Eğik geliş halinde düzlemsel dalganın üç parçalı rezistif ve kondüktif düzlemden kırınımı
Diffraction of abliquely incident plane waves by three-part resistive and conductive planes
Yazar: OSMAN YILDIRIM
Danışman: PROF.DR. ALİNUR BÜYÜKAKSOY
İstanbul Teknik Üniversitesi / Fen Bilimleri Enstitüsü
Derece: Doktora, 1994
Konu: Elektrik ve Elektronik Mühendisliği = Electrical and Electronics Engineering
Dizin: Düzlem dalgalar = Plane wavrs ; Mikrodalga iletişim = Microwave communication ; Spektral iterasyon tekniği = Spectral iteration technique

Özet:
ÖZET Bu çalışmada, rezistif, kondüktif ve empedans türünden sınır koşullarına sahip üç parçalı düzlemlerden eğik gelişli düzlemsel dalgaların kırını mı incelenmiştir. Yapılan analiz, "Spektral Iterasyon Tekniği" adı verilen yönteme dayanmaktadır. Bu yöntemin esası, herhangibir ayrıttan kırman alanı incelerken, bir önceki ayrıttan kırman alam gelen alan kabul etmek ve sözkonusu gelen alanın spektral gösterininim kullanarak olayı spektral domende çözmekten ibarettir. Bu yöntem uygulandığında, problem, hangi mertebeden kırınım incelenirse incelensin bir Wiener-Hopf proble mine indirgenir ve ayrıt koşullarının da yardımıyla sözkonusu Wiener-Hopf problemi kolayca çözülür. Tezde, bir ayrıttan ikinci ve üçüncü mertebeye kadar olan kırınımlar, rezistif ve kondüktif hallerde ayrı ayrı incelenmiştir. Sözkonusu incelemede, ardışık kırman alanların uniform asimptotik ifadeleri, yüzey dalgalarının muhtemel katkıları da gözönüne alınarak elde edilmiştir. Rezistif ve kondüktif hallerde elde edilen sonuçların uygun bir kombinasyonu yapılarak üç parçalı empedans düzlemi halindeki sonuçların da kolayca elde edileceği görülür. Elde edilen sonuçlar, sayısal olarak değerlendirilerek sözkonusu sonuçlar grafiklerle tartışılmıştır. iv

Abstract:
SUMMARY DIFFRACTION OF OBLIQUELY INCIDENT PLANE WAVES BY THREE-PART RESISTIVE AND CONDUCTIVE PLANES 1. Introduction The aim of this study is to develop an uniform high, frequency solu tion for the diffraction of obliquely incident plane waves by a three-part resistive plane characterized by resistive and conductive type boundary conditions, respectively. In both cases, for high frequency regime, par ticular attention will be given to the uniform evaluation of surface wave fields and their contributions to the double and triple diffraction. A three-part resistive plane, shown in Fig.l, is a suitable model for studying the scattering of electromagnetic waves in the presence of dis continuity in the material properties of a surface of finite width, or of composite structures on terestrial systems, aircrafts, etc. It is well-known that at high frequencies the total diffracted by a three-part plane can be evaluated as the sum of singly and multiply diffracted fields [14]. A part of the fields diffracted at one of the edges(junctions) which is illuminated by the obliquely incident plane wave propagates along the upper and lower faces of the plane and gives rise to secondary diffraction at the other edge. These waves, called the doubly diffracted fields, propagate back to the first edge to excite triply diffracted fields. In this study, higher-order multiple diffracted fields due to wave interaction between the two edges O and M up to and including third order is investigated. By using the traditio nal Geometrical Theory of Diffraction approach (GTD), singly diffracted fields from a two-part resistive or conductive planes can be evaluated ea sily by means of canonical solutions. However, in calculating the multiply diffracted fields from the edges 0 and M, GTD approach which consists of multiplying the single diffraction coefficients of successive edges fails due to the non-rayoptical behaviour of the interacting fields. That is why instead of the traditional Geometrical Theory of diffraction, a spectral iteration technique(SIT) will be employed to obtain the doubly and triply diffracted fields[14]. The Spectral Iteration Technique(SIT) developed by Büyükaksoy et all[14] consists essentially of taking the spectral(Fourier in tegral) representation of the n-tuply diffracted field by one discontinuity as the incident field for the two-part plane where the (n + l)th diffracti on will occur. Upon using the field components normal to the diffractingsurface, the (n + l)th diffracted field is obtained through the approximate solution of a pair of uncoupled Wiener-Hopf equations. There are two main objectives in this work. First one of them is to provide, via SIT, uniform diffraction coefficients for the double and triple diffraction process associated with a three-part resistive and a conductive planes, including the possible surface wave contributions, also in a uni form manner. The second is to demonstrate that a certain superposition of the solutions related to a three-part resistive and conductive planes is identical to the solution of a three-part impedance plane. It is well known that an electrically resistive surface is characterized by the relations [12] nAE\± = 0, (la) fİA(nAE) = -RrlAH\±, (16) and a magnetically conductive surface by the relations nAH\± = 0, (2a) rİA{nAH) = R*nAE\± (2b) where ft is the normal unit vector directed into the region (+), R(R*) is the resistivity (conductivity) of the resistive(conductive) surface. For example such a surface may represent a perfectly conducting plane coated with a non-magnetic thin dielectric. In such a case, the above mentioned surface resistivity and conductivity are given by (er-l)2Jfct ' = (nr - l)2kt (2C) Here Z0 = \j^f- is the intrinsic impedance of the free-space while er and t denote the relative dielectric permitivity and the thichness of the coating, respectively. For an impedance surface with equal face impedances one has [12] HA(fîAE±) = ±ZnAH± (3) where Z denotes the surface impedance. Again, it is known that a resis tive surface supports only electric currents, and a conductive surface only magnetic currents given by J = nAH\±,J* = -nAİ|î, (4) respectively; whereas an impedance surface is capable of supporting both currents. This gives the idea that, combined properly, the former two cur rent sheets can simulate an impedance. This "proper" combination can be easily shown to be [12] RR' = \;R=I;R' = ± (5) viassuming R,R* (hence Z) to be isotropic. 2. Statement of the Problem In order to obtain the diffracted fields up to the third order, the three-part plane shown in Fig.l is considered. It is assumed that the pla ne is illuminated by a obliquely incident electromagnetic plane wave whose electric field is parallel to the 0Z axis, namely; Fig.l. Geometry of the Diffraction Problem El ?M-ly Z0Hy ikz cos ö" -iK(x cos 0-\-y sin 4>o) (6a) (6b) Since the boundary conditions (la-3) are very similiar, the solution obta ined for any of them can be transformed easily to each other. For that reason, in the following, only the solution for the three-part resistive plane are given through the SIT. A) Secondary Diffraction By M Consider the geometry in Fig.l and assume that S\, S% and S3 are resistive planes whose resistivities are R\, R% and Rz, respectively. When Vllone illuminates this plane by an electromagnetic wave defined in (6a~b), a diffraction phenomena occurs at the junction 0(M). A surface wave due to this diffraction process occurs and propagates along the strip R-i and diffractes at the junction M(0). The analysis of this first diffraction are well-made in [14]. So, the attention will be given the derivation of the secondary and triply diffracted fields. An integral representation for the secondary diffracted field, say Uom and Vom can be written as follows; +00 Uom(x, y) = J A* "&)<&*"»-** du (7a) +00 VOM(x,y) = J B±M{v)e±i«''»-i'"du (76) - 00 where AqM(u) and BqM(v) are the spectral coefficients to be determined. Applying the following boundary, continiuty and edge conditions, a£Ş1±o) + w^-o)+ T T x<0 oy oy (8a) acq^+o) + a^-o) + ^ _ ^ =Qx>Q oy oy (86) - h-12-2 « - - = °> x e (-°°> +°°) (8c dy dy -[VT(x, +0) + VT(x, -0)] + -[ ^ ±-L] = 0, x < 0 (8d) 1 rTrT/ ". rrT, rtVI I TdVT(x,+Q) dVT(x,-Q), n (8e) yT(x,+0)-yT(a;,-0),a;e(-oo,+oo) (8/) U ~ O(^) (8 ) + Ugl (r, t/>) (9a) 77O) ( j,\ =^0L -if shifl0siin/> mG (j^008^) OMİrıV>) 2İ e ?72 + sinöoSin^?73G-(^,X:cosV') 0iKr [I^i-JCcos^+P^]^, V e (0,tt) (96) tU)( k_.a\. n(i)ie" "(,.), n =\/5 -»f sing0sinV> r)2G (^.£ 773G-(f,/Ccos^) [J^C-^COS^) +PÜ)]^:,^ ? (-7T,0) (9C) where (r, ?/>) are the polar coordinates associated with the junction M. Vom{^i i>) can be obtained easily by making the following substitutions in (9a-c) U ~^V, T)j -> -, Ji -> Ji By the same consideration, the secondary diffracted field from the junction O can be derived by substituting f?l- >î?3, 4> - ? 7T - (j), ^ - >? 7T - ^ in (9a-c). IXB) Triply Diffraction By O The secondary diffracted field by the junction M propagates along the strip and diffractes from the junction O. Following the same analysis done for the secondary diffraction. One can obtain the following results for the triply diffracted fields; where ^(1,1), rr(1.2), rr(2.1), rr(2,2) 'OMO OMO UoMO V"*," - V*1'1) 4- 1/(1'2) -I- V(2,1) 4- T/(2'2> VOMO - vOMO + VOMO + VOMO + VOMO (10a) (106) \/27r - i\ sinfl<,sin^ 2i V2G+(±,K cos j>) r(n,j) ri2,+am0"s'mtl) j?i G+(-±-,Kcosi/>) iKr (n,j)-\ "<».) (r 4, _ J K"'"(-K - V-) + Pr-"\ fcy, i> e (0, t)) uOMO\rıV)-\,-.".. "., G+(-±- Kcosib) y^F jz. sinfl"sin^ ^a* (£,*<*» *) 2i rj2 -sin 0" sin i> vı G+ (?£-,%. cos ij>) r(n>J)i ' V2 [i^'i-JCco^ + p^]^, v e (-7T,0)) (10c) / V^7rc-if sinflosin^ G+ (i? i,K cos ip) 2 l+»?2 sin 0« sin ^> '1 G+(î72,/Ccos ıjı) Vond)oM)=\ r(n>i) («,i)-i [^"."(-^coB^ + g^]^-, tf ? (0,*)) /Or' -?y/27rc-t^- sinfl0sin^ G+(»?i,/Ccos^>) 2 1- »72 sin 0o sin ^> '* G+(»72,/Ccos^>) [./^(-KcosVO + gM$£, tf e (-*,0)) (lOd) The solutions for the three-part conductive plane can be obtained easily by making the following substitution in (9a-b) and (lOa-b) 77 -*? \. For the three-part impedance case, the solutions are obtained by directly summing the results of resistive and conductive cases.