ■= e^ V ^ z .q ) . (6.22)
Thus, in our approxim ation function W depends on x and y only through z = x -\J~y". The fu nctio n V1(z ,q ) I s an I n te g r a l tra n sc e n d e n ta l func tio n o f th e v a ria b le z. For a p o s itiv e z we can ev a lu a te the I n te g r a l (6.20) a s a sum o f re s id u e s , and we get
V ^ z .q ) * 12 ^ L i (6.23)
. ( t . - q j w(t_) 8=1 3 (fo r z > 0), where t s a re th e ro o ts o f equ atio n (6 . 09) which were d iscu ssed e a r l i e r . The la r g e r Is z the more ra p id ly converges th e s e rie s ( 6 . 23) . For a s u f f i c i e n t l y la rg e p o s itiv e z I t s sum reduces to
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the f i r s t term. For f in ite negative values of z (e .g . - 2 < z < o) the I n te g ra l (6.20) has to be e v alu ated by q u ad ratu res. For larg e n eg ativ e v alues o f z th i s In te g ra l may be evaluated by th e method o f s te e p e s t d e scen t, and we g et
v i ( z '«> e i V w ^ T ’ (6 ' 21°
According to (6 .2 2 ), th i s gives
W = 2 / ( l + ^ ) . (6.25)
Since approximately z « - p, th is coincides with expres sion (6.05). We note In conclu sio n t h a t our fundamental formula (6.02) can be obtain ed by th e method o f p a ra b o lic eq u ation, proposed by M. Leontovich and ap p lie d by him-' to the d e riv a tio n o f the Weyl-van d er Pol form ula. The a p p lic a tio n o f L eontovich’s method (in a s l i g h t l y Improved form) to our problem w ill be given In a separate paper.
REFERENCES
1 V. Fock, C. R. Acad. S c l . URSS, >43 (19*»5). 2 B. Vvedensky, B u ll. Acad. S c l. URSS, s e r phys. , ^ 415 (19&0). 5 V. Fock, C. R. Acad. S c l. URSS, 1,. 97 (1934). u o. n . Watson, TrcaU.Be an..the..Theory o f B e m i F vnctlona. Cambridge, 1922. ^ M. Leontovich, B u ll. Acad. S c l. URSS, s e r . p h y s., fl, 16 (1 9 « ).
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... SOLUTION o p t h e problem o p pro pag ation o p electromagnetic waves *'■ ALONO THE EARTH'S SURFACE BY THE METHOD OF PARABOLIC EQUATION
The problem o f p ro p a g a tio n o f e le c tro m a g n e tic waves alo n g th e su rfa c e o f th e e a r th is so lv e d by th e m ethod o f p a ra b o lic e q u a tio n p ro p o sed by Leonto v ic h . In the f i r s t se c tio n th e su rfa ce o f the e a rth is co n sid ered as p lane and th e well-know n W eyl-van d e r Pol form ula i s d educed. T h is form ula tu rn s ou t to be th e e x ac t s o lu tio n o f th e pa ra b o lic e q uation w ith corresponding boundary c o n d itio n s. In th e second se ctio n the su rfa ce is co n sidered as sp h e ric a l, and the re s u ltin g form ula co in cid es w ith th a t o b ta in ed by Pock by th e method o f summation o f in fin ite se rie s rep resen tin g the rigorous so lu tio n o f th e problem .
A new form o f th e s o l u t i o n o f th e p ro b le m o f p r o p a g a tio n o f e lec tro m ag n e tic waves from a v e r ti c a l e lem en tary d ip o le
the f ie ld is c a lc u la te d f o r p o in ts on th e su rfa ce of th e e a rth , b u t a cc o rd in g to th e r e c ip r o c ity theorem th e same s o lu tio n gives d ire c tly the f ie ld a t any p o in t above the su rface i f the d ip o le is lo c a te d on th e s u rfa c e I t s e l f . In th e p re s e n t p a p er i t i s shown t h a t P o c k 's s o l u t i o n c a n a l s o be o b ta in e d by a n o th er m ethod, namely by red u c in g th e problem to an e q u atio n of p arab o lic type fo r the "atten u a tio n fu n ctio n ".
*In the sequel these papers w ill be re fe rre d as I .
M. L e o n to v ic h and V. Pock
s itu a te d a t a given h e ig h t above th e s p h e rica l su rfa ce o f the In th is so lu tio n
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The method o f p a ra b o lic eq u atio n was proposed by Leontovlch and a p p lied by him to th e s o lu tio n o f th e same problem f o r the case o f a plane earth . Since the considerations of the o riginal paper by leo n to v lc h ( ^ ) * * need some m o d ificatio n s, we s h a ll give In what follow s a new e x p o sitio n o f th e method, applying I t f i r s t l y to the case of a plane earth and considering then the case of a spherical earth.
1. THE CASE OP A PLANE EARTH
We assume th e tim e-dependence o f a l l th e f i e l d components to be o f the form e”lu>t. In th e fo llo w ing th is fa c to r s h a ll be om itted. Let us denote by k th e ab so lu te value of the wave v e c to r and by q th e complex In d u ctiv e c a p a c ity o f the e a rth :
I c - f S j , - e + i M . e + 1 , (i.oi)
The q u a n tity
having the dimensions of a length c h a ra c te rise s the sp ecific re s is ta n c e of th e e a rth ( t h i s le n g th v a r ie s from some ten th s o f a ce n tim ete r f o r sea w ater to te n and more m eters fo r dry s o i l ) . Let U be th e v e r t i c a l component o f the Hertz vecto r (the Hertz fu n ctio n ). This function s a tis f ie s the equation
AU + k2U - 0 . ( 1 . 03) We s h a ll w rite th e H ertz fu n c tio n In th e form
• • T h i s pap - w ill be r e f e r r e d In th e sequel as I I .
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where R la th e d ista n c e from th e p o in t o f o b servation to the *ource and th e fa c to r W le th e s o -c a lle d "atte n u a tio n fu n ctio n ". H3 i t 1® known, f o r kR-*0 th e H e rtz fu n c tio n tends to i n f in ity in such a way th a t W ta k e s a f i n i t e v a lu e . We n o m a llz e U in ( uch a manner t h a t th i s v alue s h a l l be equal to u n ity ( i t being gupposed th a t both th e source and th e o b servation po in t remain above the su rfa c e o f th e e a r th ) . In the follow ing we assume, however, th a t the source la located on th e e a r t h 's s u rfa c e . Let us introduce c y lin d ric a l coord in ates r , z w ith th e o rig in Jr. the dip o le and the z a x is drawn v e r t i c a l l y upwardb . On th e ear t h 's su rface we have z - 0 . The d is ta n c e R w ill be R - - J r 2 + z2 . Hie p rin c ip a l "larg e param eter" of our problem i s the q u a n tity h | . For la rg e In I the attenuation function Wis a slowly varying function of coordinates. In order to c h aracterize the slowness of its v ariatio n i t Is useful to introduce the dimensionless coordinates .
p - JSL . ; C = , (1-05) 2 Ini -J Ini
and to co n sid e r W as a fu n c tio n o f p and ?. The d e riv a tiv e s of W w ith re s p e c t to itB arguments w ill be then o f the same o rd e r of magnitude bb th e fu n c tio n W i t s e l f . S u b stitu tio n of (1.04) in to equation (1.03) gives fo r the fu n c tio n W(p,?) an e q u a tio n , which can be s im p lifie d i f one supposes th a t the i n c lin a tio n angle o f th e ray to the horizon Is small and th a t th e d is ta n c e from the source i s a t l e a s t equal to s e v e ra l wave le n g th s . These assumptions y ie ld the Ineq u alities:
| « 1 ; kR » 1 , (1.06) which a re eq u iv a le n t to
i « 2.JN ; P » •
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(1.07)


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Since |i)| la assumed to be la rg e , th e In e q u a litie s (1.07) hold in a wide range of the v alu es o f p and ( (and In any case fo r values of p and C of the order of u n ity ). I f the In eq u a litie s (1.07) are v a lid , th e eq u atio n f o r W (p,() assumes the form
The terms om itted in (1.08) a re of th e o rd er o f l / | q | as compared with those re ta in e d . The boundary co n d itio n f o r Won the e a r t h 's surface Is obtain ed from th e c o n d itio n f o r th e Hertz v e c to r
— * -------- U ( f o r z - 0 ) ( 1 . 09)
az
given by L eontovlch. I t has th e form
+ qxw • 0 (fo r C -0 ) (1.10) 9C where ____
ql - 1 P f el ^ (l-H)
and « I s th e s o -c a lle d lo ss a n g le , defin ed by
6 - arc tg ; 0 < 6 < | . (l .IP)
In th e lim it |n | —wo th e range o f the v a ria tio n s of p and Cls0<p<oo,0<?<®. As a " co n d itio n a t I n f in ity " we may re q u ire th a t fo r a l l p o sitiv e values of p and C [w ith the possible exception of the sin g u la r p oint p - 0 of equation (1 . 08) J the fu n ctio n V should be bounded o r such t h a t the H ertz v e c to r U Is bounded.
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We now proceed to th e form ulation o f the co n d itio n fo r p o O , Since t h i s I s a p o in t o f some d e lic a c y , we s h a ll d isc u ss I t in a more d e ta ile d way. / We must s t a t e , f i r s t l y , t h a t in the region clo se to the $ource, l . e . , f o r sm all v alu es of kR, the I n e q u a litie s (1.07) cease to be s a t i s f i e d ; th e d i f f e r e n t i a l equation (1 . 08) and the expression f o r W to be deduced from I t become In v a lid . The region o f sm all kR Is a "forbidden zone" fo r our approximate function W. T herefo re, th e c h a ra c te r o f the s in g u la rity o f the exact H ert2 fu n c tio n cannot be used fo r the purpose o f o b ta in ing the required condition a t p * 0 . For the statem ent of th is co n d itio n we have to c o n sid er the p ro p e rtie s o f the Hertz fu n ctio n fo r la rg e v alues o f kR. I t Is known t h a t f o r la rg e values o f kR the so -c a lle d " r e f le c tio n form ula" may be u sed. This formula gives an approxim ation f o r th e H ertz fu n c tio n In the whole space above th e e a r t h ’s s u rfa c e , where the In c lin a tio n o f the ray to the horizon Is n o t very Bmall. I f the Hertz fu n ctio n Is norm alized as s ta te d above, th e re f le c tio n formula may be w ritten
Is the F resn el c o e f f ic ie n t (7 Is the Incidence angle and cos 7 mz/R lr. o u r c a s e ) . The r e f le c tio n formula I s c e r ta in ly v a lid In the re g io n where th e I n e q u a litie s
(1.13)
where
<1.1*0
(1.15)
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are sa tisfie d . I f | t) | l a l a r g e and I f
TSr “ 7 “ 1 ■
th e n th e F r e s n e l c o e f f i c i e n t f i s clO Be t o u n i t y , an d we have
(1 .1 7 )
When e x p re s s e d I n d im e n s io n le s s c o o r d in a te s p ,C , th e I n e q u a l i t i e s (1 .1 5 ) and (1 .1 6 ), which a r e n e c e s s a ry f o r fo rm u la (1 .1 7 ) to be v a lid , become i « i p « 2 h l p , (i.i® )
i « | « a JpiT- (1-19)
To o b t a i n t h e r e q u ir e d c o n d itio n f o r W a t p - * 0 , we m ust c arry out a double lim itin g pro cess: f i r s t l y | q| — and then p - » 0 . I n th e l i m i t | q | —»Q vthe r i g h t- h a n d s i d e s o f t h e i n e q u a l i t i e s may be d ro p p ed an d we g e t
1 << 1 « p • (1 ‘20)
I f th ese re la tio n s are s a tis f ie d , the H ertz fu n o tio n tends to (1.17) and then W -r2 . (1.81)
In e q u a litie s (1.20) a re v a lid p a r tic u la rly f o r p-eO, i f i > 0 . Hence th e d e s ire d so lu tio n o f (1 .0 8 ) h as to s a t i s f y th e c o n d itio n
|w - 2 | —♦ 0 f o r p - * 0 a n d t > 0 .
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(1 . 22)


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However, s i n e s p ■ 0 1b a s in g u l a r p o i n t o f th e e q u a tio n f o r V, c o n d itio n (1.20) tu m e o u t to be n o t s u f fic ie n t fo r th e unique d eterm in atio n o f th e s o lu tio n . Ve re p la c e I t , th e re f o re , by • more s tr in g e n t c o n d itio n
f o r p-eO and C > 0 , (1.2 3 )
which I s , a s I t w ill be seen l a t e r , a s u f f i c ie n t one. Thus, fo r th e d eterm ination o f th e "atten u a tio n fu n ctio n " V we have th e d i f f e r e n t i a l e q u a tio n (1 . 0 8 ) , th e b o u n d a ry c o n d itio n s (1.10) and (1.23) and the co n d itio n o f fln lte n e e s o f U In the region considered (fo r p > 0 ). To s im p lif y th e d i f f e r e n t i a l e q u a tio n , we make t h e su b
stitu tio n r2 Wa . (1 . 2M
Then th e e q u atio n ta k e s th e form
• 0 (1.25)
The boun d a ry c o n d itio n f o r W1 w i l l be
ow,
The c o n d itio n a t p * 0 becomes
1 JT
(for C -0 ).
-♦0 (fo rp -» 0 ).
(1 .26)
(1-27)
Since p - 0 Is a re g u la r p o in t o f the equation fo r «1 (In d i s t i n c t i o n t o th e e q u a tio n f o r W) c o n d itio n ( 1 .2 7 ) 1« • s u ffic ie n t one.
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S o lv in g ( 1 .2 5 ) by means o f s e p a r a t i o n o f v a r i a b l e s , we e a s ily o b ta in a p a r ti c u la r s o lu tio n w hich s a t i s f i e s th e boundary c o n d itio n (1 . 26) ; namely
V1 - e ' lv 2 P (c o s vC - s i n v f) , ( 1 . 2 8 )
where v i s the param eter o f se p a ra tio n . For r e a l v a lu es o f v th is e x p re ssio n rem ains f i n i t e and s a tis f ie s a ll conditions w ith th e exception of (1 .2 7 ). For complex v a lu es o f v (ex cep t th e c ase v - ± lq ^ ) ex p re ssio n (1 . 28) becomes I n f i n i t e when C-e® and th e r e f o r e , does n o t s a tis fy th e necessary co n d itio n s. I f lq^ th is expression transform s in to th e fo ra
W1
p - c (1 .29)
A cc o rd in g t o ( 1 .1 1 ) and ( 1 . 1 2 ) , we have
$ < arc qx < f , (1 .30)
and, consequently,
Re ( q x ) > 0 ; Re ( iq * ) < 0 . ( 1 .3 1 )
Hence th e r e a l p a rte o f th e c o e f fi c ie n ts o f p and ( In (1 .2 9 ) a re n eg ativ e and exp ressio n (1 .28) a lso s a tis f ie s a l l c onditions w ith th e exception o f (1 .2 7 ). I n o r d e r t o s a t i s f y a l s o th e l a s t c o n d i t i o n ; we c o n s t r u c t a f u n c ti o n w hloh I s a s u p e r p o s i tio n o f s o l u t i o n s o f th e tw o form s (1 . 28) and (1.29)
v l m ^ • " 1V P ^c©* VC - ^ s i n v { ^ f ^>) dv + Ae .
(1 .3 2 )