New analytical Methods for wedge problems

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1 Trin, taly, September 0-4,00,pp SBN Ne analytical Methds fr edge prblems V.G.Daniele ( ( Dipartiment Elettrnica, Plitecnic di Trin c.s Duca degli Abruzzi 4, 09 Trin (taly daniele@plit.it Abstract T ne analytical methds fr slving diffractin prblems in angular shaped regins are presented. n the first e intrduce the rtating aves in a Laplace dmain; in the secnd the Wiener-Hpf technique is generalized fr dealing edge prblems f arbitrary angles. Bth the methds rk fr impenetrable edge and penetrable edges. The rtating aves methd cnstitutes an alternative and pssible mre simple methd f that f Malyuzhinets. Beside the thery, e ill deal applicatins cncerning the diffractin by isrefractive (r diaphaneus edges. The W-H technique is the mst perful methd fr dealing fields discntinuity prblems. A generalizatin f this technique slves ith success edges f arbitrary angles. We ill present this generalizatin and sme applicatins that cncern impenetrable edges.. NTRODUCTON There are many analytical methds fr studying fields and aves in angular regins. The classic nes based n the eigenfunctin expansins are described in the masterly bk f Felsen and Marcuvitz []. Unfrtunately except particular cases these methds cannt be applied t surface impedance edge prblems. Amng the specialized techniques it is ell knn the Maliuzhinets Methd []. This methd is based n the Smmerfeld integral and it is very perful fr slving diffractin prblems in edge shaped regins. ts ppularity and imprtance has increased in the curse f time and it is ell dcumented by the large number f papers recently published. The Maliuzhinets Methd is nt the nly specialized methd fr this class f prblems. Alternatively there are the t methds f Peters [3] and Williams [4]. The Peters methd is very ingenius. Hever it is based n sphisticate analytical cntinuatins that make it very invlved and difficult t apply. Even if Senir [5] used this methd fr slving the imprtant prblem f imperfectly cnducting edges, n it is frgtten. Als the technique used by Williams is ingenius but it seems specialized t slve nly the imperfectly cnducting edge prblem. Other mre general methds are based n the Wiener- Hpf techniques and the Kntrvich-Lebedev representatins. The Wiener-Hpf technique is the mre imprtant methd fr slving diffractin prblems in presence f gemetrical discntinuities. Hever there is the cnvictin that, except fr particular angles, the edge prblems cannt be studied ith Wiener-Hpf equatins. The Kntrvich-Lebedev representatins has been used in the past fr slving the PEC edge and recently als the isrefractive edges [6]. This representatin may have sme limitatins and beside, it is clsed related t Smmerfeld integral [7] and the Furier integral[ ]. Of curse every f the previus methds presents advantage and disadvantages. T this authr s pinin the mre perful methds are thse based n the Wiener Hpf techniques. Thus, a first mtivatin f his rk as that f generalize this technique fr dealing edges f arbitrary angles [8a,b]. Hever this authr encuntered sme difficulties hen, after applied this ne methd n specific prblem, he tried t cmpare the Wiener-Hpf results ith the nes btained by the Maliuzhinets methd. These difficulties derive by the use f t different spectral representatins: unilateral Furier Transfrm (r Laplace transfrm in the Wiener- Hpf technique and Smmerfeld functins in the Malyuzhinets methd. Mrever hereas Smmerfeld integrals intrduces the cmplex angular spectrum, the Furier integrals intrduce cmplex ave numbers η. n rder t vercme the difficulties f the cmparisn, this authr matured the cnvictin that, fr his purpses, it uld have been mre cnvenient t refrmulate the

2 apprach in the angular spectrum, by aviding frm beginning Smmerfeld integrals. This can be accmplished ith the intrductin f the cncept f the rtating aves. Since he believes it is interesting fr further understanding f the ave mtin in angular regins, the secnd mtivatin f this paper is the expsitin f this refrmulatin. B. Prperties f the rtating aves[9] Prperty a: n absence f surces the analytical functins v ( and v ( are regular respectively in the strips {ϕ Re[] ϕ } and {-ϕ Re[] -ϕ }. ROTATNG WAVES METHOD A. Rtating spectral aves. Let cnsider a t-dimensinal electrmagnetic field E z (ρ,, H ρ (ρ,, H ϕ (ρ, in the arbitrary ithut surce angular regin 0 ρ<, ϕ ϕ ϕ, see Fig.. Fig.. Regularity regin f v ( (//// and v ( (\\\\\ Prperty b: The functins v ( and v ( are related in all the cmplex plane by the reflectin prperty: v ( -v (- (7 Prperty c: The functins v ( and v (- are bunded as ± j fr every ϕ: ϕ ϕ ϕ. This bunded values d nt depend n ϕ : v (± j ϕ v (± j v (± j - ϕ v (± j (8 Fig.. angular regin 0 ρ<, ϕ ϕ ϕ Peters ([3], [9] shed this imprtant result: the functin: v, k sin[ ] V ( η, ( ( ϕ η k cs( here V (η, is the radial Laplace transfrm : s [ E ρ ϕ ] ρ ϕ ρ z (, Ez (, e dρ ] s jη V ( η, L 0 ( satisfies the equatin: v (, ϕ v(, 0 ϕ (3 The mre general slutin f eq.(3 is: v(, v (- v ( (4 We define v ( (clckise and v ( (cunterclckise the rtating aves in the angular regin. Anther imprtant results [9] is that the Laplace transfrm f the radial cmpnent H ρ (ρ, f the magnetic field: i (, ϕ k ( η, kl[ H ρ ( ρ, ] (5 η k cs[ ] is expressed in terms f rtating aves in the frm: i(, Y {v (- v (-} (6 ith Y being the admittance f the medium filling the angular regin. t flls that the rtating aves express the frard and the backard traveling aves f an unifrm angular transmissin line here the rle f the time is assumed by the cmplex variable. Prperty d: f the lngitudinal field satisfies near the edge ρ0 the cnditin E z (ρ, O(ρ c ith Re[c]>0, the rtating aves behave as v, (O(exp(-c m[] fr ± j. Prperty e: When the Smmerfeld integral des exist, the Smmerfeld functin s( is related t the clckise ave v ( thrugh the equatin: v ( - j s( (9 C. Presence f incident plane aves Lets cnsider an incident ave plane (Fig.3 i jk ρ cs( ϕ ϕ Ez ( ρ, Ee (0 i k jkρ cs( ϕ ϕ H ρ ( ρ, sin( ϕ e E ω µ The Laplace transfrms ( and (5 and eq.s (, (4 and (5 evaluated fr ϕ0, yield the flling rtating aves relevant t the plane ave: ( sin sinϕ i j v[ ] E (cs csϕ i j( sin sinϕ v[ ] E (cs csϕ ( Fr the presence f a far surce at the directin ϕ (Fig.3, e bserve in the regularity strip (clckise ave the ple ϕ and in the regularity strip (cunterclckise ave the ple -ϕ. Hever als fr these rtating aves prperties b and c hld again. n the flling e ill call ith v i ( and v i ( the incident rtating aves.

3 D. Presence f reflected, surface, leaky aves. The hmgeneus regin cnsidered in fig. may be bunded by ther hmgeneus r nt hmgeneus regins. n their interfaces many kind f aves can be arisen []. Frm a mathematical pint f vie these aves cnstitute ples r branch pints in the η-plane. The mapping η-k cs induces n the -plane an infinite number f images f these pints. The flling therem [9] has been shed : The functins x, s ( defined by: x, s ( x, (- x, i ( ( are alays regular in the relevant regularity strips and. This therem if very imprtant. Fr instance, it implies that the (infinite ples relevant t pssible reflected aves are all suited utside the strips and. n the flling e call scattered the rtating aves x, s (. Anther cnsequence f the therem shed in [9] is that hen the surces are plane aves and the angular regins are bunded by impenetrable edges (see fr.example Fig.3, the scattered rtating aves are mermrphic functins f. E.Frmulatin f the Maluyzhinets prblem in terms f rtating aves Fig.3. mpedance edge prblem The bundary cnditins n an impedance edge are defined by: E z ± Z ± H ρ in ϕ ± Φ. ( here the surface impedances Z ± depends n the edge material (Re.Z ± 0. ntrducing the Laplace transfrms, the eq.s (, in the -dmain, have the frm: -v(,±φ k sin V (η, ±φ ±k sin Z ± (η, ±φ± sin Z ± i(, ±φ (3 n terms f rtating ave the eq.s (3 becme: v(± Φ v (ϒ Φ ϒ[sin /sin θ ± ] [v (± Φ v (ϒ Φ ] (4 ith Z ± Z /sin[θ ± ], 0 Re.θ ± / Taking int accunt the prperties b and e, the eq.s (4 are identical t thse fund by Malyuzhinets fr the Smmerfeld functins s[]. We can use the Malyuzhinets methd fr slving these equatins [] and btain: E cs[ ϕ / n] Ψ( v( j (5 n sin[ / n] sin[ ϕ / n] Ψ( ϕ here : Ψ ΨΦ ( Φ θ ΨΦ ( Φ θ ΨΦ ( Φ θ ΨΦ ( Φ θ ith Ψ Φ ( being the Malyuzhinets functin. F. Slutin f the functinal equatins Like the Maliuzhinets methd, the rtating aves lead t difference equatin ith cnstant r nt cnstant cefficients. As indicated by Maliuzhinets, it turns useful in bth the cases, t intrduce Furier Transfrm representatins. Fr instance hen the cefficients are cnstant, e defines fr Re.t0 and φ ϕ φ the flling mdified Furier transfrms f the rtating aves: j [ v ± ], ( j V t, ϕ ] j exp[ jt] d (5,[ ϕ t shuld be bserved that in general the rtating aves may be bunded but nt vanishing fr ±j. Cnsequently e must define these Furier transfrms in the distributin space. The regularity f the scattered aves yields the flling transprt therem [9] s s V, [ t, ϕ b ] exp[ m jt( ϕ b ϕ a ] V, [ t, ϕ a ] (6 here φ ϕ a,b φ. Taking int accunt this therem, the difference equatins in the - dmain yield an algebraic system in the t-dmain: P(t X(tN (t, Re.t0 (7 here P(t and N (t are knn and the Furier transfrm f the rtating aves X(t is a tempered distributin vectr defined n Re.t0. Slving the algebraic system (7 and perfrming the inverse transfrm in the distributin space yields the final frmula: x( j x( j j j { } jt( ± ϕ x( P ( t N ( t e dt (8 j here the integral is understd t be a Cauchy principal value integral. F. Scattering f a ave plane by isrefractive edges The prblem f diffractin f electrmagnetic aves by a penetrable edge has prduced a multitude f studies, and yet the prblem is unslved. n rder t have mre insights n the behaviur f penetrable edges it is cnvenient t study the prblem f the diffractin f isrefractive r diaphanus edge. The imprtance f this prblem is due t the fact that it cnstitutes a dynamical penetrable edge prblem that e can slve in clsed frm. The slutin f isrefractive edges has been accmplished in the past by using the Kntrvich-Lebedev transfrm [6] in the frequency dmain and the Green functin in the time dmain [0]. Wiener-Hpf slutin fr the right edge are als available []. n this sectin e slve this prblem by

4 using the rtating aves methd. This apprach has t advantages. t can deal ith an arbitrary number f isrefractive edges and it gives the slutin in a frm mre suitable fr evaluating bth the near field and bth the diffractin cefficients f the prblem. Fig.4 shs the gemetry f the prblem. We have isrefractive angular regins and excited by an E- plane ave (ith intensity E plarized in the z- directin. Let s intrduce the rtating aves v q and v q here the superscript q, indicates the relevant regin. By impsing the bundary cnditins at the three interfaces ϕ0 r ϕ, ϕγ the flling system f linear difference equatins is btained: Fig.4 : scattering by an isrefractive edge -nterface ϕ0 r ϕ Electrical field matching: v v v ( v ( (9 Magnetic field matching Y ( v v( Y ( v ( v ( (0 -nterface ϕγ Electrical field matching: v ( γ v ( γ v ( γ v ( γ ( Magnetic field matching Y ( v ( γ v ( γ Y ( v ( γ v ( ( γ Applying t the previus equatins the Furier transfrms (5 and taking int accunt the transprt therem (6 yield algebraic equatins invlving these Furier transfrms. The slutin f the system has been accmplished by using the prgram MATHEMATCA. By indicating ith V (t the Furier transfrm f the ttal rtating ave v (, it yields [9]: (3 ith γ - γ The inverse transfrming f eq.(3 is generally invlved. t has been accmplished explicitly fr the sme selected cases by using the residue therem [9]. Fr the right edge (γ 3/ e btain [9]: here: Y 4Y Y 4Y Y 4Y Y 3Y c Arc tan[ ] Y 6Y Y Y ϕ i ϕ -, j (4 The asympttic behavir fr ±j is given by: v [] O(e - m. (-c t yields the near edge behavir: E z O(ρ -c, H r O(ρ -c This behavir is in accrd ith the static behavir f a penetrable right edge []. This expressin (4 cincides exactly ith the Wiener- Hpf slutin btained in []. Near and far field discussins have been reprted in []. G.Cnclusins t can be shed that the thery f rtating aves may be successful applied t all the prblems apprached by the Malyuzhinets methd [9]. Cmparing the t methds e ascertain many deep-ruted analgies. Hever t this authr s pinin, the rtating aves present peculiar prperties that simplify and make reliable their use fr studying ave prblems in angular regins. Fr instance this authr claims the flling facts must be appreciated: -the rtating aves are based n the intrductin f Laplace transfrms and nt n the smetime nt valid ansantz cnstituted by the Smmerfeld integral. - the Malyuzhinets nullificatin therem [] has been avid. Smetime this therem may be misinterpreted. -We safely deal the rtating aves in mre angular regins. nstead e must be very careful if e used Malyuzhinets methd in angular regins nt defined by - φ ϕ φ. -We can apply ith trust the Furier technique ideated by Malyuzhinets fr slving difference equatins since the transprt therem is alays valid fr the scattered rtating aves. n additin the distributin space fr slving the system in the Furier dmain assures the existence f the invlved Furier transfrms and integrals. -Many difficulties present in the Maliuzhinets methd are vercame hen e use the rtating aves. Fr instance in the sectin F e have slved (yet ith many different angular regins ave prblems invlving

5 isrefractive regins. Up t n these prblems have nt been slved by the Maliuzhinets methd. - Unifrm transmissin lines can be intrduced fr frmulate edge shaped prblems. - Rtating aves yield expressins that can be immediately cmpared ith the Wiener-Hpf results. The presence f nt isrefractive penetrable regins in general leads t very cmplicated functinal equatins invlving rtating aves r Smmerfeld functins. Even if in the past sme ingenius techniques have been ideated [] fr dealing particular cases, in this authr s pinin these gemetries can be better studied ith generalized Wiener-Hpf techniques that ill be presented in the next sectin.. WENER-HOPF GENERALZED TECHNQUE A. Mathematical ntrductin The mre simple system f generalized W-H equatins, is defined by: G(ηF (η F - (m(η F i (η (5 here F (η and F - (m(η are respectively a plus functin in the η plane (i.e. regular in half plane m[η] 0 and a minus functin in the m plane (i.e. regular in half plane m[m] 0 and F (η is a knn functin. f m[η]η, eq.s ( becme classic W-H equatins. Even ith m[η] η, eq.s (5 cnstitute a clsed mathematical prblem [8a]. Fr their slutin e ill used the same ideas intrduced in the Wiener-Hpf technique. We remember that slutin f the classic Wiener-Hpf equatins is based n the decmpsitin- factrizatin f a generic η- functin in functins that are plus and minus (i.e. regular respectively in the half plane m[η] 0 and m[η] 0. Als fr the eq.(5, the cncepts f decmpsitin and factrizatin cnstitute the key fr their slutins. Hever hereas the classic decmpsitin is explicitly expressed by the Cauchy frmulas, the generalized decmpsitin is mre difficult t btain and leads t integral equatins. - Generalized decmpsitin The generalized decmpsitin f a functin F(η (ith respect m(η is defined by: F(η F (ηf - (m(η (6 here the functins F (η and F - (m(η are respectively a plus functin in the η plane(i.e. regular in the half plane m[η] 0 and a minus functin in the m plane(i.e. regular in the half plane m[m] 0. The prblem f generalized decmpsitin is mathematical funded since it can be reduced t a Fredhlm integral equatin f secnd kind [8]: (7 F ( η j dm F ( d m( m( η d η j γ F( dm dη m( m( η d here γ runs n the real axis η leaving abve the singular pint η. Of curse the classic case m[η]η yields the explicit Cauchy expressin: F ( η j γ F( d η (8 - Generalized factrizatin The generalized factrizatin prblem f the kernel G(η is defined by the factrizatin: G(η G - (mg (η (7 here the factrs G (η and G - (m (and their inverses are respectively a plus functin in the η plane and a minus functin in the m plane. As in the classic case, e can accmplish the generalized factrizatin f a scalar G(η using the cncept f lgarithmic decmpsitin: lg[ G ( η] ψ ( m ψ ( η G( η e e G( m G ( η (8 here the functins ψ - (m and ψ (η fll frm the generalized decmpsitin: lg[g(η] ψ - (m ψ (η (9 Taking int accunt the prperty: ψ ( m ψ ( η ψ ( m ψ ( η e e e (30 e btain the generalized factrized matrices in the frm: ψ ( m ψ ( η G ( m e, G ( η e (3 The generalized (and classic matrix factrizatin prblem is instead cnsiderable mre difficult. n [8a] several particular techniques are cnsidered that apply the same expedients used fr the classic factrizatin prblems. Fr instance e are able t factrize (in the generalized sense ratinal matrices f η. Having in mind Pade representatins f arbitrary matrices this generalized factrizatin is very imprtant. - Slutin f generalized W-H equatins The slutin f generalized W-H equatins can be btained accrding t the same prcedure used fr the slutin f classic W-H equatins. First the generalized factrizatin f G(η leads t the flling equatin: (G - (m - F i (ηg (η F (η-g - (m - F - (m N decmpsing the first member G - (m - F i (η S (η S - (m yields : 0S (η - G (η F (η S - (m(g - (m - F - (m The previus equatin can be interpreted as a generalized decmpsitin prblems. Since the functin t be decmpsed is 0, the integral equatin (7 is hmgeneus. Taking int accunt that the decmpsed functins are vanishing at infinite, the slutin f the hmgeneus integral equatin is zer and cnsequently bth the decmpsed functins must vanish. t yields the slutins: F (η (G (η - S (η, F - (m- G - (ms - (m (3

6 B.Frmulatin f the Maluyzhinets prblem in terms f generalized W-H equatins. The generalized W-H frmulatin fr edge prblems needs the intrductin f blique Cartesian crdinates. Fr cmparing ith the rtating aves results, e cnsider again the Fig.3. n the regin defined by 0 ϕ φ the blique Cartesian crdinates are defined by X and Y (Fig.3: y X x y ct Φ, Y (33 sin Φ With these crdinates the ave equatin becmes [8a]: E z E z Ez cs Φ k sin Φ E 0 (34 z X Y X Y Starting by this equatin e may generalize the prcedure used in [] fr deducing Wiener-Hpf equatin fr right edges. This prcedure is lng and difficult. t requires physical and mathematical tls such that equivalence principles and characteristic Greens functin prcedures []. t has been accmplished in [8a] and yields the flling functinal equatin: (35 ξ V ( η ωµ ( η ( ξ cs Φ η sin Φ V ( m( η, Φ ωµ ( m( η, Φ here: m ( η csφη sin Φξ ξ k η, ξ η fr η 0 (36 and V ( η V ( η,0, ( η ( η,0 V ( η, L[ E z ( ρ, ] s jη ( η, L[ ρ ( ρ, ] s jη (37 are the Laplace transfrms cnsidered befre (eq.s ( and (5. Flling the same prcedure fr the regin - φ ϕ 0, yields: (38 ξ V ( η ωµ ( η ( ξ cs Φ η sin Φ V ( m( η, Φ ωµ ( m(, Φ η By impsing the bundary cnditins (, Eq.s (35 and (38 yield the generalized W-H equatin: V ( η ( m( η, Φ M ( η (39 ( η ( m( η, Φ here the matrix M(η is given by: ξ ωµ ( ξ csφ η csφ Z ωµ ( ξ csφ η csφ Z ωµ M ( η ξ ωµ ( ξ csφ η csφ Z ωµ ( ξ csφ η csφ Z ωµ Fr PEC edges (Z ± 0 the vectr prblems simplify t the scalar generalized W-H equatin: ξ V ( η ( m, Φ ( m, Φ (40 ωµ We ill slve this equatin after sme cnsideratins f the mapping ηk cs( that relates the spectral variable η used in W-H techniques ith the cmplex angular used in the the Smmerfeld-Malyuzhinets,- rtating aves dmain. C. The mapping η k cs( There are t reasns fr intrducing the mapping: η k cs(a (4 The first is related t the cmparisn beteen the W-H and the rtating aves results. The secnd is fr facilitating functin theretic manipulatins that may be bscure in the η-plane. Fr reasns explained later, the parameter a ill be assumed alays. A cmplete study f this mapping is reprted in [8a]. Firstly the cncept f prper sheet and prper slutin must be adequately cnsidered. The t branches f ξ crrespnd t the t sheets f the η-plane cut by the branch lines relevant t the branch pints ±k. The pints η f the prper sheet arisen frm the chse indicated by eq.(36 (this is a cmpulsry chice dictated by the physical existence f the Green functin. t flls m[ξ]<0 in all the real axis f the prper sheet. We call prper slutin f (39, that btained in the prper sheet. The prper slutins V (η and (η are plus functins t. t means that they have nly the branch line ηk. Fr instance if e cut the branch line crrespnding t η-k, the variable ξ changes f sign ith discntinuity,, but the plus functins hld cntinuus. Given η, the inverse f mapping (4 intrduces infinite values f. The principal value crrespnds t the principal value f the Arcs[x]. n the flling e refer alays t principal values. With this mapping (4 bth the prper and imprper sheets f the η-plane lie dn the -plane. Fr instance e can bserve that the segment f the -real axis defined by: - 0 is the image f the segment ( k,k belnging t the prper sheet hereas the segment f the real axis defined by - - is the image f the segment ( k,k belnging t the imprper sheet. t be shuld als bserved that n the images f the prper sheet: ξ-k sin (4 Since all the -plane is image f the prper, f the imprper and f ther infinite sheets arising frm the inverse f the mapping (4, e must be very careful hen e ill rerite in the η-plane equatins that have been btained in the -plane,. Fr example t deduce the eq. (38 and (35 frm the rtating aves thery, e can use the equatins that relate the rtating aves ith the Laplace transfrms: jk sin V ( k cs, ω µ ( k cs, v ( jk sin V ( k cs, ω µ ( k cs, v ( (43

7 Hever the necessity t deal the prper branch f ξ in the η-plane makes valid these equatins nly n the images f the prper sheet. t means that fr ϕφ, e can use nly the secnd equatin, hereas fr ϕ-φ, e can use nly the first equatin. t flls : jk V k Φ ω µ k Φ sin ( cs, ( cs, v ( Φ ( jk sin V ( k cs, Φ ω µ ( k cs, Φ v Φ and putting -φ : jk sin V ( k cs,0 ω µ ( k cs,0 v jk sin( Φ V ( k cs( Φ, Φ ω µ ( k cs( jk sin V ( k cs,0 µ ( k cs,0 v jk sin( Φ V ( k cs( Φ, Φ ω µ ( k cs( Φ, Φ ω (44 Φ, Φ Eq.s (44 are exactly the eq.s (38 and (35. Starting ith them, e can sh the cmplete equivalence f the eq.s (39 and the eq.s (3,4 [8a]. Cnversely e cannt assume fr all the pints f the -plane, prperties that are valid nly fr the images f the prper sheet. Fr instance the functin ξ in the - plane is expressed by: ξ k η k ( cs k sin At the same time, it lks an even functin f (secnd member and an dd functin f (third member. The reasn f this paradx is due t the fact that the secnd member is valid nly fr values f that are images f prper sheet. Changing in e instead g int the imprper sheet. Of curse it turns very useful deal functins f defined in all the -plane. We can alays d it thrugh analytical cntinuatin. Hever e must remember that fr arbitrary values f, the analytical cntinuatin may invlve imprper value di ξ. t shuld be remarked that the W-H equatins refrmulated in the -plane may yield difference equatins. The prcedure fr ding it is based n the eliminatin f the minus (r alternatively the plus functins thrugh the intrductin f a further mapping beteen cmplex planes. t may be very invlved and it is described in [8a]. The decmpsitin-factrizatin and the slutin f difference equatin cnstitute then t different aspects fr slving diffractin prblem in angular regins. Hever hereas the decmpsitin-factrizatin cnstitutes a clsed mathematical prblem, difference equatins instead generally invlve many slutins. Additinal cnditins must be taken int accunt fr btaining the real ne. Fr instance if these equatins invlve rtating aves r Smmerfeld functins e must cnsider slutins f the difference equatins regular in suitable strips f the -plane (fig.. The plus and then minus functins instead have mre stringent prperties in the -plane. Fr instance hen e are dealing slutins f eq.(39, the mapping (4 carried ut n plus functins f η yields functin f that are even and regular in 0. Cnversely -even functins regular fr 0, crrespnds t plus functins in the η-plane [8a]. This prperty justifies since e chse a. Other chices f the parameter a yield prperties f plusfunctins less cnvenient in the -plane. Minus functin may be cnsidered as plus functin evaluate in: -η. Cnsequently their images in the - plane have the prperty f being invariant hen e change ith - -. When there are mre slutins f the difference equatins relevant t plus r minus functins, e must ascertain thse satisfying the previus cnditins (and eventually ther regularity prperties and t discard the thers. D. Slutin f the PEC edge prblem. Fr slving the generalized W-H equatin (40 e first cnsider the generalized factrizatin f ξ k η g ( m g ( η (45 nstead t use the general technique f factrizatin, it is mre cnvenient t reduce this prblem t a difference equatin prblem. T this end e can eliminate the minus functin by intrducing the mapping : α sin φ η cs φ ξ, η sin φ α - cs φ τ (46 ith this mapping: m k α τ, ξ cs φ α sin φ τ (47 and eq.(45 becmes: cs φ α sin φ τ g - (τg ( sin φ α - cs φ τ Changing α ith -α yields: -cs φ α sin φ τ g - (τg ( -sin φ α - cs φ τ Eliminatin f the functins g - (τ yields t the flling difference equatin: X(4φ- X(0 (48 here: g (k cs ( X( sin (49 The mst general slutin f (48 is peridic functin f 4φ. Since the lgarithmic derivative f g (η is a plus functin, g (-kcs must be even in and nt vanishing in 0. Taking int accunt als the asympttic behavir f g (η the nly acceptable peridic functin is (nφ/: sin g ( k cs sin n t yields: k η g ( η η k sin[ arccs( ] n k

8 m arccs( g ( m k cs k n t is cnvenient separate in the functins: (-m,φ and (-m,-φ, the diffracted (subscript d and the ptics gemetrical (subscript g fields: (-m,φ d (-m,φ g (-m,φ, (-m,-φ d (-m,-φ g (-m,-φ (5 n the flling e cnsider the case (fig.3 ϕ > /. t means that the face ϕ-φ is in the shad regin: g (-m,-φ0. On the illuminated face ϕ-φ instead there is the incident plus the reflected field yielding: g k sin( Φ ϕ (5 [ η, Φ] j E ω µ [ η k cs( Φ ϕ ] ith E intensity f the incident ave plane. Further applying W-H technique requires the generalized decmpsitin f: k sin( Φ ϕ g ( m [ j E ] S ( η S ( m S( m (53 ω µ ( m m Again using the mapping (47 eliminates the minus functin S - (m and yields the difference equatin: Y(φ-Y(0 (54 ith Y( S (kcs ( t is readily seen that Y( must be have a ple in ϕ ith residue: R Y ( j E kω µ sin( ϕ / n The mst general slutin f (54 is a peridic functin f φ. Taking int accunt the asympttic behavir f S (η, this evenness in the -plane and the presence f the singularity ϕ the nly acceptable peridic functin is: R Y ϕ cs( cs( n n here R -(/n sin(ϕ /n R Y ( t yields the slutin: η sin[ arccs( ] ω µ n k R k V ( η k η η cs[ ( arccs( ] cs n k t has been verified that V (η is a plus functins and cincides exactly ith the expressin btained ith the rtating aves (eq.5 ith Z ± 0 E. Slutin f the Malyuzhinets prblems The prcedure flled fr the scalar case can be used fr the vectr case (eq.39, t. The generalized ϕ0 n factrizatin f M(η invlve vectr difference equatins in even functins f. Fr the Malyuzhinets prblem, the system can be reduced t a first rder difference equatin and it alls the use f the the Malyuzhinets prcedure fr slving difference equatin. Hever it is mre interesting t slve the vectr prblems by a direct generalized factrizatin f the matrix M(η. We remark that by this ay, the general prblem f decuple vectr difference equatins (see the recent imprvement [3] is reduced t a matrix generalized factrizatin. Generalized matrix factrizatin have been btained by this authr fr the flling cases: aratinal matrices, b matrix cmmuting ith mnic plynmial matrices [8a]. n the Malyuzhinets prblem e are dealing ith matrices cmmuting ith arbitrary plynmial matrices. Unfrtunately e ere nt able t generalize the methd ideated fr the classic factrizatin f these matrices [4]. Hever in the next sectin e ill sh that ith the mapping m(η defined by eq.(36, the vectr W-H generalized equatins can be reduced t a classic Wiener-Hpf equatins. F Equivalence f generalized and classic W-H equatins in edge prblems. The difficult prblem f the generalized factrizatin f the matrix M(η may be vercame by intrducing a map η η(η that reduces the generalized W-H equatins t classic W-H equatins. Fr m(η defined by (36, it may be accmplished by bserving that in the -plane eq. (39 has the frm: G X Y ( Φ (55 here: V ( k cs G ( M ( k cs, X (, ( cs k ( m( k cs, Φ Y ( Φ ( m( k cs, Φ N the mapping: Φ (56 yields: G X Y ( The definitins f X and Y and the eq.(56 assure that X ( and Y ( are even als in. Cnsequently in the plane η defined by: η k cs, (58 they are plus functins. t yields a classic W-H equatin in the η -plane: G ( ( η X ( ( η Y ( ( η (59

9 ~ Expressins f the matrix G ( η are very invlved. We ere nt able t perfrm their classic matrix factrizatin in general. Hever fr right edges and half-planes, these matrices d cmmute ith matrix plynmial f η and they may be explicitly factrized [4]. V. Cnclusins We have seen t different appraches fr facing field prblems in angular regins, The first (Smmerfeld- Malyuzhinets-Rtating aves is based n the intrductin f the -plane. The secnd (W-H classic r generalized is based n the intrductin f the η-plane. The first leads t slutin f difference equatins. The secnd t the decmpsitin-factrizatin prblem. Even if they are ambiguus, the difference equatins may be slved by technique cnsiderable mre simple f thse required fr factrize r decmpse matrix functins. Hever this authr claims the imprtance f the generalized Wiener-Hpf technique fr the flling reasns: - Wrking als n arbitrary angular regins, the W-H technique n dubt cnstitutes the mst general methds fr slving discntinuity prblem. - The thery f the factrizatin is better mathematically funded f the thery f the difference equatins. This aspect is imprtant hen n clsed slutins f the difference equatins are pssible and apprximate techniques becme necessary. T this authr pinin apprximate factrizatins face better these prblems. - Using different media in angular regins yields t functinal equatins in the η- dmain that can be classified as mdified generalized W-H equatins. The same ccurs hen are dealing anistrpic r general linear media in angular regins. The slutins f the dminant part f these equatins can be btained by factrizatin techniques [5]. Even if in the past, methds have been ideated fr penetrable prblems r general linear media als in the -dmain [,6], in this authr pinin, these prblems remain better studied in the η-dmain. ACKNOWLEDGEMENT This rk as supprted by the talian Ministry f University and Scientific Research (MURST under grants MM [3] A.S. Peters, Water aves ver slping beaches and the slutin f a mixed bundary value prblem fr φ- k φ0 in a sectr, Cmmunicatins n pure and applied mathematics, vl.5, (95, 97-08, [4] W.E. Williams, Diffractin f an E-plarized plane by imperfectly cnducting edge, Prc. R. Sc. Lnd. A vl.5, ( [5] T.B.A. Senir, Diffractin by an imperfectly cnducting edge, Cmm.Pure Appl. Math.. vl.,( [6] L.Knckaert, F.Olyslager and D. De Zutter, The diaphaneus edge, EEE Trans. Antennas Prpagat.m vl.45, 997, , [7] G.D Maliuzhinets, Relatin beteen the inversin frmula fr the Smmerfeld integral and the frmulas f Kntrvich Lebedev, Sviet Phys. Dkl. 3 ( , [8a] V.Daniele: Generalized Wiener-Hpf technique fr edge shaped regins f arbitrary angles, Rapprt ntern ELT Dipartiment di Elettrnica- Plitecnic di Trin. Settembre 000 [8b]V.Daniele: Generalized Wiener-Hpf technique fr edge shaped regins f arbitrary angles, MMET 000.Kharkv. Ukraine, September -5, 000. pp [9] V.Daniele: Rtating aves in the Laplace dmain fr angular regins, Rapprt ntern ELT Dipartiment di Elettrnica-Plitecnic di Trin. Dicembre 000 [0] R.W. Scharstein, A.M.J. Davis, Time-dmain three-dimensinal diffractin by the isrefractive edge, EEE Trans. Antennas Prpagat., vl. AP-46, 998, 48-58, []V.Daniele, P.L.E. Uslenghi: Wiener-Hpf slutins fr right isrefractive edges, Rapprt ntern ELT Dipartiment di Elettrnica-Plitecnic di Trin. September 000 [] V.Yu. Zavadskii: Certain diffractin prblems in cntiguus liquid and elastic edges, Sviet Physics- Acustic, vl., n., ct-dec., 966; p [3] T.B.A. Senir, S.R. Legault: Secnd-rder difference equatins in diffractin thery, Radi- Science. vl.35, n.3; May-June 000; p [4] V.G.Daniele: "On the slutin f vectr Wiener- Hpf equatins ccurring in scattering prblems," Radi Science, vl. 9, pp , 984. [5] V.G.Daniele, R.D.Graglia: Apprximate Diffractin cefficients fr the right-angled penetrable edges, this Sympsium [6] B.Budaev, Diffractin by edges, Lngman Scientific & Technical, UK, 995 REFERENCES [] L.B. Felsen, N.Marcuvitz, Radiatin and Scattering f Waves, Prentice Hall, Engled Cliffs, USA, 973 [] A.V.Osipv, A.N. Nrris, The Malyuzhinets thery fr scattering frm edge bundaries: a revie, Wave mtin, n.9, (999,

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