AN APPROXIMATE SOLUTION FOR THE PLANE WAVE DIFFRACTION BY AN IMPEDANCE STRIP: H-POLARIZATION CASE

Transcription

1 Ayd, E A; İk, T Okvro rjšj a dfrakcju ravog vala mpdacjskom rakom: slučaj H-polaracj ISSN (Pr), ISSN (Ol) ID: TG-63 AN APPROXIMATE SOLUTION FOR THE PLANE WAVE DIFFRACTION BY AN IMPEDANCE STRIP: H-POLARIZATION CASE OKVIRNO REŠENE ZA DIFRAKCIU RAVNOG VALA IMPEDACISKOM TRAKOM: SLUČA H-POLARIZACIE Em Avşar Ayd, Turgu İk Orgal scfc papr Absrac: I hs sudy, h dffraco of H-polard pla wav by a fly log srp whch has h sam mpdac o boh facs wh a wdh of a s vsgad by usg a aalycal-umrcal mhod Th dffracd fld s obad by a gral quao rms of h lcrc ad magc currs ducd by h cd fld Ths gral quao s rducd o wo ucoupld gral quaos ha clud oly ducd lcrc ad magc currs sparaly Boh of h currs ar dfd as a sum of f srs of Ggbaur polyomals wh ukow coffcs sasfyg h dg codos Th gral quaos ar rasformd o lar algbrac quaos by usg aalycal mhods ad h ukow coffcs ar drmd by solvg umrcally obad marx quaos Numrcal xampls o h RCS (radar cross sco) ar prsd, ad h far fld scarg characrscs of h srp ar dscussd dal Som of h obad rsuls ar compard wh h ohr xsg mhod Kywords: mpdac, srp, aalyc, umrc, dffraco Ivor asv člaak Sažak: U ovom radu sražuj s dfrakcja H-polarraog ravog vala od bskoačo dug rak koja ma jdak opor a obj sra šru a koršjm aalčkog-umrčk mod Dfrakrao polj dobvo j pum gral jdadžb u smslu lkrčh magskh sruja ducrah upadm poljm Ova grala jdadžba svda j a dvj odvoj gral jdadžb koj asbo uključuju samo ducra lkrč magsk sruj Obj sruj dfra su kao broj bskoačog a Ggbaurovh poloma s poam kofcjma koj adovoljavaju rub uvj Igral jdadžb prvor su u lar algbarsk jdadžb pomoću aalčkh moda poa su kofcj uvrđ rjšavajm umrčk dobvh marčh jdadžb Prdsavlj su umrčk prmjr a PRP-u (površa radarskog prsjka), a karakrsk raspaja a dalkom polju kod raka daljo su raspravlj Nk od dobvh rulaa usporđ su s drugom posojćom modom Ključ rjč: mpdacja, raka, aalčk, umrčk, dfrakcja INTRODUCTION Th scarg of lcromagc wavs from gomrcal ad physcal dscous s o of h mos ssal flds h lcromagc wav hory Howvr, h frs cosdrabl sps hs ara of rsarch ar du o Lord Raylgh ad ASommrfld Almos a cury ago, Raylgh [39] (from Bor M [5]) vsgad h problm of scarg by a prfcly coducg sphr; ad Sommrfld [5] ovrcam h problm of dffraco of pla lcromagc wavs by a absoluly coducg sm-f pla Ovr h las fw dcads, h rsarchs h scarg of lcromagc wavs by svral objcs hav b dvlopd as a rsul of s drc applcably o cvla applcaos, cludg mlary os, such as rmo ssg, o-vasv dagoscs mdc ad odsrucv sg Th rcvd sgal, whch s sprad by a objc, ca b usd o rsolv som of h gomrcal ad physcal proprs of h scarr Sc h rcvd scard sgal powr s drcly corrspods o h scard fld or radar cross sco (RCS) of h objc Thrfor, mlary applcaos, ordr o avod h cho sgal, h RCS of args lk ar crafs mus b cu dow o a mmum lvl Addoally, ordr o dcras h frc causd by obsacl as buldgs for sac, h RCS of such obsacls ha ar clos o radars should also b rducd Corrspodgly, groups of grs ad scss ar also rsarchg o xrac as suffc formao as possbl from low RCS valus for boh mlary ad cvla applcaos Svral mhods xs, whch ar applcabl o rduc h RCS of som obsacls For sac, f h shap of h arg s o b modfd, scard rgy somhow may b drcd oward som dsrd rgos Bu som ohr suaos xs whr shap modfcao s Thčk glask, 3- (6),

2 A approxma soluo for h pla wav dffraco by a mpdac srp: H-polarao cas Ayd, E A; İk, T cofd by h arodyamc srucur of ha arg, ad aohr chqu such as absorbg layr s appld for h sam purpos Th absorbr may coss of dlcrc or magc marals, parally dlcrc ad/or magc marals or a umbr of layrs of such marals Ev hough such mxurs of shaps ad marals propos mor dgrs of frdom rms of dsg wh corollg h RCS of h arg, smlarly, h complxy of h soluo procdur s also crasd Soluos for rcogd problms cludg halfpla, cyldr or sphr ar apparly ssal rgardg h dffraco hory, ad srp s cosdrd o b o of h mos mpora famlar srucurs du o s gomry, srps ar usually accusomd o vsga h mulpl dffraco phomo Furhrmor, a larg amou of applcabl ssus, spcally rmo ssg, modlg by coducg, mpdac or rssv srps s possbl Addoally, by usg h dualy prcpl dffracg a sl a absolu coducg pla ca b rducd o a prfcly coducg srp problm Scarg of cracks or gaps ha may occur o h obsacl s surfac, whch s complly or parally flld wh som maral, may provd a sgfca corbuo o h ovrall scarg par I such ssus h gaps or cracks may b modld by srps ad/or sls Accordgly, du o s adjusm o may praccal problms, srps hav b broadly rsarchd by may auhors by usg dscv aalycal ad umrcal srags Lraur Rvw A larg umbr of aalycal chqus hav b dvlopd cosdrg dffraco by scarrs wh svral shaps, ss ad cosu marals Ths mhods ca b cagord wo groups, amly, xac ad approxma mhods Usg a xac mhod s possbl, wh h obsacl's gomry corrspods o a coorda sysm havg h wav quao sparabl Furhrmor, xac soluos usually clud sophscad gral xprssos, whch ar rqurd o apply ar-fld, far-fld or hgh-frqucy asympoc for grg purposs Du o hs cosras o may praccal problms hav xac soluos, ad mosly svr asympoc xprssos ar acqurd Approxma aalycal chqus wr obad from h xso of classcal opcs by gomrcal Thory of Dffraco (GTD) roducd by Kllr [3] ad Physcal Thory of Dffraco (PTD) roducd by Ufmsv (from Bhaacharyya AK []) I gomrcal opcs ro wavlgh approxmao s mplmd ad spac s dvdd o dsc llumad ad shadow rgos I s cosdrd ha rgy sprads ub of rays, obvously h dffraco mpacs ar dscardd prvous smas; whr hs ffcs wr frs sudd by GTD A boh cdc ad rflco boudars, h dffracd fld bcoms f ad also a h dg whch s causc for h dffracd fld Th rm of a dffraco coffc ca b acqurd smply by xprssg h dcal soluo of a rcogd problm GTD form I ordr o hac GTD ad ovrcom som of h mhods drawbacks wo fudamal chqus hav b dvlopd amly, Uform Asympoc Thory of Dffraco (UAT) ad Uform Thory of Dffraco (UTD) roducd by Ahluwala al [] ad Kouyoumja ad Pahak [] rspcvly I UAT s cosdrd ha h fld soluo ha occurs a dg dffraco problm ca b xdd a spcfc asympoc srs cludg a Frsl gral, whras UTD Kllr's dffraco, h coffc s mulpld by a facor volvg a Frsl gral Th mulplcao facor has a faur such ha h fld po approachs h shadow boudary ad a f fld s obad a h shadow boudars whl approachg ro I physcal opcs, surfac currs ar assumd o duc oly a h llumad par of h scarr ha fucos as h supply of h scard fld Th accuracy of h chqu ca b rasd by hacg h assumd curr dsrbuos, howvr, may o b abl o accou urly for h prsc of dscous o h obsacl By PTD, Ufmsv roducd h frg curr cocp as a rsul of h physcal or gomrcal dscous I was assumd ha h r curr o a coducg surfac cluds a frg curr (ouform par) alog wh physcal opcs curr I dsco o GTD, PTD ylds h f flds vrywhr cludg h shadow boudars a h causcs Sc s gomry s smpl, h srp ssu was rsarchd by may scss May chqus wr proposd, bu o yldd a ffc soluo So far, a svr soluo for h problm of dffraco by a srp dos' xs Howvr xplag h amps o solv hs problm whch had yldd sgfca soluos, s worh Som of h cosdraos mad wll b gv a squal ordr as follows Mors ad Rubs [33] usd a Mahu fuco xpaso for lcromagc flds o oba h xac srs rprsao for h soluo of h dffraco by srps Accordg o h rapd covrgc of h Mahu srs, hr soluo ylds mprovm for low frqucs Howvr, wh coms o hgh frqucy rag hr proposd mhod s o accura sc hr s a cssy o clud larg umbr of rms whl compug h f srs Grbrg [6] proposd shadow curr cocp for dffraco ssus rgardg lcromagc wavs by a coducg sl by mplmg h gral quao mhod Howvr, h sam mhod could b appld o solv h problm of dffraco by a coducg srp, whch s achvd by usg h dualy prcpl Th problm wh hs mhod s rducd o h soluo of a scod kd Frdholm quao Accordg o hs formulao, h asympoc form of h soluo for h hgh frqucy cas ca b obad asly I should b ak o accou ha h applcao of gral quao mhod hav b usd by ohr rsarchs bfor Grbrg, prvous suds, g Copso [], Lv ad Schwgr [7] ad Mls [3] Igral quao basd formulao was frsly mplmd by Copso [] (from Sor, TBA []) for a spcfc cas of h Sommrfld half-pla whch was a quao of h Wr-Hopf yp 8 Tchcal oural, 3- (6), 79-97

3 Ayd, E A; İk, T Up o 95s, approxmaly all rsarchs wr focusd o h dffraco by prfcly coducg srucurs Sor [] proposd h frs sudy for h ssu of f coducvy, by usg a sm-f mallc sh as h dffracg srucur Th spulao usd hs sudy was h sadard mpdac codo For E-polarao, Sor dfd wo ducd currs amly lcrc ad magc, ad by usg Gr's hory Sor had acqurd h oal fld a ay po off h pla rms of hs ducd currs ad cd fld Th dg codos for curr compos wr bg gv as O(x / ) ad O(x / ) for, for h agal ad ormal compos, rspcvly, of h lcrc ad magc currs Th asympoc bhavor roducd by Sor [] for h currs corrspod o asympoc bhavor of h fld compos proposd by Mxr [3] Ths s cosdrd as o of h mos ssal formao for our mhod Accordgly, h formao bhd hs bhavor of currs a h dgs gv us h possbly o xpla h currs a srs of som spcal fucos wh som ukows I may b od ha h srp s usd as a basc lm h formao of grags Dffraco by a mpdac srp was rsarchd by Faulkr [] (from Bowma, [6]) va a Wr-Hopf prmarly basd mhod, whr h wdh of h srp was cosdrd larg compard o boh h wavlgh ad h magud of h surfac mpdac such ha was bg rsrcd o lmd valus Maluhs [8] (from Bowma [6]) hough-ou h wd srp drawback, furhrmor, whr h muual raco bw h dgs was glcd: h, h wo dgs wr assumd as dpd sm-f half plas, wh h absc of h ohr bg xcd by a quval fld alo Bowma [6] has also rsarchd h hgh frqucy back-scarg from a absorbg wd srp wh radom fac mpdacs by usg a approxmao basd o h kow half pla soluos obad by Maluhs [8] Mulpl dffraco or h raco amog h dffracg dgs of a srp s of gra applcabl mporac addoally, dffraco problms whch volv boh sl ad or srp gomry rprs a hrpar mxd boudary-valu ssu whch srcly may b rprsd va rpl gral quao approach ha grally ylds corrspodgly o a modfd Wr- Hopf quao Possbly, h soluo for mulpl dffracos ar obad by a rpv procdur prsd by os [] (from Srbs AH ad Büyükaksoy A []) Kobayash [5] had rsarchd umrous yps of modfd Wr-Hopf gomrs ad cojoly cofrrd h soluo for H ad E polarao cass Aohr approxma mhod s prsd by Tbro ad Kouyoumja [7] ad s calld xdd spcral ray mhod whch cluds rprg h cd fld o h scod dg as a sum of homogous pla wavs ha ca b rad sparaly By usg hs mhod, Hrma ad Volaks [8] had rsarchd svral dffraco by a coducv, rssv ad mpdac srps whr h asympoc soluos prsd for hs cass also volvd h x Okvro rjšj a dfrakcju ravog vala mpdacjskom rakom: slučaj H-polaracj mpacs of surfac wavs Th spcral rao chqu roducd by Büyükaksoy al [9] s cosdrd a alrav spcral doma mhod furhrmor, assumg ha h dg s llumad by h fld dffracd from h ohr dg, s possbl o oba h doubly dffracd fld addoally, cosrucg aohr Wr-Hopf problm for hs cofgurao A dald dscusso of h mod mhods ar gv by Srbs ad Büyükaksoy [] Th dvlopm umrcal chqus for h soluo of h scarg problms has always b paralll o h voluo compur chology Alhough umrcal mhods may b cosdrd as mor obvous compard o aalycal mhods, sc h marx vrso procdur for h aalyss, compur capacy lms h s of h ssu ha ca b dal wh Grally, for h barrrs havg a maxmum dmso of a fw wavlghs, umrcal mhods ar abl o maa prcs soluos Th Mhod of Moms (MOM) ad Galrk's Mhod [7] ar h popularly usd umrcal chqus I MOM boh h basc fucos ad h wghg fucos ar dsc howvr, hy ar cosdrd smlar fucos Galrk's Mhod I umrcal mhods, h scarg drawbacks ar prsd as gral quaos of boh ukow fld quay ad ukow ducd surfac curr dsy by applyg Gr's horm: H G(r,r ) E(r) EG(r,r ) ds () s ( E) G(r,r ) or rms of surfac curr dss E(r) E { (r ) M (r ) } G(r,r )ds () s Th fudamal fucos ar broadly usd for rprsg h currs o a gv surfac Howvr, mos suds h gral quao s achvd drcly by applyg h wll-kow mhods such as Mom Mhod, Galrk's Mhod or F Elm Mhod Wadura [53] vsgad h curr fudamal fucos for curvd surfacs gral Volaks [5] rsarchd h scarg by a arrow groov a mpdac pla, ad solvd h gral quaos by prsg a sgl-bass sudy of h quval curr o h arrow mpdac sr Basd o hs suds, h obad gral quaos ar solvd drcly by applyg hr o of h abov- mod umrcal mhods Th GTD was usd by Sor [3] o oba h xprsso of h scard fld by a rssv srp whr h rsuls m h os obad by h umrcal soluo of h gral quao cludg srps a arrow as a sxh of a wavlgh Sor also appld MOM o spc h corbuo of fro ad rar- dgs o h far flds has b rprsd ha, for srps wh a wdh grar ha abou a half wavlgh, h corbuo rgardg h fro dg s smlar o a half pla o, havg smlar rssac ad h corbuo of h rar dg s proporoal o h squar of h curr a ha Thčk glask, 3- (6),

4 A approxma soluo for h pla wav dffraco by a mpdac srp: H-polarao cas Ayd, E A; İk, T po o h half pla corrspodg o h rar dg of h srp Grally, h gral quaos Eq () ad Eq () ar usually solvd by umrcal mhods Howvr, hy ca also b rformd o a s of algbrac quaos by applyg som aalycal chqus Addoally, s possbl o solv h marx quao obad, by usg sadard marx vrso algorhms Th m dd for h soluo of hs marx quao s proporoal o h s of h dvlopd marx I h cas of larg bods, h m rqurd ca b ormously larg parcularly for RCS smao: Accordgly, h s of h marx mus b maad as small as possbl Ems ad Rogowsk [3] solvd a wo-dmsoal problm of lcromagc wav dffraco by a pla srp wh dffr boudary codos o s surfacs Th dffracd fld was xprssd by a gral rms of h ducd lcrc ad magc curr dss Th rlad boudary-valu problm h doma of h shor-wavlgh was occurrd as a marx Wr- Hopf quao whch was solvd hrough h Khrapkov mhod Ths aalycal soluo s a powrful ool o sudy h ffcs of a sgl srp udr h cdc of pla wavs I ca b usd o valda h umrcal mhods Apaydı ad Svg [3] sudd o mhods of moms (MoM) for modlg ad smulao of scard flds aroud f srp wh o fac sof ad h ohr hard boudary codos Scarg crosssco was calculad umrcally ad compard wh hgh-frqucy asympocs (HFA) modls such as physcal hory of dffraco (PTD) ad hory of dg dffraco (TED) Accordg o auhors kowldg, hs mhod s h frs applcao of MoM o h srp wh o fac sof ad h ohr hard Th H-polard pla wav dffraco by a h maral srp has b solvd usg h Wr-Hopf chqu ad approxma boudary codos by Nagasaka ad Kobayash [3] Employg a rgorous asympocs, a hgh frqucy soluo for larg srp wdh has b obad Th umrcal xampls hav b do o boh h radar cross-sco (RCS) ad h far fld scarg characrscs of h srp Som of hs xampls hav b compard wh xsg mhod ad rsuls agr rasoably wll wh xsg mhod Comparso of Numrcal-Aalycal Mhods Commoly, h lcrcal s of h body rsrcs h racably of umrcal chqus, howvr, h gomrcal complxy of h objc rsrcs h applcably of h aalycal mhods Furhrmor, hybrd mhods ar usd alog wh chqus basd upo h xso of classcal opcs ordr o oba a asympoc soluo for problms rgardg hgh rgo frqucs Hybrd mhods cludg boh umrcal ad hgh frqucy asympoc chqus may hav h ably o xd h class of lcromagc scarg problms ha ca b hadld Th hybrd approach ca b rformd as a fldbasd aalyss whr h GTD soluo for h fld assocad wh dg or surfac dffraco ar usd as h al po Ths soluos fuco as h asa o h MoM formulao ad rprs h lms of a scarr o mg h rqurms of a rcogd gomry whch s o cooprav wh a GTD soluo slf (Bursd al [7] ad Sahalov ad Thl []) Alravly, curr-basd formulao s possbl a suao such ha h aalyss procds from asa soluos for h currs obad from physcal opcs ad PTD (Thl ad Nwhous (975), Eklma ad Thl (98) (from Mdgys- Mschag, L N, 989)) ad Mdgys-Mschag ad Wag (983) (from Mdgys- Mschag, L N, [9-3])) Th opo of a mxd fld-curr basd approach also xss Grally, rgorous scarg hory, rls o h hory of boudary-valu problms whch h progrss s closly dpd, o mahmacal brachs Th ma approach for h soluo bhd wav scarg s o rform h problm rms of h gral quaos sasfd by h ukow fucos alog wh h curr ducd o h scarr surfac Numrcal chqus ar uld o shrk hs gral quaos o a sysm of lar algbrac quaos Th prsc of dgs o h grao coour produc rrors curr dsy calculaos I ordr o hac h accuracy of h soluo, som approachs accoug for h Mxr's dg codo xplc form hav b suggsd by Shafa (973) (from Vlv, I E [5]) A smlar chqu was usd by Bulr [8] whch h aalycal soluo of h gral quao was obad for h curr ducd o a cofd coducg srp I hs soluo h curr mass was xprssd as a produc of a Chbyshv polyomal ad a wghg opraor whch fulfll h dg codo ad afr spcfyg h Fourr yp coffcs, h xprsso of far flds wr obad So a gral quao whch coas h curr dsy ad xcao was obad as follows, k E(x) j { (x )l x x dx (3) Whr k (c l ) (x ) dx} x s h ukow curr dsy / ad c j( / ), (c = 577 s h Eulr s cosa) Th soluo for h curr dsy was xprssd as, x f x () (x) [ / ( ) ] ft ( ) whr T dos h Chbyshv polyomals of h frs kd ad f s ar h ukow coffcs By usg h orhogoaly proprs of h polyomals, h ukow coffcs ad so h xprsso of h curr dsy wr obad for boh E ad H- polarao cass 8 Tchcal oural, 3- (6), 79-97

5 Ayd, E A; İk, T A subsu chqu was proposd by Vlv al [9-5] whr h soluo compass ay prassgd accuracy Th dsrbud fld was rprsd usg h Fourr rasform of h corrspodg surfac curr dsy whch offrs a umbr of hacms o ovrcom h problm A hybrd chqu basd o h sm-vrso procdur for quao opraors ad h mhod of moms was usd o yld h opmal soluo Th ssals of h soluo ad s applcao o h wav scarg by polygoal cyldrs ad fla coducg srp srucurs ar proposd by Vlv ad Vrmy [9] Th prvous vrsos of hs mhod wr suggsd by (Nomura ad Kasura [36], Hogo [9] ad Osuk [37] (from Vlv, E I [9])) for complly coducg srp ad sl cofguraos Ths aalycal-umrcal mhod, whch uls spcral approach, somhow solvs h problm o a sysm of lar algbrac quaos for h ukow Fourr coffcs of h curr dsy fuco Appropra rucao of h f sysm of quaos ca oba h rsuls wh ay dsrd accuracy I should b rald ha h applcably of h rucao mhod cao always b modfd, addoally, h marx lms corrlad wh h sysm of lar algbrac quaos usually collaps slowly wh a cras of hr dx 3 Ams ad h Scop of h Sudy Dsp hr ar varous powrful aalycal chqus, h ma fluc of umrcal chqus s ha hy may b appld o a scarr of radom shap ad ar grally oly rsrcd by h s of h scarr Bu hs lmao s a ralsc problm Apparly, a s of lar quaos whch do h scarg ssu ca b grad bu h obad s may b oo larg o b solvd Forualy, h dvlopms compur chology mak h soluo of may lcromagc problms possbl for a rqurd dgr of accuracy I coras, h asympoc chqus work bs wh compard o h wavlgh h scarr s s larg Howvr, h dffculy of h problm crass wh h complx shapd bods ar of rs Th us of aalycal mhods alog wh umrcal mhods may bypass hs rsrcos Th umrcal mhods ar lmd o h bods havg a maxmum dmso of lss ha a fw wavlghs, whl h aalycal mhods produc accura oucoms for h scarr much lar ha hos of o wavlgh Thrby, hs mhods may b combd o ovrcom h scarg problms volvg scarrs of rmda s ad s h rsoac rgo Addoally by usg aalycal-umrcal mhods h compuao m rqurd may b dcrasd o a accpabl lvl I hs sudy, dffraco by a mpdac srp s rsarchd by usg h aalycal-umrcal chqu proposd by Vlv ad Vrmy [5] I Chapr, h formulao of h problm for H-polard ras s gv By xprssg h lcrc ad magc currs as f srs rms of Ggbaur polyomals, wo gral quaos spcral doma for lcrc ad Okvro rjšj a dfrakcju ravog vala mpdacjskom rakom: slučaj H-polaracj magc currs ar drvd I Chapr 3, h gral quaos ar rducd o a sysm of lar algbrac quaos for boh currs wh som ukow coffcs I Chapr, som physcal quas ar rprsd rms of h ukow coffcs whch wll b dod by solvg h sysm of lar algbrac quaos I Chapr 5, h soluo of brach-cu grals ar obad ad fally Chapr 5, h curvs for boh currs, far fld ad RCS ar rprsd Th rsuls ar rvwd ad compard wh som prvously obad rsuls Th ams of h curr rsarch ar: o mploy a w aalycal-umrcal chqu o solv a rcogd dffraco problm ha sll dos o hav a propr soluo, o oba a accura soluo for h mpdac srp whch ca b usd o smula may praccal obsacls, 3 o rrv a soluo whch may work a wd frqucy rag, o s a bas soluo ad wr compur cods whch may b usd o aaly som complx srucurs ha could b rad as a combao of mulpl srps FORMULATION OF THE PROBLEM Th scarr of h dffraco problm ha wll b formulad hs sco s a srp of wdh a whr h sam mpdac s assumd o b mposd o boh sds Th gomry of h problm s llusrad Fgur ad η dos h ormald mpdac of h srp Fgur Gomry of h problm Sc h srp s uform alog h -axs, h problm ca b rducd o a wo dmsoal problm Th m dpdc of h flds ar assumd o b xp( ω) ad supprssd hroughou h aalyss Th cd magc fld s gv as a larly polard pla wav H x, y k(x y ) () whr cos () Th oal fld s, s H x y H x, y H x, y (3) whr H s s h scard fld Thčk glask, 3- (6),

6 A approxma soluo for h pla wav dffraco by a mpdac srp: H-polarao cas Ayd, E A; İk, T O h srp, h oal fld mus sasfy h Loovch boudary codo whch s frquly calld as mpdac boudary codo, gv by, H x, y y k H x, y y So h oal fld ca b xprssd as [] H x, y H x, y a () kyi m x I x (5) y a H k ( x x) y dx whr I m ad I ar h quval magc ad lcrc curr dss rspcvly Exprsso of magc ad lcrc curr dss ca b dfd as follows rspcvly, I ˆ ˆ a y H H Ha I aˆ H (x, ) H (x, ) y ˆ H (x, ) H (x, ) a f (x) aˆ ad x x (6) (7) I ˆ m a y E (8) E H E H (9) aˆ ˆ ˆ x a y a E x y H H H E aˆ ˆ x ay y x H H I ˆ ˆ ˆ m ax a y a y y x H(x, ) H(x, ) I ˆ m a y y Z () () () (3) ky () ky Im H(x, ) H(x, ) aˆ y y H(x, ) H(x, ) y y f (x) aˆ So Eq 5 ca b rarragd as a H ( x, y) H ( x, y) f( x ) f( x ) y a (6) H ( k ( x x) y ) dx () Applcao of Boudary Codos If w rwr h Loovch boudary codo for ad, ad f w subrac ad add hs wo quaos w ca oba h followg xprssos: H (x, ) k H (x, ) (7) y ad H (x, ) k H (x, ) (8) y Th dffrc of Eq (7) ad Eq (8) s H(x, ) H(x, ) y y y y k H (x, ) H (x, ) (9) Ad h sum s H(x, ) H(x, ) y y k H (x, ) H (x, ) a lm f( x ) f( x ) y () y a H( k x x) dx(5) I Eq(), w rqur h sum of hs wo xprssos as () Usg h xprssos lcrc ad magc curr dss f ad f Eqs (9) ad () f (x) k H (x, ) H (x, ) () f (x) H(x, ) H(x, ) () k y y ar drvd Th xprssos of oal fld a y ca b obad from Eqs (3) ad (6) as H ( x, ) H ( x, ) lm ( ) ( ) y H ( k x x) dx ad H ( x, ) H ( x, ) a f x f x (3) y a 8 Tchcal oural, 3- (6), 79-97

7 Ayd, E A; İk, T Okvro rjšj a dfrakcju ravog vala mpdacjskom rakom: slučaj H-polaracj H ( x, ) H ( x, ) H ( x, ) H ( x, ) a lm f( x) f( x) H ( k x x ) dx (5) y y a a lm f( x ) f( x ) H ( k x x ) dx y y a For h cd fld w hav ha H(x,) H(x, ) (6) kx H (x, ) ad from Sor [] w hav ha a () ( lm lm ) f( x) H (k x x )dx y y y (7) a ad a lm f ( x) H (k x x )dx y a a y a () lm f ( x) H (k x x )dx a a () f ( x) H (k x x )dx () (8) So by subsug Eq (6) o Eq (7) ad Eq(8) o Eq (5) o ca g ha; f(x) H (x,) k (9) a () f( x) H (k x x )dx a or f(x) k a () a kx f ( x) H (k x x )dx (3) Eq (3) s h gral quao for h magc curr f (x) A hs sag w wll drv h gral quao for magc curr spcral doma usg h gral rprsao of Hakl fuco gv as; k ( xx) () H ( k x x ) d (3) Usg Eq (3) Eq (3) a k ( xx) kx f( x) f( x) ddx(3) k a ad chagg h ordr of grao kx f( x) k (33) a kx kx f( x ) dx d a s obad Whr a kx F ( ) f ( x) dx (3) a s h Fourr rasform of h magc curr dsy f (x) So kx f( x) k kx Fm () d (35) s drvd Th followg varabl chags ar rqurd o b abl o xprss h curr dsy fucos rms of Ggbaur polyomals Sc, f w us h varabl chag x a, x a, ka a x a ( / a) ( a) ( ) ( ) ( ) F f a ad F ( ) af ( a) d (36) ad l f( ) af( a) (37) So, F () f ( ) d (38) s obad Mas ha Eq (35) ca b arragd as; f( x) a ( a ) a ( a a ) Fm () d or af(x) Fm () ad f( ) d Fm ( ) d If w mulply boh sd of h quao by gra bw ad + f ( ) d d d Fm () d (39) () () ad () Thčk glask, 3- (6),

8 A approxma soluo for h pla wav dffraco by a mpdac srp: H-polarao cas Ayd, E A; İk, T s drvd From Eq (33) s obvous ha h form o h lf had sd ca b xprssd as F ( ) f ( ) d (3) m Th frs rm o h rgh had sd s s ( ) d () ( ) ad h scod rm o h rgh had sd by chagg h ordr of grao; Fm () ( ) d d ( ) Fm () ( ) d (5) Fm () s ( ) d ( ) s obad So Eq () ca b rarragd as s ( ) Fm ( ) ( ) (6) Fm () s ( ) d Ths quao s h gral quao for h magc curr dsy spcral doma I a smlar way, h gral quao for h lcrc curr dsy spcral doma s obad Ths quao ca b xprssd as s ( ) F ( ) ( ) (7) s ( ) F () d 3 REDUCTION OF LINEAR EQUATIONS TO THE SYSTEM OF LINEAR ALGEBRAIC EQUATIONS I hs sco, h soluo of h gral for lcrc F ( ) ad magc Fm ( ) curr dss wll b rducd o h soluo of wo ucoupld sysm of lar algbrac quaos Th frs sp of h rduco procss s o xprss h curr dss Fourr rasform doma ad oba a gral spcral xprsso for currs Th by usg h cosras mpld by dg codos, lcrc ad magc curr dsy xprssos wll b obad rasform doma Fally, hy wll b wr h form of f sysm of lar algbrac quaos volvg Ggbaur polyomal coffcs as ukows 3 Gral Exprssos for h Curr Dsy Fucos h Trasform Doma Sc s cssary o xprss h curr fucos spcral doma, h Fourr rasform of h F( ) curr dsy fuco f ( ) mus b foud as F( ) f ( ) d (3) Th curr dsy fuco f ( ) s dfd for ad s ro lswhr L f ( ) by a uformly covrg srs, such as, fc b rprsd f ( ) ( ) ( ) (3) whr C ( ) do h Ggbaur polyomals ad s a cosa rlad o h dg codo Th valu of Eq (3) wll b drmd by forcg h fucos such as o sasfy h dg codos for lcrc ad magc curr dss sparaly For h lcrc curr dsy fuco, ad h magc curr dsy fuco f ( ) f ( ) m, from Mxr s (97) dg codos, v ca b drmd for as, / / f( ) O( ) ad fm( ) O( ) (33) Th ordr rlaos gv abov for lcrc ad magc curr dss ca b obad rspcvly as / ad / by cosdrg h asympoc bhavor of Ggbaur polyomals oghr wh Eq (3) Th grao Eq (3) s dvdd o wo pars, as follows (3) F( ) f ( ) d f ( ) d By rplacg wh ad by chagg h ordr of uppr ad lowr lms h frs rm o h rgh-hadsd, Eq (3) wll b wr as: (35) F( ) f ( ) d f ( ) d Now o drm f ( ) h followg formula (Prodkov, A P, 983, p 73) wll b usd C ( ) ( ) C () (36) ad, f ( ) ca b xprssd as: fc f ( ) ( ) ( ) ( ) (37) By subsug Eqs (37) ad (3) o Eq (35), fc F( ) ( ) ( ) (38) ( ) d s obad If s v, = p, h Eq (38) aks h form of, 86 Tchcal oural, 3- (6), 79-97

9 Ayd, E A; İk, T F or F v ( ) ( ) p C p ( ) v ( ) ( ) p p f p p C ( )cos( ) d f d By chagg h ordr of grao ad summao v p p p (39) (3) F ( ) f K (3) s wr wh p p K ( ) C ( )cos( ) d (3) O h ohr had, f s odd, = p+ h Eq (38) aks h form of, F or od ( ) ( ) p Cp ( ) f p od ( ) ( ) p F f p p C ( )s( ) d d (33) (3) Aga by chagg h ordr of grao ad summao, ylds F ( ) f K (35) od p p p wh p p K ( ) C ( )s( ) d (36) I ca asly b s from Eq (38) ha, F( ) F ( ) F ( ) (37) v od K p Now, ordr o xprss ad Kp a form cov for umrcal calculaos, h followg xprssos [38] ar bg usd: ( x ) C (x) s(ax) dx ( ) (a) ( ) ( )! ()(a) ( x ) C (x) cos(ax) dx () (a) ( ) ()! ()(a) (38) (39) Okvro rjšj a dfrakcju ravog vala mpdacjskom rakom: slučaj H-polaracj As usual, () dos h Gamma fucos ad () dos h wll-kow Bssl fuco of h frs kd I Eqs (38) ad (39), by sg x,, h followgs ca b wr ad a K p Cp ( ) ad K ( ) ( )cos( )d p ( p ) ( ) p ( p)! ( )( ) p C p ( ) ( )s( )d ( p ) ( ) 3 p p ( ) ( p )! ( )( ) From Eq (3), Eq (35) ad Eq (37) p p p p p p F( ) (3) (3) F( ) f K f K (3) ad from Eqs (3) ad (3), ca b wr as p F( ) f ( ) p p p p ( ) ( p ) p ( p) ( )( ) p f ( ) 3 ( ) ( p ) p ( p ) ( )( ) or p F( ) ( ) p ( ) ( ) ( p ) p (3) f p ( p ) ( ) 3 ( ) ( p ) p fp ( p) ( ) By rarragg h las quao, h Fourr rasform of h curr dsy fuco ca b obad as follows: s (33) Thčk glask, 3- (6),

10 A approxma soluo for h pla wav dffraco by a mpdac srp: H-polarao cas Ayd, E A; İk, T F( ) ( ) f ( ) (35) ( ) ( ) () ( ) Ths compls h calculao for h Fourr rasform of a curr lm rprsd rms of Ggbaur polyomals gv by Eq (3) Sc o rsrco s mposd o h curr srs xprsso durg h drvao, s obvous ha hs rprsao s vald for boh lcrc ad magc currs Now cosdrg h dg codos sparaly for lcrc ad magc curr compos, h corrspodg spcral xprssos h Fourr doma ca asly b obad Frs, h magc curr dsy wll b obad smply by subsug whch ylds F m m( ) ( ) ( ) f / (36) / Smlarly, by srg Eq (35) s possbl o oba h Fourr rasform of h lcrc curr dsy Bu, s obvously s ha du o h ( / ) rm h xprsso, h curr dsy wll hav a sgulary hs cas Thrfor, ordr o rmov hs sgulary h srs xpaso gv by Eq (3) wll b usd for / : prsc of f f ( ) ( ) C ( ) (37) I [38] s gv ha, C ( ) T( ) (38) whr dos h wll-kow Chbyshv T () polyomals whch s vald for ; hrfor, Eq (37) ca b wr as, f( ) ( ) f T ( ) (39) From Eq (3), h Fourr rasform of f ( ) for s, F ( ) ( ) f T ( ) d (33) or F ( ) f ( ) T ( ) d (33) Th followg qualy s gv for [38] a x px T (ap) dx a a a x (33) so h gral Eq (33) ca b calculad as, ( ) T d ( ) ( ) (333) ad by subsug Eq (333) o Eq (33), F ( ) ( ) f ( ) (33) s obad Now, ordr o calcula h corbuo for =, l h aalyss sar wh h srs xpaso for h Fourr rasform of h curr gv by Eq (35): F( ) ( ) f ( ) ( ) ( ) () ( ) Th rm ( v ) ca b wr a mor cov form: ( ) ( ( )) (335) whch gvs (Prudkov, A P, 983, p 7) ( ) (( )) (( )) (336) whr () dos ha ( ) (( )) ( ) (337) Th followg dfy [38] () () ( ) (338) s usd wh Eq (336) ad ( ) (( )) ( ) ( ) ( ) (339) s obad By subsug Eq (339) o Eq (35) ad cosdrg ha wh / for h lcrc curr F ( ) f ( ) (3) s drvd Hr, h followg qualy s bg ak o accou ( ) ( ) (3) ( ) ( ) ad h aalyss gvs h compl xprsso for h lcrc curr dsy F ( ) f ( ) f (3) ( ) ( ) for ay valu of 88 Tchcal oural, 3- (6), 79-97

11 Ayd, E A; İk, T 3 Sysm of Lar Algbrac Equaos for I h prvous sco, h Fourr rasform of for was obad as f f ( ) F ( ) X ( ) (33) whr X f for (3) ad f X ( ) for (35) If Eq (33) s subsud o Eq (7), h problm s rducd o ha of fdg h ukows wh,,, as follows: f s ( cos ) X( ) cos ( cos ) (36) s ( ) X ( ) d By rarragg Eq (36) s ( cos ) X( ) cos ( cos ) (37) s ( ) X ( ) d s obad I ordr o b abl o xprss Eq (37) a mor cov form for umrcal calculaos, boh sds of h quaos wll b mulpld by l ( ) for l,, ad by grag ach rm wh rspc o from o l ( ) X( )d s ( cos ) l ( ) cos d l ( cos ) ( ) X s ( ) ( ) dd (38) s wr If Eq Pogrška! Ivor rfrc j proađ s rarragd by chagg h ordr of grao ad summao, ylds Okvro rjšj a dfrakcju ravog vala mpdacjskom rakom: slučaj H-polaracj X l ( ) ( ) d cos s ( cos ) l ( ) cos d (39) s ( ) l ( ) X ( ) d d Th grals h frs ad scod rms o h rghhad sd of Eq (39) ca b calculad as follows from h qualy gv [38]: l( ) s ( ) l( ) d (35) Ths qualy wll gv h valu of h frs gral o h rgh-had sd of Eq (39) by sg whch ylds l ( ) s ( cos ) d cos l ( cos ) ( cos ) cos (35) By usg h srs rprsao of Bssl fucos, h gav argum fucos ca b xprssd as follows rms of Bssl fucos wh posv argums: l ( ) s ( cos ) d cos ( cos ) l l ( ) cos (35) Now, subsuo of Eq (35), Eq (35) o Eq (39) ylds ha, l ( ) ( ) X d cos l l ( cos ) ( ) cos (353) l ( ) X ( ) d Th gral o h lf-had sd of Eq (353) s dod as E l ( ) ( ) dl d (35) ad h gral o h rgh had sd s amd as E l ( ) ( ) Dl d (355) So, by subsug hs wo quaos o Eq (353) E E E X d l X Dl (356) s obad wh cos ( cos ) ( ) (357) E l l cos Thčk glask, 3- (6),

12 A approxma soluo for h pla wav dffraco by a mpdac srp: H-polarao cas Ayd, E A; İk, T Now Eq (356) ca b arragd smply as E E E X dl Dl ( ) (358) whch gvs a f sysm of lar algbrac quaos for f Th dals of h umrcal compuao wll b show h followg scos whch rqurs o mapula aalycally h gral s abov 33 Sysm of Lar Algbrac Equaos for m f I h prvous sco, h Fourr rasform of f ( ) m for was obad as, ( ) Fm( ) Y (359) whr m Y ( ) ( ) f (36) If hs quao s subsud o Eq (6) ( ) s ( cos ) Y ( cos ) (36) s ( ) ( ) Y d ( ) s obad By chagg h ordr of grao ad summao ad by rarragg h quao ( ) s ( cos ) Y ( cos ) (36) s ( ) ( ) Y d ( ) s wr By mulplyg boh sds of Eq (36) by ( ) l ad grag ach rm wh rspc o from o l ( ) ( ) Y d s ( cos ) l ( ) d ( cos ) l ( ) s ( ) ( ) Y d d ( ) s foud Or Y l ( ) ( ) d s ( cos ) l ( ) d ( cos ) (363) (36) ( ) s ( ) l ( ) Y d d ( ) Each rm Eq (36) wll b wr lar a form mor cov for umrcal calculaos For usg smlar oaos wh h prvous sco d E l l ( ) ( ) d (365) s wr Now o valua h gral o h rgh-had sd of Eq(36), Eq (35) s usd wh whch gvs cos l ( ) s ( cos ) l ( cos ) d cos cos By subsug, Eq (366) o Eq (36) E l ( cos ) dl Y cos Y s obad, or ( ) ( ) d l E E E dl Y YDl, ad (366) (367) (368) s wr, whr E l ( ) l ( ) (369) D d E l l( cos ) ( ) (37) cos Fally, Eq (368) ca b xprssd as, E E E Y dl Dl (37) whch s h sysm of lar algbrac quaos for FIELD ANALYSIS m f Th oal fld xprsso Eq (6) volvg h lcrc ad magc curr dss as ukows s h fudamal formula for fld aalyss I h prs sco, frs h fld aalyss wll b prsd a formal way by assumg ha h curr dss ar kow Th, h sps for umrcal drmao of h ukows wll b gv Scard Far Fld ad Toal Scarg Cross Sco As sad Eq (5), h scard fld xprsso was gv as H x, y H x, y a kyi x I x y m a H k ( x x) y dx I ordr o oba h scard far fld, h frs sp s o subsu h asympoc xprsso of Hakl for larg argum Whch s gv as [5, p337] H () ( ) () () 9 Tchcal oural, 3- (6), 79-97

13 Ayd, E A; İk, T Usg x rcos, y rs () h followg xprsso whch s a par of h argum of h Hakl fuco: (x x ) y r cos r xcos x r s or r rxcos x x x (x x ) y r ( cos ( ) ), r r s wr Th, by srg o h argum x x k (x x ) y kr cos ( ) r r (3) () kx kx kr cos ( ) kr kr ad by usg h bomal xpaso formula kx k (x x ) y kr cos O( ) kr (kr) (5) x kr( cos ) O( ) r kr s obad So as Eq (3) ca b xprssd as, () () x H ( k (x x ) y ) H ( kr( cos )) r (6) kr x kr( cos ) r x ( cos ) O( ) kr r kr ad by rarragg smply () H ( k (xx ) y ) ( ( x (7) kr cos )) r O( ) kr kr s wr If Eq (7) s subsud o h quao of scard fld h followg s obad s s H ( x, y) H (r, ) a kr (8) k xcos f( x ) f( x ) dx kr y a Th followg of h paral drvav polar coordas r (9) y y r y wh Eq () cos s () y r r ad cosdrg for larg r, h scod rm o h rghhad sd s obvously ro So, Eq (8) ca b wr as Okvro rjšj a dfrakcju ravog vala mpdacjskom rakom: slučaj H-polaracj s H (r, ) kr s cos f( ) f( ) d Th, l h scard fld b xprssd as kr () s H (r, ) A(kr) ( ) () whr, A(kr) ( kr ) kr (3) ad, ( ) ( ) m( ) () I hs cas rprss h far fld radao par Th grals Eq () ar h Fourr rasforms of h curr dsy fucos By usg Eq (33) ad Eq (359) h ad ca b ( ) ( ) ( ) m obad as follows: ( ) X ( cos ) (5) ad, s ( cos ) m( ) Y (6) cos As kow h oal scarg cross sco ca b calculad as [5, p], s R ( ) (7) a whr s h cd agl I s obvous ha h calculao of h RCS rqurs o kow h valus of X ad From h aalyss accomplshd h prvous Y scos, should b clar ha h drmao of ad Y s rducd o umrcal E E E l, l, l D D d ad d E l 5 NUMERICAL ANALYSIS X valuao of As show h prvous sco h aalyss of h scard fld s rducd o h umrcal valuao of E, h fucos ad D Th gral xprssos l of hs rms gv by quaos Eq (35), Eq (355), Eq (365), ad Eq (369) ar o cov for umrcal calculaos Thrfor by usg som aalycal mhods hs grals ar valuad as follows: E l d E, l d l l (5) l l l Thčk glask, 3- (6),

14 A approxma soluo for h pla wav dffraco by a mpdac srp: H-polarao cas Ayd, E A; İk, T E d l l 8 l 3 l l (5) l l k E l, k Dl hkl k l k l l k k k { k l k k l k k l l k k l k, k l l hkl l k k l k l l k k k l k k l } k k l (53) D l E { Rl, E l l l k, k hkl } k l k l l k k k { k l k k k l l k k l k l k h l k k W, k l kl l l k l l k k k l k k l } k k l whr, W, l W, l () dos h Gamma fuco ad, h kl k l l k k k k l k k l 6 RESULTS (5) (55) Th approach mhod usd hs hss s a hybrd mhod amd as aalycal-umrcal mhod I gral, h am of usg hybrd mhods s o lma h dsadvaags of h aalycal mhods whch opra wll a hgh frqucs ad of h umrcal mhods whch opra wll a low frqucs Sad ohr words s am s, o oba a accura soluo for a wd frqucy rag By comparg h rsuls obad by Vlv al [5] ad h os obad by usg hs mhod ca b asly cocludd ha our rsuls ar much clos o rsuls obad by Vlv al [5] Fgur 6, Fgur 6, Fgur 63, ad Fgur 6 llusra h moosac RCS as a fuco of cdc agl for ka = 5, 5 I ordr o vsga h ffc 9 Tchcal oural, 3- (6), 79-97

15 Ayd, E A; İk, T Okvro rjšj a dfrakcju ravog vala mpdacjskom rakom: slučaj H-polaracj of h surfac mpdac o h scard far fld, wo dffr cass hav b cosdrd as η = 5 ad η=3 I vw of h wo RCS curvs for h mpdac srp, h backscard fld s o affcd by h mpdac of h srp surfac h shadow rgo O h ohr had, Fgur 65, Fgur 66, Fgur 67, ad Fgur 68 llusra h bsac RCS as a fuco of obsrvao agl for θ = 6 ad ka = 5, 5 I s s from h fgurs ha our RCS rsuls agr qu wll wh h rsuls of Vlv al [5] I addo, hr ar dffr rals Fgur 69 - Fgur 65 Fgur 63 Moosac RCS [db], 6 o, ka=5 : ; : 5, 3 ; : 5 ; : 3 (Vlv al [5]) Fgur 6 Moosac RCS [db], 6 o, ka = 5 : ; : 5, 3 ; : 5 ; : 3 (Vlv al [5]) Fgur 6 Moosac RCS [db], ka=5 ( 5ad 3) Fgur 6 Moosac RCS [db], ka=5 ( 5ad 3) Fgur 65 Bsac RCS [db], 6 o, ka=5 : ; : 5, 3 ; : 5 ; : 3 (Vlv al [5]) Thčk glask, 3- (6),

16 A approxma soluo for h pla wav dffraco by a mpdac srp: H-polarao cas Ayd, E A; İk, T Fgur 66 Bsac RCS [db], ka=5 ( 5ad 3) Fgur 69 Moosac RCS [db], ka=5 ( 5ad 5 ) Fgur 6 Moosac RCS [db], ka=5 ( 5ad 5 ) Fgur 67 Bsac RCS [db], 6 o, ka=5 : ; : 5, 3 ; : 5 ; : 3 (Vlv al [5]) ( 5ad 3) Fgur 6 Bsac RCS [db], 9, ka=5 Fgur 68 Bsac RCS [db], ka=5 ( 5ad 3) ( 5ad 3) Fgur 6 Bsac RCS [db], 9, ka=5 9 Tchcal oural, 3- (6), 79-97

17 Ayd, E A; İk, T Okvro rjšj a dfrakcju ravog vala mpdacjskom rakom: slučaj H-polaracj wav Th caocal srp srucur s chos rms of s coformy o may praccal problms Ths mhod s applcabl for h aalyss of mor complcad srucurs whch may b cosdrd as a combao of dffr srp cofguraos A ha ra, h mahmacal ss of h soluo wll o chag Howvr, hr wll b som xra rms h marx quaos Bcaus of hs, h compur cods hs hss mus b modfd for dffr srp cofguraos ( 5ad 5 ) Fgur 63 Bsac RCS [db], 6, ka=5 Fgur 6 Th oal radao par for ka=5 Fgur 65 Th oal radao par for ka=5 7 CONCLUSION I hs sudy, h dffraco of h H-polard pla wav by a fly log srp, havg h sam mpdac o boh facs wh a wdh of a s cosdrd Applyg h boudary codo o a gral rprsao of h scard fld, h problm s formulad as smulaous gral quaos sasfd by h lcrc ad magc curr dsy fucos Th gral quaos ar rducd o wo ucoupld f sysms of lar algbrac quaos ad physcal quas ar obad rms of h soluo of sysms of lar algbrac quaos Numrcal xampls o h moosac radar cross sco (RCS), bsac RCS, ad h oal scarg fld radao par ar prsd Som obad rsuls ar compard wh h ohr xsg rsuls Th sysm whch s cosdrd hs hss s h smpl mpdac srp llumad ormally by a pla 8 REFERENCES [] Abramow, M; Sgu, I E, 968 Hadbook of Mahmacal Fucos Dowr Publcaos, Nw York, p 3 [] Ahluwala, D S, 97 Uform Asympoc Thory of Dffraco by h Edg of a Thr- Dmsoal Body SIAM Appl Mah Vol8, No, pp 87-3 [3] Apayd, G; Svg, L, 5 Mhod of Mmos Modlg of Backscarg By a Sof-Hard Srp IEEE Tras o Aas ad Propagao, Vol63, No, pp [] Bhaacharyya, A K; Sgupa, D L, 99 Radar Cross Sco Aalyss ad Corol Arch Hous, Lodo, p 89 (Uffmsv, P I, 958) [5] Bor, M; Wolf, E, 965 Prcpls of Opcs Prgamo Prss, Frakfur, Grmay, p88 (Raylgh, L, 87) [6] Bowma,, 967 Hgh-Frqucy Backscarg from a Absorbg If Srp wh Arbrary Fac Impdac Caada oural of Phys, Vol 5, pp 9-3 [7] Bursd, W D; Yu, C L; Marhfka, R, 975 A Tchqu o Comb h Gomrcal Thory of Dffraco ad h Mom Mhod IEEE Tras O A Ad Proc, Vol AP-3, pp [8] Bulr, C M, 985 Gral Soluo of h Narrow Srp Igral Equaos IEEE Tras o A ad Proc, Vol AP-33, No, pp 85-9 [9] Büyükaksoy, A; Srbs, A H; Ugör, G, 989 Scodary Dffraco of Pla Wavs by a Impdac Srp Rado Scc, Vol, pp 55-6 [] Casro, L P; Kapaad, D, 8 Th mpdac boudary valu problm of dffraco by a srp Mah Appl, Vol 337, pp 3- [] Chakrabar, A, 977 Dffraco by a udrcoally coducg srp Id Pur Appl Mah, Vol 8, pp 7-77 [] Copso, E T, 96 A gral-quao mhod of solvg pla dffraco problms Procdgs of h Royal Socy A Mahmacal, Physcal ad Egrg Sccs, pp -8 [3] Ems, V; Rogowsk,, 5 Dffraco by a Srp wh Dffr Boudary Codos o Is Surfacs 6h Iraoal Cofrc o Compuaoal Problms of Elcrcal Egrg (CPEE), Lvv (Ukra), pp 39- [] Faulkr, T R, 965 Dffraco of a Elcromagc Pla-wav by a Mallc Srp Thčk glask, 3- (6),

18 A approxma soluo for h pla wav dffraco by a mpdac srp: H-polarao cas Ayd, E A; İk, T IMA oural of Appld Mahmacs, Vol, pp 9-63 [5] Fls, L B; Marcuv, N, 973 Radao ad Scard of Wavs Prc Hall, Nw rsy, p 883 [6] Grbrg, G A, 96 Dffraco of Elcromagc Wavs by Srp of F Wdh Sov Phys Doklad Vol, No 6, pp - 5 [7] Harrgo, R F, 968 Fld Compuao by Mom Mhod Macmlla, Nw York, p 85 [8] Hrma, M I; Volaks, L, 987 Hgh Frqucy Scarg by a Rssv Srp ad Exsos o Coducv ad Impdac Srps Rado Scc, Vol, No 3, pp [9] Hogo, K, 97 Dffraco of Elcromagc Pla Wav by a If Sl Embddd a Asoropc Plasma oural of Appld Physcs, Vol 3, No, pp [] Imra, A; Naqv, Q A; Hogo, K, 9 Dffraco of lcromagc pla wav by a fly log coducg srp o dlcrc slab Op Comm, Vol 8, pp 3 5 [] Imra, A; Naqv, Q A; Hogo, K, 7 Dffraco of lcromagc pla wav by a mpdac srp Prog I Elc Rs, Vol 75, pp [] os, D S, 96 Th Thory of Elcromagsm, Prgamo, Lodo [3] Kllr, B, 96 Op Soc Of Amrca, Vol 5, No, pp 6-3 (Kllr, B, 953) [] Kouyoumja, R G; Pahak, P H, 97 A Uform Gomrcal Thory of Dffraco for Edg a Prfc Coducg Surfac Proc Of h IEEE, Vol 6, No, pp 8-6 [5] Kobayash, K, 993 Th Wr-Hopf Tchqu ad Is Applcaos o Dffraco problms Ivolvg Two Dmsoal Obsacls wh F Cross-Sco Lcur Nos, p 5 [6] Lawr, B; Abrahams, I D, 7 A brf hsorcal prspcv of h Wr-Hopf chqu Eg Mah, Vol 59, pp [7] Lv, H; Schwgr,, 98 O h Thory of Elcromagc Wav Dffraco by a Aprur a If Pla Coducg Scr Comm Pur ad Appl Mah, Vol 3, pp [8] Maluhs, G D, 958 Excao, Rflco ad Emsso of Surfac Wavs from a Wdg wh gv Fac Impdac Sov Phys Doklad, Vol 3, No 3, pp [9] Mdgys, L N; Wag, D S, 989 Hybrd Soluos for Scarg from Prfcly Coducg Bods of Rvoluo IEEE Tras o Aas ad Propagao, Vol AP-3, No, pp [3] Mdgys, L N; Wag, D S, 989 Hybrd Mhods for Aalyss of Complx Scarrs Proc of h IEEE, Vol 77, No 5, pp [3] Mxr,, 97 Th Bhavor of Elcromagc Flds a Edgs IEEE Tras o A ad Prop Vol AP-, No, pp -6 (Mxr,, 99) [3] Mls, W, 99 O Cra Igral Equaos Dffraco Thory Vol 8, pp 3-6 [33] Mors, P M; Rubs, P, 938 Th Dffraco of Wavs by Rbbos ad Sls Phys Rv, Vol 5, pp [3] Nagasaka, T; Kobayash, K, 5 Wr-Hopf Aalyss of h Pla Wav Dffraco by a Th Maral Srp: h Cas of H-polarao Elcromagcs Advacd Applcaos (ICEAA) 5 Iraoal Cofrc, Tur, pp [35] Nobl, B, 958 Mhods Basd o h Wr- Hopf Tchqu Prgamo, Lodo [36] Nomura, Y; Kasura, S, 957 Dffraco of Elcromagc Wavs by Crcular Pla ad Crcular Hol Phys Soc apa, Vol, No, pp 85-3 [37] Osuk, T, 978 Dffraco by wo paralll sls pla Mah Phys, Vol 6, pp 9-95 [38] Prudkov, A P; Brckov, Y A; Marcv, O I, 983 Igrals ad Srs of Spcal Fucos Nauka Prss, Moskow, p 785 [39] Raylgh, L, 87 O h scarg of lgh by small parcls Phlosophcal Maga, srs, Vol, pp 7-5 [] Spgl, M R, 968 Mahmacal Hadbook of Formulas ad Tabls Mc-Graw Hll Compay, Nw York, p 7 [] Sahalos, N; Thl, G A, 98 O h Applcao of h GTD-MM Tchqu ad Is Lmaos IEEE Tras o A ad Proc, Vol AP-9, pp [] Sor, T B A, 95 Dffraco by a Sm- If Mallc Sh Proc of h Royal Socy, Vol 3, pp [3] Sor, T B A, 979 Backscarg from Rssv Srps IEEE Tras o A ad Proc, Vol AP-7, No 6, pp [] Srbs, A H; Büyükaksoy, A, 993 Som Approxma Mhods Rlad o h Dffraco by Srps ad Sls Aalycal ad Numrcal Mhods Elcromagc Wav Thory, Scc Hous, apa, pp 9-56 [5] Sommrfld, A, 896 Mahmasch Thor dr Dffraco Mah A 7, pp [6] Thl, G A; Nwhous, T H, 975 A Hybrd Tchqu for Combg Mom Mhod wh h Gomrcal Thory of Dffraco IEEE Tras o A ad Proc, Vol AP-3, a, pp 6-69 [7] Tbro, R; Kouyoumja, R G, 979 A Uform GTD Soluo for h Dffraco by Srps Illumad a Grag Icdc Rado Scc, Vol, No 6, pp [8] Vlv, E I; Kobayash, K; Ogaa, M, Koshkawa, S, 998 Dffraco by a Srp wh Dffr Surfac Impdacs: Th Cas of H Polarao VII Iraoal Cofrc o Mahmacal Mhods Elcromagc Thory, Vol, pp [9] Vlv, E I; Vrmy, V V, 993 Numrcal- Aalycal Approach for h Soluo o h Wav Scarg by Polygoal Cyldrs ad Fla Srp Srucurs M Hashmoo, M İdm, OA Tryakov Aalycal ad Numrcal Mhods 96 Tchcal oural, 3- (6), 79-97

19 Ayd, E A; İk, T Okvro rjšj a dfrakcju ravog vala mpdacjskom rakom: slučaj H-polaracj Elcromagc Wav Thory, Scc Hous, apa, pp 7-5 [5] Vlv, E I; Vrmy, V V; Shsopalov, V P, 989 Elcromagc Wav Dffraco o Polygoal Cyldrs-Nw Approach Proc of 989 URSI I Symp o Elcromagc Thory, p 68 [5] Vlv, E I; Kobayash, K; Ogaa, M, 998 Dffaco by a srp wh dffr surfac mpdacs: h cas of H polarao Procdgs of 998 Iraoal Cofrc o MMET, Kharkov, Ukra, pp [5] Volaks, L; Bdgaaval, S S, 996 Scarg by a Narrow Groov a Impdac Pla Rado Scc, Vol 3, No, pp -8 [53] Wadura, S, 99 Elcrcal Curr Bass Fucos for Curvd Surfacs Elcromagcs, Vol, Np, pp 77-9 Coac addrsss Em Avşar Ayd, Asssa Profssor Adaa Scc ad Tchology Uvrsy, Dparm of Aroaucs Egrg Syha/Adaa-TURKEY ayd@adaabudur / rasvam@gmalcom Turgu İk, Profssor Çukurova Uvrsy, Dparm of Elcrcal- Elcroc Egrg Adaa-TURKEY k@cudur Thčk glask, 3- (6),