212 High-Frequency Scattering from a Wedge with Impedance Faces Illuminated by a Line Source, Part I: Diffraction ROBERTO TIBERIO, sexe wensen, et, GIUSEPPE PELOSI, wensea, ts, GIULIANO MANARA, MEMBER, IEEE, ano PRABHAKAR H. PATHAK, Fetiow, REE ohem of eating the seatered id 2 fave dstance from the edge of an inpedance medge which minted ly atin source is considered. The preentation ofthe ret dvd for thei tracted forthe directed Fld erections of incidence willbe examined In the ‘expt exis reciprocity with respect and satering, The surface wave comb Second part of this paper 1. Inrropucton Tit HIGH-FREQUENCY description of the searing from edges in nonperfectly conducting surfaces is of in- terest in a variety of applications. Surface impedance bound- ary conditions may provide a useful model for several practi- cal configurations. Hence, the scattering from a wedge with impedance faces is an important canonical problem inthe geo- metrical theory of diffraction (GTD) analysis of practical structures involving edges in impedance surfaces. An exact solution of this problem when the wedge i illuminated by a plane wave perpendicularly incident on its edge, was given by Maliuzhinets [1]. This solution is based on a generalized reflection method, which was recognized to be more general than other methods. Also, it was found useful to extend the Solution to treat some special cases ofthe more general prob- lem of the scattering of an electromagnetic plane wave at skew Incidence [2}-{8}. Manuscript ceived uly 22,1987; revaed March 23, 1989. This work was supported inp yh a Serv Escttons Progam unde Coc Noaoie-009 beter be Oar o Nal Rec and The Os Se Univesiy Reseach Foundation a npr y IDS Tngegne et Siem SpA "Tiberi swith he Deparment of Electonics Engineering. Universi atime ar S01 rc aye Ecce Laboratory: Deprinet of Ele Engineering, The Okie Sie Ushers. (Columban OF 2518 ar '. Peo swith he Deparment of letooisEaneing Unversity of ence, Via S. Mana, 330139 Fores aly ‘. Mevra with uty rnin ¢Telcomunicaion, Unive si of Pha, Vis Dasa, 2 36100 Ps ay "PH. Pabak with he Deparine of Ele Eagiering, The Ohio Ste Unverin.Cotumtus, OF 301. EEE Log Nebo asi, Dilferent procedures have been adopted to derive asymp- {otic high-frequency expressions from the above mentioned~ ‘exact integral representation ofthe scattered fied [2], {4 [5}, 7}-[11]. All these expressions apply to the case of a plane ‘wave illumination of the wedge, except for that in [5] where the high-frequency solution is heuristically extended to treat ‘other ray optical iltuminations, ‘This paper deals with the evaluation of the field scattered from 2 two-dimensional impedance wedge when itis illumi nated by a ray optical field with a curved rather than a plane ‘wavefront. In examining this problem, we found it useful 0 begin with a rigorous analysis of the simple canonical prob- Tem ofa wedge illuminated by a uniform line source. Our final goal isto obtain uniform high-frequency expressions for the total field in the special format of the uniform GTD (UTD) [12]. To this end, there are two important aspects to be dealt With. The first one is that of deriving, on a rigorous bass, a ‘uniform asymptotic solution forthe diffracted field which has the same structure as that for a perfectly conducting wedge, including the pertinent distance parameter which oecurs a5 a part of the argument of the transition function in the uni form solution, The second aspect is that of examining the effect of the wavefront curvature on the excitation of surface ‘waves, while retaining the continuity of the total asymptotic high-frequency field at their shadow boundaries. “The presentation ofthe results ofthis work has been divided ito two parts. Part I is concerned only with the calculation of the field diffracted into space by the edge. To this end, 8 double spectral integral representation for the total field is ‘used. This is rigorously obtained by applying reciprocity and superposition of plane wave spectra to the far-field response ‘of the wedge when i i illuminated by a plane wave; the later plane wave response is based on the Maliuzhinets solution. This integral representation yields exact explicit expressions forthe diffracted field and for the surface wave contributions. ‘This shows that three field contributions may arise when sur- face waves are excited. Next, & convenient reformulation of the integral expression for the diffracted field is asymptoti- cally evaluated to give the UTD solution. It is seem that this solution is the same as that which ean be obtained from a perfectly conducting wedge solution, by multiplying 1) frst ‘each ofthe four characteristic terms of the UTD formulation by simple factors which involve trigonometric functions, and. 2) then, this resulting expression by a factor which involves (0018-926X/89/0200-0212801.00 © 1989 IEEETIBEIO 0 MIL EREQUENCY SCATTERING PROM WEDGE ‘nz Fig. 1. Geometry for he sifstin st woe, the standard Maliuzhinets functions, This solution has useful symmetry properties and also satisfies reciprocity with respect to the directions of incidence and seatering. When the dis- lance of either the source or the observation point is assumed practically infinite, the present formulation may be regarded as an improvement of that given in {11} due to the addtional twseful symmetric nature ofthe solution; this aspect is of im Portance in the future generalization ofthe present solution to ‘even more general situations. The effect ofthe surface wave poles on the diffracted field is momentarily neglected. Ths subject wil be discussed in the second part ofthis paper, namely in Part Il: Surface Waves. [a Part I, the high-frequency expressions forthe surface wave tions are presented. Also, difracted wave contribu- tions related tothe surface wave poles inthe integral expres- sions forthe fields are obtained, which provide the continu ity ofthe (otal field at the shadow boundaries of the surface ‘waves. Its shown that this result is achieved by introducing dona ems ito the present slain forthe diffated field 1. Forsutation ‘The geometry for the problem of diffraction by @ wedge with surface impedance faces is shown in Fig. 1. The exterior wedge angle isn. A line source is located a 0 and the obser vation point at 0; their locations, in a polar coordinate system with its origin atthe edge Q, are denoted by (7, $') and (r, 4), respectively. By virtue of reciprocity which is satisfied by the present solution, the locations of source (0) and observer () may be interchanged, Two different uniform, isotropic, arbitrary impedance boundary conditions may be imposed on the two faces. The surface impedances of the 0 and mx faces are denoted by Zp and Zy, respectively; Zo, are complex numbers whose real part, because of energy considerations, ‘must be nonnegative The following notation will be used to describe the bound ary conditions for an electric (TM, e) and magnetic (TE, f) line source: Sin 8, = Ze/Zon, Sin Oy = ZanlZe in which Z, isthe free space impedance. Throughout the pa per superscripts e and h have been suppressed. The transverse magnetic (TM) and transverse electric (TE) cases are treated 213 together, and the expressions presented later apply to both ‘eases provided thatthe proper value for @" is used. In this paper, the total field 1 is represented as the sum ‘of a geometvical optics field u! and a diffacted field u! In adition, as will be shown later, three field contributions, wl, u!¥ and w, may exist when tee pertinent surface wave is excited. ‘The format ofthis paper i as follows. In Section ML, frst a convenient, symmetic expression forthe plane wave response ‘ofthe wedge in the far zon is found [1]. Next, the plane wave spectral integral of line source at is weighted by the above response ofthe wedge, ad the resulting expression is used to ‘weight the plane wave spectra integral of another line source at 0’. Thus, an exact double spectral integral representation for the field is obtained. This procedure enables one to introduce ‘pattern funetio both at 0 and 0". In Section 1V, the integral represcatation is asymptotically evaluated to obtain a UTD formulation forthe desired diffraction coefficient. To this end, 4 convenient variable transformation is introduced into the above integral representation, inorder to rewrite it in a form tore suitable for our purpose UL, rca. ResassexrariOn ‘An exact integral representation was given by Malivzhinets for the fel scared ata finite distance r from the edge of the wedge in Fig. 1, when i is illuminated by a unit plane wave (P77 ce). The diffracted field i! is described by an gral of complex variable defined along a tepest descent path, In the far zone limit (7 — 02), ths integral may be replaced by its value atthe sale pot. This expression is the far-field response of the wedge illuminated by a plane wave However, duc tothe symmetry of the problem we found it more desirable to modify that expression ito a form which is cexplcly symmetic in & and 6. For the sake of convenience, this expression is given next in terms of its pattern function PQ, 62,50 that ul = P(6,8') exp(—jkry/ Vier @ in which & isthe wavenumber ofthe medium surrounding the wedge, and an exp( at) time dependence is suppressed and understood. In (2) (Appendix 1) exp(-ia/4), P@.9) M4(6, 8°) where afb. 8°) = Walx/2/8Vne/2 — 6 ¥(nx/2 — 8°, (4)214 in which ¥ denotes the auxiliary function and Ys the spe- cial function introduced by Maliuzhinets [1]. Details for the derivation of (3) are given in Appendix I, together with some useful expressions and properties of ¥. Furthermore in (3) [H(*) C5") -oe(3]-a(3)m() satay) “C2 fo) (522) Before proceeding further, itis intresting to examine the behavior of the coefficients Q, Ay of the four cotangents in {G) at their pertinent shadow boundary (SB) and reflection boundary (RB) which are defined by: 6 — 6! = x (SB): +9" = x (RB)p when the O face of the wedge is illuminsted, and 6-6" = —1 (SB)q; 6 + 6° = Qn~1)r (RB), when the ‘rx face ofthe wedge is illuminated. As can be easily inferred from the relationships (31) and (28) given in Appendix I, Ante a) ,(6, 8°)An(-#,6") = 1 a) when d — $" = x, at (SB), (6, 6°)An(@, -6') = 1 (6) when 4 — 4° = —m, at (SB), and, denoting by To, the re fection coefficients ofthe O and nx faces ofthe wedge, 4(6,6')Ag(-8,-69) = —To (8) when @ + 6° = x, at (RBY, 2,(6,8°0An(6.6°) = Tn om when & + 6" = Qn ~ I), at (RB), Also, it ean be shown that when Zp = Z, = 0, expression (@) reduces to the known results for a perfectly conducting wedge illuminated by either a TE (sin @) = 0) or a TM (sin Gon = &) plane wave ‘The expression of the plane wave, far-field response of half-plane (n= 2) with equal impedance faces (sin 6 = sin 6; = sin 6) reduces to _exo-ir/4) P66) = ? 0506.69A-4.6 6.0") Te OPAL 9.89 ® in which 2 sin(a/2) sin(a’/2) ~ sin 8 Maas = 5 in a/2) sin(a/2) ¥ 5m 6 o ‘The above expression provides useful physical insight into the description of the high-frequency scattering phenomenon. It is seem that Q, Ay = 1 in (8) at any SB and RB; furthermore, A, = | for a perfectly conducting half-plane in both the ‘TE and TM cases. This suggests thatthe coefficient 0, Az is 4 well-behaved and slowly varying function of the incidence ‘and the observation aspects, and of the electrical properties ofthe surface. The plane wave far-field response given in (3) is now em- ployed to derive an integral representation of the field when ‘both the source 0” and the observation point 0 are located ata finite distance from the edge of the wedge (Fig. 1). To this end, the plane wave spectrum representation of two line sources, ransmiting at 0" and receiving a O, is used. For the sake of simplicity in the notation, but without loss of gener ality, the two line sources are assumed to be uniform with a unit amplitude so that their spectrum representation is given by (13) HQ = SPIED | exot-sho cosy adn (10 in which denotes Hin the TE cae and Ein the TM cie,p = 7, a= 6,1 = forthe line source at 0, and pn ras dea = Efrat a0, The contour of iegration Cis cite (je, ¥ + Jo) when @ = mx/2 or (— = Jom “ic) when @ = n3/2. Ths, the following exact inegral ‘Rpresetton is obttined forthe problem depicted a Fi. ni {J Pe 89 expt coset 0 treosté—ol}dyds, UD In order to derive a high-frequency approximation which is ‘the main purpose ofthis paper, itis desirable to deform the ‘contours of integration in (11) into two steepest descent paths (SDP) S(@) and S(¢') through the saddle points at and 6", respectively. For the sake of convenience in the explanation to follow, iis useful to employ the expression (26) (Appendix 1) for ¥s(x/2)/¥(nr/2 =n) in (L1) and to introduce the notation Main) exp {Jkol ~ cos(n ~ a)]} etna) a 2s ‘where y(n) is defined in terms of Yx in (27) of Appendix 1. So that, by using in (12) the same 9, c and p as in (10), and ou. a3) the integrand in (11) can be conveniently rewritten as AG) a(8)GE, £'), which puts explicly the surface wave PE EVM (EE,TIBERO ea: HIGH FREQUENCY SCATTERING PROM WEDGE Poles into trigonometric factors. Also it is useful to define ao = Re {Mon} + 2d {tm{Gn}) (14) in which gd denotes the Gudermann function. Is seen thatthe surface wave poles, ether 9 = no = —4 ory = 1m = Mx + Oy, are captured in the contour deforma tion process if ether < ag of a > ne ~ ay. Thus, after deforming the two contours of integration in (11) and then by transforming both (6) and S(6') into two SDP's $ and S” through a saddle point atthe origin, the following expression forthe total field w is obtained wee fe yh jeereo SBF OIG + OF +o dtaET + MO. ap téy 9) | a0” + 6 690Ep.F +6d8" + MO apalkin 9 | eh + 6.060 + 6.840 + nj EEp. OEE. IGE p, BIAS, ap IAL + expl—Jk(r! + DL where p = 0,1, and nsx Likolt ~ cos (ny ~a)]} G+ oY 2 Bp. Walp). (16) denotes the residue at the pole either 1p = 9 OF np = ne and : It is noted that expression (15) is symmettie with respect to O and 0’, as expected. Furthermore, (15) provides a useful and clear physical interpretation of the scattering phenomenon; it shows that when a line source is located at, is total field at 0 consists a four field contributions in addition to the GO field u®. Their behavior can be easily identified by considering the exponential factors in g and g. It is apparent that g is, associated with a space wave and g with a surface wave. Thus, itis seen thatthe second term in (15) is a field u! diffracted imo space due to a space wave incident atthe edge. The third term wis a surface wave (>, ¢) launched at 0 from the ledge by a space wave from 0°. The fourth term w behaves at Oas a space wave due to a surface wave 2(E;, ¢") incident atthe edge; its strength increases as 0’ approaches the edge. Finally, the fith term u’* behaves at O as a surface wave (Gp, &) due to a surface wave g(Gp, 6”). OF course, the last thee contributions may or may not exist according 0 (17). ‘A suitable asymptotic approximation of the second term is for p = Ova < ag: Ha) pe na>ne-a, (1) otherwise. 215 ‘resented in the next section, In evaluating the diffracted field contribution, the GO poles which may occur in a neighbor: hood of the’ saddle points are properly accounted for. The effect of surface wave poles that may cross the paths of inte- ration, is momentarily neglected, This problem will be con- sidered in the second part of this paper as mentioned earlier. ‘There, high-frequency expressions forthe surface wave con- \wibutions from the third and fourth terms in (15) will be pre- sented. Also, the surface wave pole effect will be included imo the solution forthe diffracted fel. IV. Tus UTD Sowwtios ‘The purpose of this section isto derive a uniform asymp- totic high-frequency expression forthe diffracted field win the UTD format. To this end, consider the second term in (15). Expression (3) suggests & representation of the pattern function P(g, $°) as a sum of four p, (i = 1,4) terms, so that =D <8) with f= ( [pores +9 + expl=Jk(r cos f+ 1 cos F)Ndbdg". 19) Due to the form of the exponential factor therein, (19) is not suitable fora direct asymptotic approximation. Its shown in ‘Appendix II that by applying a sequence of convenient variable transformations, an alternate exact integral representation for vf = expt —aker + of” | pee expl-k VF +r? + 2rr’ cosh x cosh yldydx (20) in which x, y are real variables, and Pile.y) = PAL + OS + OOM). AN In QI), the functional dependence off, t” on x,y is implicit, and 1 (2 2) 1 sin? (9/2) It is important to note that the validity of (20) is not af- fected by the presence of complex, surface wave poles, and the GO pole is independent of the y variable. Assuming that (7 + 7) is lange, a first-order saddle point asymp- {otic approximation on the y axis, for the large parameter K(P #7? + 2’ cosh 2)", yields uf ~ Vin expl-jktr +r) a Bil 0) ne VERE PES Br cosh 8) ste ds @)216 Next, (23) is asymptotically evaluated via the modified Puuli-Clemmow method of the steepest descent (14). Retai ing the frst nonvanishing term, of — SHE 9g, vee a Ver POO ae exp(ikr) ir co in which Fis the UTD transition function and a, denotes the pertinent a*(6 +6") in [12, eg. 27) Finally, the desired high-frequency solution is obtained by insering (24) imo (18). This solution is valid for both an electric and 2 magnetic line source illumination of the wedge, provided thatthe proper value of sin 8s i introduced therein. Also itis symmetric with respect to (", 6°) and (F, @) as expected iF thas to satisfy reciprocity. It is worth pointing out that when the observation point lies on a face of the wedge, a vanishing field is obtained from the present solution except for the case of a perfectly conducting face with # magnetic line source illumination ofthe wedge. Such a result i to be expected since only the lading term of order A-"? is retained in (24). However, a more accurate evaluation ofthis field is easily obtained by directly introducing the present formulation into the solution given in [15], [11}; there the first nonvanishing term is retained to evaluate the diffracted field coateibution on the impedance surface. V. Discussion ‘The two-dimensional problem of the diffraction from the edge of a wedge with surface impedance faces has been con- sidered, when both the source and the observation points are located at a finite distance from the edge. The plane wave, far-field response of the wedge together with the plane wave spectra of two line sources, have been used to derive an ex- act integral representation forthe total field, including surface ‘wave contributions. ‘A UTD solution forthe diffracted field has been obtained. ‘The formulation adopted in this paper exhibits a desirable, ex- plicit symmetry with respect tothe incidence and the observa- tion aspects. This rigorous asymptotic analysis demonstrates that the UTD formulation is directly applicable to extend the plane wave, far-field response of the wedge to teat the con- figuration considered here. High-frequency expressions forthe surface wave contribu- tions will be presented in the second part of this paper. In asymptotically evaluating the diffracted field contribution, sur- face wave poles have been neglected. However, in the second Dart of this paper it will be shown that they ean be properly ‘accounted for by simply introducing the relevant additional terms into the present solution, Arrenons I AA direct asymptotic evaluation of the expression for the diffracted field given by Maluzhinets in (1) yields exp(—i/8 ss —(¥ine/2 8-2) Viewer? % “on(@!, + 6) —Wina/2 ~ 6 + )on(6',x -6)]. 25) In (25) ¥ is expressed in terms of the special function Yq introduced by Maliuzhinets [1] as Yne/2 0) = Valx/2) Boome) 26) ‘ies with Yalnrr ~ 9 ~ 2/2 ~ OoWbaln ~ ¥/2~ On) Yul = 9 = 72 + Bo Wdal— #/2 + Hp) It is useful to remind that ¥ satisfies the fundamental rela- tionship Max/2 ~ 2px +0) Yorn — 9) where Ty denotes the reflection coefficient Py of the O fa and T, of the mr face of the wedge. Furthermore, in 25) sing) eostaa/n) ~ costar) = fe(25")-«( Inthe plane wave, far-field limit, (25) is an exact solution ‘Thus, if it has to satisfy reciprocity, the required symmetry With respect to ¢ and ¢ must be implicitly embedded into the ¥ functions. However, itis desirable to have an expression which is explcily symmetric. This may be particularly useful when such expression is used in an integral representation Which has to be asymptotically approximated. To this end, it can be shown that a convenient application of the recurrence property Yala + #/2W ala ~ 4/2) = V(H/2) cos (/2n) Yields the relationship VEC /2) { en( Ror -9) Meala) = en (p= 0,n) 28) Toe rnan(niem) = G0) Yor 6 =n) = an Using (1) in (25) puts the ¥ functions into the expression Sy given in (4), which is explicitly symmetric with respect to @ and 6". Consider now the resulting expression for P(@, $')/Mx(6, 1°) which involves only simple trigonometric functions. In order to render it symmetric, itis convenient to rewrite the term in brackets of (1) as (1 fo +efin(2#) ain(2)] sa) (Sm