ILLINOB INSTITUTE OF TECHNOLOGY INTERACTION OF ELECTROMAGNETIC WAVES WITH A PLA.MA SURROUNDING AN INFINTE LY CONDUCTING WEDG BY JAYANTILAL G. SHAH Submitted la Partial Fulfllnst of O R p- r for the Degrm af Doctor of y in Electrical ]gmm iab OX 8at. School of f11m1s a Ib of T. CHICAG ILL Jm, 1962 tk ho low's RL-211 = RL-211

ABSTRACT The problem chosen is "ihe Jmtb mtion of Eleetreumagmetic Wave with a Plasma Surrounding an hfmitely Coduatifng Wj ". With a simple moLl of plasma, representing it as a saar permittivity, presnt only on one side of the wet., this problem becomes esesal a three region problem, with Dirishlet and Jemmama. beimury eudi os on the $ -pLane. The wave equations in the two regions containing non-zere fields are transformed in the v -plane by meano of Kmterwich-Li s. This method has been found effeetive in the Moietie of VW"e eIem wboundary conditions on thw 0 -plnes. Theb ry mlein ariig from the requirement of matching non-zero fl rene the of.1 ~t of one pair of medi, give rise to am Inpa e 4ism6 A gest- all the information of the probWem. The sk te in w Ir than Ne Wair of regions exist In which mm-nero feIC mm rAlIu M to be mh to be described by a system of intepal - 1T pem r m therefore,, is a three regis geeintry, ev o rrms bel an iuflmey conducting Mil Om, In wbhIh the ftlek am emeIdazed zero. The sIea of the Integral ete hue been coneidamed in fur MrI Icve of the -rs ehmw. - lntquL Of fuisns tabulated in Takes I md IV g oo the rIe of the p seauuters,

for which a particular class of solution is onverget. Erita* ooae ditioe for the lowest class, gives the locations of the swre r Iwd this lowest class of solution holds. Cosideration of a three dlinhi-l wave prpagation is expected to yield a richer and mere prastWal variatiem of parameters. ntrodstion of simple dyadic permttivity dMs not nvolve any new technique as long as two dimeasioo problem is oasidered. A simple problem on the aplication of this ntd rreeals it as an extension of method ot inmal s. The rang of the praeter and a1 give all informatio regardng wave propagation.

TABLE OF CONTENTS Page ACKNOWLEDGEMENT.......................... LIST OF TABLES............................. LIST OF ILLUSTEATWIOS........................ CHAPTER I. INTRODUCTUOS......................... IL SETTlINGm UP THE INTGRZAL EQUATION OF OF THE SIMPMIFED PROBLEM...... IlL SOLUTION OF THE INTEGRAL EQUATWaI. IV. INVESTIGATION OF DIFFENENT CLAES OF SOLUTIONS..................... Thle Clam. k* 1) The CIO" k*O The Clasakg Conditlems for IVrWbalk wet t9.- VarIibh V. COMPLET OF AND YAT Il OK TRE B~~IaIIB~w........~~............. VI. IWIUAL 'UATEK WW3 PLAM 4 we-'l1 l~bll I~bS EaStrie~ Cwr Lim 4. VL AN APPUCATUK OF TM USI~l)....... V.II. CONCLUU1 AID) PUm FlSTUEI~ WUK~ISC................~.. Coalesim~rJ Sugget im for Further Work iii V vi vii 1 6 25 47 74 83 of la

TABLE OF CONTENTS (Cont'd.) APPENI )Ix A. PO1 PERTIES OF ThA.WCED ETAL FUICT.......... *.. B. BEHAVIOUR OF Tf, J) AS EITHm || OR Iv BECOMS LAGE.......... C. EVJAATIOn OF INTmERL.5...... D. RELATIONS AMONG THE PAMABMT3BW k kl a............. * * * * * * * Page 113 117 118 128 BIBLIOGRAPH...................... 131 iv

ACKNOWLEIDGEENT The author wishes to exprea his grsitsde to Professor K. M. Sepli for the invaluable help he hl given th i rsearch. He suggested the thei problem ad too a persemal iiret in every stap of its preparation. To Dra. Ra E. Ktlanm sd 1U b et hmA1 of the Raition Laboratory of e Universty of M- Ip e r is q-q for their very valaable a eareaive -i -M d.. U. Special thal are de h'. E. R. W-beA e toi Uleotrkl Enirin Deartmet of the MiO is In- d Ta-mO e h atl s co-advisor at a orieal time n th progr. The rearoA r eoed in this thesis was sn-_Pp -d 13 part by Air Force contrats at the -beratry. V

LET OF TABLES Table Page L Foram of V1v,) and f1(j) for te FC" r........... 72 IL Order of Z1(v,) as Im M - f Xo * Fear (2MO.......00F 73 HI. Forms of F1 and F2 for the Four Cl................ 77 IV. Order of Fl( nj) as m rn - bo for the Four Clam........ 77 vi

LUT OF ILLUSTRATIONS Figure Pago 1. The Geometry of the* problem...................0 *0 00 00 0e 6 2. Contour to Helpzvauatlea of oanthenYz yy+Jti~ plow......~ 39 3. Geometryof the simplifl~d Preblem......9....... 000 **0 99 4. Contour to H*lp Evalwation of ItrieS-mla toI sa - 117 vii

CHAPTER I INTRODUCTION Problems of electromag etic wave pWpngtiii in a two region geometry have been usually solved by expressing the fileld In each of the two regions as a sum of a coipletemet of fmm - a msan atohing the fields at the boundary of the two reglem. 1U b..inlry oonditiems thus give rise to a Dirlohlet, oeUan, or hobtk bandary valu preblem. This method becomes more and more dificult as the ambeIr of regims increases, especially when different kiand of hmry oodt sare enoala red; but seems to be moy diffioWt wI thw- met' GI Crbega fiaatftas- for tb s f IeI n *es e I bwrIs incomplete. The problem of c~fhastie by a wed, inin~1y a t regiem problem, has been t8lsd by eerII rs a a 3 bmm*ry vabu problem by gmsdering *we to be of M eeIieg material. Seaior L h~ * Ie I.w Y th" A1 V -IIr l a tiom by a sml- inla mtdi fc ee onduativf i 4 e meutal is tUaws taoo " ike to a c ge in the The numbered s reb in b."iU "*I-tbe at 1

2 impedawc at the surface. The prepod rte effbot of Os Impdano at two boundaries of a transLision line is very wUlknown In trbmam aims line theory. Senior [16 ha also slvod th problem of diffraction by an imperfectly oontl ng w.n., by repre enting the suracse impsdanee a a boundary iondlt in and ha shown that his solution coincides with that of Sommerfeld for an infinitely osetn we~e. Feien [i7] has considered a w.~e and a oone with a iMnearly vry rime impdane. The solutions of Somra feld and Saicr one the metbod Of funotlens of complex variable and ar expree as interal T or a complex GrXameter. Felsein'a soluAti a "obtined in terms of JGPORIMsIs f using orthogonal sets of Ifutes. 7%* author was h — InroatdJg I reading the article of Kontorowlekh an IPbev who have sowan t the transform which they iItrodn their rtisle "es the 0asitien of the problem of diffraion by a half pim to be. of the two forms - a Sommerfeld tal er a I eYI nl ~mehe. Baker and Co 's [is] km r a ~ t th by a black half-plane an a sam of twV pstW. r ee the effc0 t accor to thlaws of geometrigal e adm * ether being a orrlem term wvehis hts accout of the di~mntiea. 'Pe difration tern! santts of 'out' BmLU~ fUS~tir5D ~ which leek term ielving gative pmWse i.WW

I the argument. These 'cut' Hankel functions oannet be formed from normal orthogonal functions. Both the part of ti lio have n_- _t-, S across the geometrical hadow the totl tle ia iss, hwver, rotinuous - Whipple [19] has observed similar phom a ia his wek rati by a wedge. The problem is set up as a three region boundary value problem in Chapter IL by takinf a simplified model of plasma representing it as a scalar permittivity. This leads to the singular integral eqaen (1.85). The integral equation (2. 85) holds for ay suitable three r sometry in the shape of wedges and an electric line -re excttio (2. * it has been set up from the omidewation of oad ef the regioms einsmg.1 plasma, describable by -m s d a -ear p{ m ttt (2.5). the media must be isotropic a"d lbar, p o - beg M s by dam et sealar parameters. The non-lianmaw ef the plasm rgs { en bwee considered negligible dee to low ief w - J -, -tAoe pw be present at least me of th, -the tMl at the surm- ef whic muwt be bu. The s tak e m e tng regio order to the t ral eq. two regions may hare1 r "m b pmrmeters desserl by f(. and (7. S). The excitation sroo is iit io di eamr medium, so that the -wa

4 originate in the denser medium and propaate into the rarer medium. If there are more than one pair of regions, at e surface of oa t of whie non-zero fields are required t e h e atced t tieoz h to be de cribed in terms of a system of sular inteal ea rather thl a single oe. The geometry has been further ese ea g to Ifiity in both the p uad Z directions, with no variation of either the parame ers or the *itate in the Z-direction, in order to redee the problem to a two diaemal one. The only approximate solution of the problem of a three regio b ry value problem in the shape of wedges tt te author a en, isthat by Karp and Sollfrey [2 of New York Uniersity. Besides their olutio being a good approxiatio Sfor the case in whLc the difference In the dielectric constants of the two media in o t i ery small, the geestry which they have considered has furher rFritiems of 0 beig equl to and of the source of waves being in the rer amedim. The solution of the i (2.) hs be- - In class h(l) in Chtr BJ by re Ieing Ito to equivalet Frde equation of the seeond type. Th remaining three clases me di"sussed i Csr IV. Chapter V aims at s -uriui sing and making qsu 1prists eI vati on the investigations of Che n m and IV d th teti e solution to the problem set up In C -AIL 3 cVIr VI t-e into consideration the changoes ntredeced i- the lsqra- l equation (. 88) de to

5 simple anisotropy of medium 2. A simple problem as bee worked out in Chapter VII to indicate the physical slafiMafe f thie method, fro the points of view of images and wave prpatltie. The raFe of the parameters 9k, a ad al for the four claes of solto give all the information about the wave propagatio. Comolusins of Chapter vmI show the effectiovee of the Amthd ued in the iaalMi. Sugteti for further work ae also oir in the same a-.

CHAPTER II SETTING UP THE INTEGRAL EQUATK)DN OF THE SIMPLIFIED PROBLIM The problem hin is to study the 1*rftion of eLs*1atzaapt Waves with a plasma surrrnmdizu an Infinitely iwne. ~rcp Fcgure 1. Tb gat17 d the pib7Lm. The w* e Is tis baying a igI NW Ireu~ byr amtrm plama as shown. Th sybm is m I tem to A- y in both the a and p direottin... The nW e1 So~ The axmes wtgkith $til- that = 0 bsts aiw St te w, s*e p0 a-rm es out at right ofe r Md kim o II.. Of th wege. The plasmte l*d infw is bs rpm~u so tha thep - etes representing thr phin d ot ly w the coordinates. No.1). C. sppLdpI rd~ u 1mM he ain O th do Wpm on of tho penm am be f wg a a" sad not a *'adlo am&. 'Th lesity 6

7 of e1.&vromagnstic waves is tknlow so tbA the ncm-4laear terms.U-V and x.3 In the eqtiua of motion of thnartieles o00Rnstitug3 the piasma carn be neglected. The 'weige Is assmewd to be made of a perfectly mIrguerle so thtthe fieIub ~msbdb w~emrzero(O* < 0< ). An elcrcUn i.mew. te to iniity aud having no var~iatlrs I ea4wciinissumd j~1 -~(2.1) p whre J in the stremgh of the elatoerocsd 2ir - 02 > O 0 (2.2) so that the euciting sooe la isoterge fwn The holds In the v y h i~ by a subseript 1, (2'- 2 ~:0 ~A1),O ON" aJ p~aem as atr of thewo by a subscript 2, (01i > 0) m ithese in he plmmumie s at the WA of goeweby as pu 3, (Or-~~ plsaWilli be mu ob presuit only m,* r*M ofthe 2 to Awid-* possibility ofeltgrle n Insead~a i~am. The si nIof p Osam be m as aWV' ess a seelar peritivtyr- gh -

8 wl2 1- P, r 1a (p = (2.3) P 2m 0 gives the pglsma ree frequency, u s th fre jmeiy of the exciting source, N is the volume daslty ef elestrems, e is the elneabromie charg, m Is the electronic mass, co i the permiM of V um, mE v is the effetive collision freqimeey. Since the medium and w MTnrem h so w Var$* in lt Z-direction, the preblem reduces to a twe diemslmai NW, an &Z =0 (2.5) If VI is taka. to qU2 e 3-9 o e e K1 the problem can be formulated In terms o ) - ui eii to be sartsfied by the f GUnti Ine tV e WSO, ml he u d a follews: 2 1 _ _ _ _ 2p 1k~ 1 OP P =P h+0. (2. 7) 2 2 (2.8) 2w (0jh0

9 k=W 2 c (2. 9) 2 P 0 in which a time dependence of e is assumed. The boundary conditions oni (i = 1, 2) are, 1=o ( 2r-l o) (2.10) 2 = (t 3,) (2.11) p1 2= (02 I 1) (2.12) a*1 = 8a '2 (. = 1) (2.13) Both qi1 and O 2 am required to to zero at p -0 beoause of the presence of the oaucter at p 0. Multiplyig (2. ) and (2. 7) thro by r tm to a name aitable form to whiob theo twtk LKb trmfrm ]m be a -1d. 0 80*+ a pl + a O + ') St 3) (2.14) MultIpl~ B.14) thruIh by. d daqtogat 0 to give.s, +Sp + IP+...+'.+... ] '*U Go p' ), -,- (2.16)

10 Integrating by parts, p p a2 Hj(klp) dp = p Hv (klp) 8ap o 0 - ap l [Ha V (kip)+P ap Hv (kip)] dp (2.17) o + O [ k+ap H(k1P) d (218) ^ p 1 H((kip) + p a H (kip) + + =f j2H a(kip) +P a Hp 8 kip (k)) ] (.9 + HV(kip) + p ap Hv (klp)) d (2.18) Substitut (2. 18) into (. 17) gives. 2 I~) a Pq1 H (klp) dp | IH (kl, ) i)l - P ip, k, -;, H, (k) + 0 0 J['JI

11 00 2 Ht(kp)2 a8 j, -dp =8 0 v -dp. p 0 (2. 21) Define the transform of q,1 (, P) a 00 ~ H (2(k~p) h v.00 5 ko "' "p dp 0 Substitutin (2. 22) into (2. 2 1) gives., (2. 22) 0 82 v ~dp = a2 ~p p (2. 23) Substitutin (2. 19)., (2. 20) and (2. 23) into teL. H.S. of (2. 18) gives, 00 [p a34ii + Poao ~+a / ~i+ kP2 2 H (2 klp) p (3) (2)1 =LHv (k 1p) 8pj'I - po' H~ Jl(kip)J 00 0 + +2( 2 12) J 1 Hvk p k Rv kp 0 - d*+a2 4+ W (2. 24) Inmthe bracketsanSeR. H. S. of (2. 24), Hs)(k p) -pP -+ a, (2.25) a H (k~p) -+ - pp [Ali +tj (2.26)

12 Thus, pH (klp) a pl - Plp 8p H P) (k e -jklp ep 1 + j klp0 (2.27) if it is assumed that i1 satisfies the Sommerfeld's radiation condition. In a similar way the brackets may be assumed to tend to zero as p -. 0 for a suitable behavior of the function 1' The brackets inside the integral on the R. H. S. of (2.24) is simplified by using, (2) 2 (2) v2 (2) op a H (kip) + k( p H(k1p) = - H (kp) (2.28) pP V p Using the definition of transform of 1 (2.22) and (2.27) and (2.28) simplifies the R. H.. of (2.24) to, (o a) S LP l+pp+%1+k12p2] i ( dp z 0 = 1a2 1 + v2 (2.29) The istegral on the R. H. I. of (2.16) evaluas to, oo HPP J HO)kk u ) J.Jo P J (p-p,) -) D zDi(-P,)) (2.30) 0 P

13 where D = Juo J HO (Ik1p) (2.31) Substituting (2.29) and (2. 30) into (2. 1)) gives the differential equation for i1 d 1 (2.32) — 2- + v2 1 = D ( -Os) (2.32) d 2 For solution of the inhomogeneoum equation (2. 32), the procedure indicated by Collin [2] in followed. Two indpndnt olutiona, i11 = cos v p (2.33) and 12 sinv 0 (2.34) of the homogeneous equation obtained from (2. ) by ptting the R. H. S. equal to zero are chosen. The Wronskian of these two futiom is equal to, w = 11 12 - 211 v (2.35) The particolar Itgral of tie i _-aeejmeus dfbrential e -jtlon is then given by, 11 do 11 J 12. d (2.36) lp 12 W 11 where f is the forcin funotin, f = D< ( -,. (. 2.37)

14 Substitutin (2. 3 3) through (2. 35) and (2. 3 7) into (2. 36) aadcoosing appropriate lower limits gives for the particular itgral, ilp v1 and YO comyoa -Cos, v 0 sin v (O > 6p I sin v 0 cog v O's - coo v 0 sin v 05(O < ) (2. 38) (2.39) The complete solution of (2. 32) is given by, =I A1 co V + B1 minv~ ~sn ~-~ the positive sign bein used fore0> 00 and the negatiw Ib f Using the boundary condition (2. 10) ino(2.40)0, Al coosv(2 - 00) + Bsin v(2 - 0) + -a in v(2 - 00 - 0) zO V (2. 40) (2.41) Cos v(2,r - 00 D fin v(2rf- 0) min Y(2ir-* -0) sin v (2i 0) (2.42) Sustttig(2. 42) lae (2. 44~ giveso min VW (2w.0)sav so f -0-8 + - sin v ( - ). Takin the invrere trmsoru [JOf Vq ( v, O)o gives (2.43)

15 (p,)-(. ) J 1(,0)J^(k f) u (2.44) or j a 1(p, 0) 4 ( s) e-(,nar H (k' ) d (2.45) -J a The two forms of inversion are written dwn, sinoe it will help to visualize the poles of the integrand better, and thus the behavior of the function. The two forms also lead to different forms of reprueenttia of the funation. In (2.43) if the function A (v) is knwn, the inversion (2. 44) or (2.45) will give the field functiona in vacuum. Use the beadary conditions (2.12) and (2.13), leads to an Interal equation for a similar funtion A2 (v) appearing in the expression for the field funtion 2 ia th plasma region. It t the purpe of this hapter to set p this Integ equation. The following quantities required in (2. 12) and (2. 13) are oailated below. A sinY D sin..1 stv<2-.-.) _ _ _ _ + (v,* a v(l ) - D<2)-^ ) t sin.v(2-) in v(2 - ) + D5 si(^- ) (2.41) where v 2 -0 ^ (2.47)

16 is the SOgla serocpebyvacuum regim. aWk I-A I sin Y(2ir D oos vo min(2r- - ~) du v(2w - 0) 0 + D cosy(o - ) (2.48) aW 1'Al AV Cosyvov ~ ~slayv(2s - ) D coosv 1sv(2s-o..~' sisav (2 - V - Dcosiv(O 5 The functlon 0' in reg-ion 2, obeys tbo eqsls (2.49) 2 2 &022l~P12 =0 (2.50) Mul -ym (2.5 r by -J -- -- mdi —& I- from 0 to m, and follow Sin siia ruet i~g(2. 19) r I - (2. 29),q gives the eqm es "w-2 2 d +/22 Wa (3. 51) where 00 (a) *2 (Ps, 11) my 04 do 0 (3.52)

17 General solution of (2. 51) is, 72 A2 cov v) + B2 sin v. The boundary condition ona from (2. 11) is, 2 = (o = A2 oA v o + B2 in v 0 con v B2 A2 sin v 0o _ A2 uin v ( -Po) *//2 (v, ~ ) = - rs v 00 Takl the nverse tranokrm of / g1ves, - a 02(pt,,) 1 i,dA -jc (2. 53) (2. 54) (2.55) (2.56) (2.57) (2.58) nw erdr to mmW m d i X os (ll 12) and (2.13),jht rMeI d o *1 mi lt m,d thie i l,rmvsr at =, - 1 it is ear to obtain re ntioms be.te I- 1w 5ad *1 respe to te saue wav nnreber.

18 From the boundary condition. ip, (P.. 0) 0 =01 = ~'2 (p 0) (2.12),4) (khp) Multiplying both side by - - and nitegr-t from 0 to oo, P gives, o 0 H() (klP) p dp p 1a,1 3 /2 i(Pt -) -. (2. 60 p =1 0 The L. H. S. of (2. 0) is 1 (v, ) " dfed by (2. 22), ad subtuti for k2 (p, ) in the R. S. of (2. 0) from (2. S) gfve, 01(v s ) 1 2 o H (m) (kip) 0 Jco -J. i (2.61) ie, -' the order of klqrate mr p mi M, ^1,3,1 *(Vo 0)I W(v zl *1 - 1 2 1 2 -J 0 SA s*02 do)i0 1do (2.1 -1l 92) 63)

19 where OD Ii p (2.64) 0 11can be b u JAo WoIegas ODi k2p) JV(k~p) dp- j 0 % kp)Y kp dp I2~- iI3 (2.66) where =2 ~ I do and 3OD J(k2p) YV(k p)dp 0 The value of I2 s takes from Watson, (2.66) (2.67) '2= k~(74:, 2 F1 2 2 [0< k2 < k1; Re(pA + + 1 )> -1 (.6 (2.68)

20 htorder to eyahiat.1, use Inmadesofanbr forula la Wato. which reads, 0A2P 2V2 j 2 2 1 iNr (i)(k ) 13 L1l> k2 > 00. 1I.G)> IRe(v).2](89 To compare wfth the integrals 12 m 3r~rd in thb deeop1 etmimt of I inataken ansr -0. 3 1i 3 o 1 1; Y i!; 4 r 40 2 2 1 Urn ii )(2.70) Nlew, JlA m Li.2r NOlk0 r-+ 0 + 1)(p + k) 5Wnttubg (2. 71) laLL.H.S. of (2. 74*ghoso

21 Cg-ct ($1 - v ) T 2 sukwS~tttuti~ (2. 72) ifteiii~~~~~iiiii~~~~ (2. 65) gib s* - 1 (A V) Ij Jo J 2 'l = [2 1+i coo V) I X fia (jA - v ) 1r substititlr the value of 12 from (2. U) I **S (2. 73) gtv",, (2.72) (2. 73) je 2 2r f 22C 11 JA + v 2(~ (i. g. &. _ 7W+ 1 x x 2 1 2 0 2 pC+: 2 asp > a* yl (2. 74) substittt~ the vrrlues of 4/1 ( v, 0 ) M)d rtl 0 0)~ and (2. 57), into (2. 63) givwow fromr (2. 46) Al sin v OV *inv (2s - 06) AV04w i vOia n (W -Os DItvMwnvmr mC~ v fftv(ft. D + - slay th. 01) IF = where r A2 (A )p GiR A 110 r ILdo - (2. 75) (2.76)

22 in th. angular sector occupied by the plasma region on the right of the concdcting wedge. In the boundary condition, aIso P1( ) f 1 = a02 (Pi,) 0 (2.13) the expression (2.58) for 1ii2(ps ) is subetltutd on the R. H. S. giving, j OD 8ip(Ps )Jt =8 -4 5 (2.77) Mltiplying beth sides of (2. 77) by V and iltegratlag fro 0 to W gives, a dp [i t(kip) 5 0 -j. SUbstitting (2. 21) on the L. H. 8. =d keft a ngthe ordr of Lutgrat m mr And 31 In the ALH.5. of(2. 7S) gtve.

23 a i. (v, 0) 0= o 3 1 2 -J co 0 JM(k2P2)H (k~p) d p ~~ (2. 79) From (2. 57), a0/ (,0) 0 z 1 A2(y) vomm v -a.P min v 0 (2.80) Substitutin (2. 4)) and (2. 8* and (2.64) Into (2. 79) gives, miny (2uf- 00) Doomsv W iv (2vr -0-. ) 513 iv (2. - ) - Domy(h 0) 3 j So A2A20 4s M 1d (2.81) From (S. M Al1 D siny 01 stnv (sr - O's - N) v stav ov Dmfbvtj.1 fyWO + +Ks? Aol (~p si p ait.k 'ido. (.2 (2.82)

24 sin v(2s 0 sin iv(2w - 0 - Dsuw(og - 1) cotA41+ I~ icOD SjO A2G*I)MA Vslnbs cot4yo SiX A0 I11dp, (2.83) EquwAg the values of IVobta 6) in-ed from (2. 81) and (2. 83) gives the integral eqiaon fer A2 (es) S O -jOD A2 (P) F2oivsde4]4 sin v(2v-x 0) This equation can be fuahe spiled to, j0D S A2()A) IIF 2 co i AVWJ O vd -co2s aim*00I oMI,5fVvMsovPJ4 (2.84) -2D sin y(2r.0.0 ). 0 a (2.85)

CHAPTER m SOLUTION OF THE INTEGRAL EQUATION In thU chapter, the iateral equation otaied in (2.85) will be solved. The integral _qrite for A2 ) il repetod here with all the neessary details. J A2 ) 22 = "Il 1[ t slMpini~ +.rii/eeIv d~ = i 2JA = -2D sin v (2r - 0o - ) (3.1) jlp= i - 0o (3.2) v 2- - 0o 1 (3.3) D = jwpo JH1 (2) k1p(3.4) - -sin J - r-v 1 2s9in -V)4 *< r{+l, x 2F1( 2 j-i;+l; rA) [B~> sv] (3.5) The itegral eq In (3. 1) i a stIar Interal i b of the ple at A= v (3.6) in I~ noemr of -'I A Norpalo f thI l i an Othe 25

26 contour of integrtion. This type of sigular integral equation is susly solved by reduing it to sn equivdaist Fre*ei eWtin. But before reducine it, it Is necsary to pat (3. 1) in" the &sa rd singular aernter form. For this purpose, (3. 1) Is rewrittnu as hi11ews, where 2_ )A+ V 1 ' J( 2 2FI (~'2 k2 1 [ ~)F,+l') k) pco 4~~n+V sa es#A H(v, )= (39) 2 sin Moe f (v) = 2D sin v (2i1 - AD - 08) (3.10) In erO te mmuss (5. 7) in 4 sas ainm ed by Mmmkhellalvl 1 the Miewing trmrgms m ed ~o ct (3. 11) and -ji r e (3.12) In T r(3.13) V=L in'to (3.14) If

27 2 2 1 2 If J*+ v = Ia turt-c) Jr 0 I (I.) 9- -I f2(7 -r 'rS (3.15) (3. 16)' (3.17) (3.18) (3. 19) d 7' O-jew -ej-vw MFultiplying the umesrato and d~n h. et PR. R. S. ( (3. 19) by ii 0 gives -if. -1(, )r Nj r-r I doe 5131 (M -vOrv 0 -e CD T(C,T')WV iT) 42 uW-,~ A f)*- f -z- -f(* 2 7 e -or (3. 21) where T(v * ) — + I i 3,t. T(rZ )- k2I 2 1 kn(*rr-) 0 -1 fito, - +I 6, (O x Ir 2 2 1 \2u * 2w Vo r ' (3. 22)

28 sinh2(3.23) 0hC ohP sn _+(a)o ~ V 15n 7 __ _ _ (3.23) an IsubstitutingO)six (3.21),gies D0)IAJ ( Ds (3.2! JrJ Now in ordr klm fs 3m~ w- y n(.2) rs rw tranformtlai ae u3sed; 1 t t 1 * (S. 28) 1to d'r -2(3.30) (1-t)

29 dTr -0 (3. 31) (t -t0 )(1 -t) Substituting (3. 28), (S. 29) nd(3. 3 1) into (3. 2 1) tkrwugh (3. 24) and (3. 27.), gives, 1 A (t) 2 2 If (1 -t 0) (t -t) )(1 -t) T (too Q MKtox Q dt z - ff" (t ) 0 (3. 32) k2I H(?C Z~) -4 H(te t) u o 0 s t 2n '31 14I (14f n 1 - t (I s' S If 1-t 2 kI2 1 (3. 33) 1 [f-ta ~-t L~t 2 L 1-t f'(zV) 4f'(t ) -4 1677] aE4aTvij (3.34) IAr - 00. og a 2i Do ) mu& 1% --- -) In r 4] (3.35) sa" I**~it) -jkp comhnit 0 ( 1 -t ) df. (3. 36)

30 The qutl ca (3. 32) can mwbe put Wea & atmadoetrfom irJff(t) (3.37) 0 t -t 2j (1-t)0 K~t GO TOt) w t).(3. 38) (1-) * The frIIowiug mUe is usmd byMuw M[] fa'rw ea a 8iiD4U1S 1pr la tdf tol fb-t kin to an qisa Fim qatmi. BOt ) K(t,at) =2j T(tw t 0) HOOt to) (3. 39) 0 0 02~ 0 k', 0 2~n 1 1nmu - t ot H(t0 ) t — 0-mha(3.41) tt 3 4 t ) - _ _ _ _ _ _ _ _ _ I n (3.42)

31 t,t) - K(t,t ) lt 0,t) t- t~) 0 In terms of B(to) and k(t. t), n (3. 37) an be ressed as, Bdt ) tA (t) 1f 0 0 EquatIe (3.44) is m ro-w d to an 1l- Frdhim equattin. By the equivalene df the Fiw Sdi ar e - the raduond Frdholm equaton is mat thalt th s=etloms bel ae tl do dam h amf being soat t[]. For a singular its l quatin of the first tye i- by (3.37) r equivalently by (3. 44), all tb *sisf - nm f Isp1- aA- a -qal ends. Thus the tottil number ofm —ta * "a a quto is, 2p = 2. (3.45) Since the t (M. ) an (L t) W an a1 singularity at t = 1 in th - tt Is d t - se e be — - to a class h (1) that will be bMId at t = 1. Far go dass, q = 1. (3.46) THMa the index of the problem is, p -q -)X 0 (3.47) The fimdmma funmte. of the oass h (1) emr s1 *l f t 43.44) is iveM by

32 0fF Z(t) =B(t) Define aeratewa (3.46) 1 IL 0 Ir1t Tj a Ft- I t- *)15C (3. 4*),0t) 0t) dt (3.50) Then lb. Fr~delem qun -- m a to (3.44)15s given by, A 2(t0) + K*kA 2 = -K* f' (3.51) In (3. 5 1), h fa1 - K*f' sa" perwate K*RMM kA are WM - as f ollows: - K*f =fAt )= — 0t 0 ~a [!a9~1dt ft t-I Num (3.52) vWere, 9 2w -0_ (3.53) The Intompt4 wal taI4A 4 a (3. 54)

33 appearing in (3. 52) is more easily evaluated in& the MA - ape". D(t) -4 D(M.) zj4JwO H(3)(k p) 7 1t 7 1-t 7 0 1 i;A J si [ 22 n t -*sin M( -3r k In k, k1 (3. 55) (3. 56) (3. 57) (3.58) (3. 59) (3.60) I 1-t 1 t - to -+ (1-t)dAL (1-t ) d -t4 e Sk -4 T T I + e'0';V gs~1 (3.5U) Ahr. (3.62) lot(. (50 giws, (3.61) (3. 62)

34 2wnko6J e I = 1..t HpM (k1ps) sin IA9 a A 0. Mu( 1"slp(2v- 20.) Wla jAM )w (13b. (3.63) i 2 I =A e '5 where 2wwM J 1 -to (3.44") (3.465) and '54= H,~ (kips)hminl 5lU of jAg@ j4 (-#' Sla,s (2W- 20) a&40~ -V~w(le60 - MI) (3.66) Using H()(k~p8) &a ~ + f p (3.67) the integral 15 can be beika p We tw 7~ 15=18~S+1'7 (3.68) where, ~.=isaft * A I&is%) mi~co E(-.fi M-Mk.- ) gbSI.vfi+i r dMA (3.68)

3 l, and '7= S o)4'A (3. 70) The integral 15 is evaluated in Apon adlx C. Substitutin (3. 54)s, (3. 64) and (3. 65) IMOt (3. 52) givesp T.V 1t) 5 (3. 71) This funotiea can be in -terms of th vaiable tv, or =are cwu'vemIbently in terms of the variabl v by means of transformiations (3.12) and 43,219). Thus, It is obsorvle4 * Sere is a Psie In Ev) atv 0,9 bt thsCarl 1)0 avoided by using *e lbs ef 1qra1 a ldu-e at et, shw In Appendix C. lb kVatem br ad 02 give by 43. 4* and (2. 53), lhwevro do not held br v 0, m-d dde point may be made a siglrpoint (d the problem. Meeyr v uNpmeraly Iessno variation In the 0 diovnmWka and wth the oefr et preblem, no va~alnin the 0 domain 1t, *ot adissible.

36 Now the operator K* k A2 is evaluated. 1 K*k A2 = i~ N(t0,t) A2(t)(it (3.73) 1rj Q where., N(t03t) = 5o fI k (t1,t) (3.74) k~t1t) =K(t1g t) - K(t1,f t1)(.5 t - t K(t13t1) ~B(tj) (3. 76) 2j (1 -tl) 1-t Substitiatl (3. 76) md (3. 77) 1." (3. 76) gives, ~ (3. 7*) Joe (3. 74) gives, N~t0, t)= 1 3. 79) where

37 1 4t7 [J(l1-tlp T~t1, t H(tj, t) - (1 -t)B(tp] (3.80) This 1i~.groa can be mere eaily evaiustod in the ri pis". by definin transformatlons connecting ti to Ti, siiar to those emastla t and t0 to;i and ve respectively. tI -jyR 1-ti (3. 81) ______ (i-t 1) 1j-t ( -0 1 0 1 1 (i-t0) - dn (3. 82) 2 1 -+ 1 - t -tl (i+ e -J r)( +e On) -j -jT~v (3.83) j (l-t0) j(i-t0) -4 (1 + 'Ci) j(1-t0) e (1 +etir (3.84) 2j (I - t1)T(t13t)O H(t3 2 k1 Pt) 4 2j (3.85) (A~ ) k 2 JA + r) go n + le. 2 ) sk. 2 F1 ( 2 P; #A 0 3 '(1 -?7 ) (u + 1) kl 2 (3. 86)

38 Hi(fl,;A ) = la2 Csp ami TI +;Mfl si4p Co 7o 2 s;A 0s (3.8IT) (I - O B(td 1 1 k27 ~~ BWi = - I 1-e 2 k ala (2r- 200) ala n)00 (3.58) Substltutla (3. 82) tkrmw* (3. 86) We. (3.8SO glvesa, * IV z e 18 7 jOD -un r (*ib 3l)1(10 K -jW +1Idr (3.80) su-bstitut~n (3. 58) is" (3. 75), glvs,9 e 1 where 19 = (F + F2) dn -jOD (3.9* (3.91)

39 F1 = F 2= (i + -Jpw) T~r~jjB(Tns) (3. 92) 2(e -JAlO-". sa n )1 2 (3. 93) In order to.valimt (01, I, the following @- - rIs tsa 'I. Vn=n~r+jRt (0, (2t, -R) n7ur jAt Flgur 2. Co-.stoi -9 evaluation Of 19 ea th. 17 =TIr+ jnil plane Set up a coatur I"-a gra, 110 = S(F + F2)d?7 C (3.94) where C is the oo —.A ur shown in the Figure 2.

40 On the horizontal portions of C, r = + j R (3. 96) where Or is the real part of q, and the positive and negative signs refer to the top and bottom portions of C -plvel. Consider the funaions F1 and F2 as R -+ c. Using the asymptotic behavior of T(rL, ) given by (B-2), gives, C1 e l(,r, + jR ) ( -r + J)e -R(2r-2+io + + -- -2 + ) = Oe 2 asR - (3.96) where, 2 (1+0-^)1-eq) p- (1+e- e 2 e 2 *g in; k2 (3.97) X being some - of r ad |k; i aand are inflm (A-5)4 and, ek = Arg. - (3.98) k2

41 R( +, + + k) C2 0 (J r -R) e- ~ 3w -R(2v-2o+ 2 -^- 2' - +e 1) 0 e 2 R -2 a (3.99) where, C2 z Cle (3.100) Although in (A-5), positive u well as negative valm of 9 are admitted, the geometr of the problem reairee bet a a d ek to be positive. This was taken into cousideration wlle obtaining the aymptti c behavior of F1 in (3.96) aud (3. 9). Th1 F1 will tend to aero em the tpr- m portiom of C, au R -+ oD, provided, 3e a +- + 1 > 20+r S (3.101) and 71r i fiite. The elaes of soluttiom i-nal in this adca r i valid only r the geomety r wh (. IS) blds.

42 F2 0 - - 3tjt. j(n j(3.10) which tendi to rro u R t to to a. Thus for, o0< n< 2 (3. 1o) the integrand of I10 vanVihe on the hori a port-em of C, whA (3. 101) holds, and I10 can be evaluated for only the vertial prtu of C. On the left vertical side of C, r7=0+j i1 (3.104) where Ui is the imaginary part of r7. On the right vertical aide of C, rn = 2 + j ri. (3.106) Then, takit a ee t that Ft mad r te to ero R Ite to infinity on the horirl portieon of C, g em boe wrt den as,

43 1lo 5 [1+ F2| d1 + 1 +F2j di -jD r7 O+ji 72+j = 2j [Sum of residues witbin C106) The numbers of peles of F1 within C, dw to theotor of sin r2v- 24o) i t dsh minato r will vary with 0s. For t ss of cooaretenoss, let, I > o > - (3. 107) 2 4 in which case, ths number of poles of F1 due t the te ftor will be 2. The number of poles of F1 wkt l C au tlsM, rl = 1 (3. 10) r = v (3. 106) 7 = n ( = 1, 2) (s. 110) where 17z= a. 111) 2r - 20o The pole of F2 t C is, l =v. (s. 11l2)

All these poles are simple poles., and the residas at these poles are as follows: j T(1, AA) Hl, JA) R= - I CO-B(TV1) 2 (3. 113) 1 F2(1i+ ')Ttfr) nfr) I (. -oi. e-v) (1+ ~ ~Irv) ( +j.-VI (3. 114) Thefirstit-egraloua TVth.L. H.5. of(31sin.pltI k, adi odrToeaatthe secotdinteg ral an the L. H. S. of (3. 10(e),emdt ~1 sa" Fa are evaluated at (3. 105). 115) F1 I1 " - *5 -j)WnrSa (1+e )( e ) 8*-ri.V)L"B~n+ 2) 2 (3. 116) F i 2+2 2(e-JS ~eW9r) sin(rl-v) L 2 21 (3. 117)

45 Substituting (3. 113) through (3. 117) inte (3. 10) give, g =2-j [RI + +R + Q(vp+) = 1,2) (3.118) where jco Q1 (,) F21] F (3.119) Substltuting (3. 11) (3.9d),giv -Jeo N(t0, t) = 4 - (1 ---t) L' J1 + +a W 1 (3.120) In terms of N(t, t) ad f(to) giva by (3. 13 m- (3. T1) rFzp- vly, the Fredholm equatio to pt ortghsal -a Mr ea- (3.1), ia which a class of solutio at re am bem-dl, -lboet at _ _1 —.-_ (3.101), is sought, is give by, 1 A2(t) + - J N(tot) A(t) dt = f(to) (3.121) 2t0 ~) J 0 2 The solution of i equation is givn by, [J. oo 1 A2to) ft - I ' [ltet t) I 18{..122)

48 N1(t0,t) = N(t0. t) (3. 123) 1 N3(yt) = 5N1(tO,,t1) Na (t OA (3.124 a is mre ocw te o we~m the fr.g) variabLes imm.ad of (ty t) variables by m Oan f transferm lm (3. 11), (3. 11). (3. 2S) an (3. 29). The variable of litgaist1 may a&so be.1rWs in terms of th. variable run by means of tramsierm xiesila to (3. U~. Them, (3.122)eytsd in terms Of6 (, )A) Variables r A ()fv-21 N6,js P f(Us~ 4o (3.125) where 1(v) is gtvea by (3. 7S), and (3. 126) NnNJ 'to ) Ni(L du% (3. 127)

CHAPTER IV INVENTIGATION OF DIFFERENT CLAE8 OF SOLUT)IO There are in general four laees of seltioas to the migular integral equation (2. 8). hI ChapL r Im the elav of seatita reualan bounded, subject to the ooadtlo, (3. 101), ha bo ba-i-ed. Tht os of solution has been obtained as o la (l) -pta-al te (3.4 ) that is boaad at the non-special end t = 1, and having an inax givre by (3.47). 4The eer three olasses of eltl is are, a) hO, 1) which is obtained by aoliderA cmg ta*t s wi bouided at beth t 0 IO t = 1; b) (0) which remain b-d at t = 0; c) ho wkic is not i_-eeearily r ire to be eaded t either t = 0 or t = 1; this is the hieat olass of solutions. The olam M s, 1) is the lowest lass of solutions; its ldex is equal to ). = p- - (4.1) sinae i the i olaaa, q=2 =4 2} and p is given by (3.). 47

48 The fundameatal function of lass h(O, 1) is, z(t) = CO, t t-iT B(t). (4.3) The operlors similar to these deftind in (3. 4) and (3. 5) are now defined for this class as filow. -,,. (o) ( - ) -.. (4.4) 1 I(t) ( J 0o Jt-it (t-to) B ) 1 k0 w - \J k(to t) p(t). (4.5) where Bt) and k(t, t) are defined by (3. 42) and (3.43) reeotively. The Fredholm equation, giving a solution belbelng to the e — h(O, 1), equivalent to the aular integral equation (3.44) Is gives by, A2(to) + K k A2 =- K If' (4.6) The existence of this class of solution is subject to the n-eoery sad sufficient condition [5], 1 - r (t) dt ) —. 'In= (L 7) 0 ft f1 BDt)

49 Transformin Iii to t;Aj UP",, bY then u~s. of (3. 11) and (3. 28),p givesp Sw ilz 2j woo i 2 k A do (4.8) Expressin Iu as a "am Of twe I4qa& s II= 2ww 0 112 + 113] (4.9) where, 1123 S j 113 S eJMsison 4Gh I SJAJk1, 14Sin Pe~j *1U14. (kips) ~ jfl$I* mm ( —m) dtA k2 (4.10) (4.11) MaAgin WeOM sr sn A(2u~V -2 tve~e a ad8 ~ 3 by the e* of residues tasbg do~.r SimIar to theee is - p i C,9 givens, I12 7- 3 iWj + + 12 k-i ~3 2 + i - + 0E8 k=l (4.12) (4.13)

50 where, (12) 2kfo k (12) (-1.k ^in Pk dsin k2 t (kI) )1 'k Pk kr a ^ ^2 k (13) (-1) sin 2k J2k(kp,) k^ -2k 2 -2k:2kr oa an $k (4.14) (4.15) (4.16) R(13) a -~k k (-1) oia e'k *t Oi JAk0 Oklkjps) k2 k I k vsia^T;sjoo hky~iL G 1 (4.17) where k s dfian dby (C. 13). R(12) - 2k (-1) siak-l) J l(p) k k- 1 2i r (2k- 1)X2 coo (2k- 1) ~ + ~ocot(2k-1)0 + - j-l ikl ) c+^ 2241 k-1) fo -(2,- 2o) cot 2(21 (4.18)

51 (-1)k sin2k-1) 9 J (k~kps) k 1-2k (2k -1) wr @..(3k4-) 00 k a k1pe)Iju X k1 1 + L + log (.. + (r209) set)0100t (4. 19) 9~k,J0(k~p5) (13) - 00J kp)(4.20) 0 j~r (2ir-200) Swtstutiu (4..12) ad(4. 13) in" (4. 9) gives, (1) 1) (2) (13) -R2+R]1 k=1 k Ak 0 (4.21) The necessary and saufflcisnt oon -dit io (4. 7) fer the.ieof ti class of solution yields the values of 9 adPs, givin the of t h excitin souree, nefoer r th.um of this eh~s.4 b~y equatin aeparAsJy to zwro, Ow real sad iqnzyPM"t of (1) (12) (13) (12) (13) (12) (13)1(.2 a _R +R -R +R -R + R jO422 -w2k 2k 1-2k 2k-i 0k14 21 (13 1) 3+R (1 (13) (RI (13=0 4.3 I[ Zk~14k. 2k-i+ R.~ R k 0 j 23

52 For the location of the exciting oure given from 9 and pa by (4.22) and (4.23), this clas of solution in giTn from the mltati of the Fredhelm equation (4. ) *equivlntt to the linilar *atiom (344), pertaining to which the two oprators l* Ft and k are defined l (4.4 and (4. 5) respeatively. Evalu atinc te ft -K* f is (4. ) gives, 1 -K* tf(to) f - Rt= I14 (4.24) where 4 D(t) inh o i J C= — < it (4.25) Tranaformi 114 in the -space just as in the eae of I4, grie, l v 14 =Ae j 15 ( ) I14 = A e a I (4.26) 15 where joD -JaO H (I(-klaI) -I *l - )vA I in; (3-l) 4 2 I w v)r C2 kd (k dA (4.27) and A is defined in (3.65).

53 Expressing I15 as the sum of two integrals gives, I15 = I16 + I17 where (4. 28) j3o I = -joo JaO 11 7 -jO j s.n mi sin sA J. i (klp.) j lnl sA m in.A (P J- (k1Ps) I p s i sn p,(2 - 20o) mtin 2 -,) v. k dp. k2 kl f do ~ k2 (4. 29) (4. 30) Using the method of Appendix C to evaluate 116 and I17 except that now all the poles are simple poles, gives, 00 16 2 [R16)+ R()+ R(16 +R16 + R2k+) (4.31) k=l k a) I17= 2wj R(17)+ C R(17)+ R(17) +R(17) + R(17) (4.32) 7 -2k 1-2k -"k v -2k] in which (C. 10) Is am to hold. The various resl*- are gi as Ifiwel:

54 R(16)= R 2j e JVW &ina sn a00jv(kl )k - - -oom1 (:m-) I v si Yv sinv Ob-200) k2 - (4.33) (4.34) I k R2k - 2ki acm2oe sin ( l)2k (16)= 2k-i Pk (2k-1)r e(2k-1)~ Cosr 2 kI 2k-I wommuma) k2 (4.35) (4.36) ~j(- -1k)k mi ala al sinkPo J (k p ) k Ivy ai&k 51U4SkV) I- k2 R(16) (17) 0 2j(-1)k e JWI sir42k+ v )ala(WU+v)ooJak+(kip.) 2k~.L (2k +v) r min iv sin (3k+v)(hr- 20) Joe0J01t lr(2r-SVm.f (4. 37) (4.38) (4.39) R(17) -2k akr 2W X0af at (17)= 1-2k (2k-1)r.suft4 * Sk. 2a-1 (4.40)

55 (17) J(.l) kxla(2k-v)Osln(2k-v)~ J k~(kjp) c2 R 0 27-vk ( 2k~) (4.42) v -2k (2k - Y) sintav isin2k- Y) (2- 20) kl1 Sust uig(4.31) and (4.32) Ino(4. 28) gilves, (L~7)'8 >11) (17) (16) (17) (16) (17) k IV 2kk_ 2 A"k i'-2k 2k+v In which the vaurious rosldws mm gv by (4.23) h 46 (4.0). Substitutin (4. 2X) int. (4. 2*,g0 s fMt 2l.T1(4.) 0 - 0 It5 Exrp f fft) In th vrw L f (to) -4 frl:z2w 315 (4.45) NW seUlm& K* ILA2In(4. 6 m Is

56 where t ot-1 \ (tlp N(t,t) = — ktt) dt (4.47) in which k(tl, t) and B(tl) are defined in (3. 75) and (3. 76) respectively. Substituting (3. 78) inte (4. 47) gives, N(to,t)= 18 (4.48) iJ (1-t) where [2 (1l-tl) T(tl, t) H(tlo t)- (l-t) B(tl)] I18 '.............. (4.40) Jt-I (tl-t0 t-tl) Transforming the variable of ntegration te n by mans of (S. 81), gives. Ti 18 1-t -A Le x t -f r (1+ *.^) T()H(< ) + 1nl 1 lf )( j di~ (4. * Sibstitutlng (4.M.t (4. 4s)(ve, e,, 1t8 (1-t)