>1< The Mehod of Images in Velociy-Dependen Sysems Dan Censo Ben Guion Univesiy of he Negev Depamen of Elecical and Compue Engineeing Bee Sheva, Isael 8415 censo@ee.bgu.ac.il Absac This sudy invesigaes he applicaion of image mehodology o velociy-dependen wave sysems. Special Relaiviy is used fo he analysis of waves scaeed by abiay moving objecs in he pesence of a pefeclyconducing plane-ineface. The vaious scenaios consideed involve geomeical, maeial, and kinemaic symmeies. Cases discussed include fee-space, maeial media a-es, and maeial media in moion, wih espec o he plane-ineface bounday. The las configuaion is elaboaed fo wo diffeen scenaios: he fis assumes he same medium velociy houghou space when he plane bounday is emoved; he second inoduces wo symmeical velociy-fields in he half-spaces involved, wih a jump in flow diecion a he ineface. Whee he mehod applies i simplifies he analysis, and he esuls enich ou ye limied epeoie of canonical poblems fo elaivisic scaeing. 1. Inoducion 2. Relaivisic Elecodynamics 3. Plane Wave Scenaios 4. Images in Velociy-Independen Scaeing 5. Velociy-dependen Images in Fee-space 6. Velociy-dependen Images in Maeial Media 7. Concluding Remaks Refeences 1. INTRODUCTION The mehod of images consiues a useful albei limied echnique faciliaing soluions of he Laplace equaion, e.g., in elecosaics and magneosaics, and o a lesse exen of he veco o scala Helmholz wave equaions. Essenially, he appoach seeks o eplace given boundaies and he condiions on hem, by viual souces o scaees which mainain he oiginal bounday condiions, hence also peseve he fields in he iniial egions of space. Addiional fields ae ceaed in egions of space which ae ielevan o he oiginal poblem. The ensuing configuaions ae usually easie o analyze. Pesenly image echniques ae sudied in wave sysems involving scaees and inefaces in moion. This equies he use of Special Relaiviy in ode o faciliae he ansfomaion of fields and wave paamees fom one ineial efeence-sysem o anohe, i.e., aking ino accoun boh kinemaical and dynamical effecs of moion. In addiion o he maeial and geomeical symmey
>2< consideaions involved wih saic configuaions, hee poblems also involve kinemaical symmeies dependen on velociies. In he following, we conside hee efeence-sysems: Γ is associaed wih he plane bounday a-es; Γ chaaceizes he scaeing objec a-es; Γ involves he image objec a-es. A feaue common o all scenaios discussed below is a plane pefec mio, i.e., a pefec conduco, o alenaively a pefec magneic wall, chaaceized by ε, µ, especively. The plane-ineface, a-es in Γ, is defined by is oienaion in space, specified by a uni nomal veco n, and by he oigin = locaed on i. In view of he symmeies involved, his choice of oigin is conducive o simple expessions. Fo simpliciy only he case of a pefecly-conducing bounday is consideed, he analogous case of a magneic mio leads o simila esuls. Fuhemoe, examples ae limied o wodimensional geomeies involving cylindical scaees. The symmeical siuaions discussed subsequenly involve pais of vecos. We deal wih a pai of vecos, say a, b, symmeical wih espec o he ineface n, saisfying N ( a b) =, N = I nn (1) In (1) he dyadic, N = I nn = n n I, wih he uni dyadic I, sos ou he componens paallel o he plane, i.e., pependicula wih espec o n. Obviously his equies a, b, o be co-plana, ohewise he angenial componen vecos will no be idenical. The complemenay case, whee he componens paallel wih espec o he bounday add o zeo, is given by pais of vecos, say A, B, saisfying N ( A+ B) = (2) Below we also deal wih many cases whee he componens pependicula o he plane ae equal in lengh, augmening (1), (2), wih nn ( a + b ) =, nn ( A B ) =, especively. This will happen in isoopic media, applying in ou case o fee space and maeial media a-es wih espec o he bounday in efeence-sysems Γ. 2. RELATIVISTIC ELECTRODYNAMICS Conside an ineial efeence-sysem Γ, chaaceized by a quaduple of spaioempoal coodinaes R = (, ic) = ( x, y, z, ic) (3) whee c is he univesal consan, usually efeed o as he speed of ligh in vacuum. Mahemaically, R denoes he locaion fou-veco in he Minkowski fou-dimensional space. Fo ealy efeences see e.g. Sommefeld [1]. The Maxwell equaions fo souce-fee egions, e.g., see Saon [2], ae given by
>3< E= B, H = D, D=, B = (4) see also [3, 4] fo pesen noaion. In geneal, all fields in (4) depend on space and ime, i.e., E= E( R ), ec. Einsein s so-called pinciple of elaiviy [5, 6] asses ha he Maxwell equaions ae fom-invaian in all ineial efeence-sysems. ( ) Thus in a efeence sysem Γ, we have like in (4) E = B, H = D ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) D =, B = ( ) ( ) ( ) ( ) (5) ( ) ( ) ( ) wih E = E ( R ), ec. The spaioempoal coodinaes of Loenz ansfomaion ( ) Γ ae elaed o hose of Γ by he ( ) ( ) ( ) ( ) ( ) ( ) 2 = U ( v ), = γ ( v / c ) ( ) ( )2 1/2 ( ) ( ) ( ) ( ) γ = (1 β ), β = v / c, v = v ( ) ( ) ( ) ( ) ( ) ( ) ( ) U = I + ( γ 1) v v, v = v / v (6) ( ) In (6), when choosing ( = we obain ) ( ) = v, o ( ) v = d / d, hence v is ( ) ( ) he velociy of he oigin of Γ as obseved fom Γ. The dyadic U muliplies ( ) he veco componens paallel o he velociy by γ. Le us symbolize (6) by ( ) ( ) ( ) R = R [ R ]. Is invese R = R[ R ] is eadily deived by simple subsiuion ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 ( ), γ ( / c ) (7) = U + v = + v ( ) Replacing in (7) v = v yields he same funcional fom as in (6). This inveibiliy is an impoan popey of Einsein s heoy, ofen efeed o by he somewha vague phase all ineial sysems of efeence ae equivalen. As a coollay o (6), (7), he applicaion of he chain ule of calculus yields ( ) ( ) 2 ( ) ( ) ( ) = U ( + v / c ), ( ) = γ ( + v ) (8) consisenly denoed by ( ) = ( )[ R ]. Similaly o (6), (7). The invese of (8) R R R = R[ ( )] eadily follows. Thus new Minkowski fou-veco diffeenial R i i ( ) opeaos ae defined, R = (, c ), ( ) = ( ( ), c ( ) ), in ΓΓ,, R especively. Combining (4)-(8), Einsein [2, 5] has deived he field ansfomaions ( ) ( ) ( ) ( ) ( ) ( ) 2 E = V ( E+ v B), B = V ( B v E/ c ) ( ) ( ) ( ) 2 ( ) ( ) (*) D = V ( D+ v H/ c ), H = V ( H v D) ( ) ( ) ( ) ( ) ( ) V = γ I + (1 γ ) v v (9)
>4< ( ) ( which may be geneically symbolized by ) ( ) F = F [ F ]. The dyadic V muliplies ( ) he veco componen pependicula o he velociy by γ. Eihe by manipulaing (9) o diecly fom he pinciple of elaiviy, he invese fomulas of (9), ( ) [ ( ) ( ) ( ) ( ) F= F F ], eadily follows, e.g., E= V ( E v B ). By subsiuion fom (5) ino (9), diffeenial opeaos wee defined and used [4, 7-9] ( ) ( ) ( ) Γ, Γ ( ) Γ, Γ E = W E, H = W H W = V ( I v I ) ( ) Γ, Γ ( ) ( ) 1 (1) ( ) 1 elaing he fields in Γ o hose in Γ in a vey compac noaion. In (1) denoes he invese ime deivaive, which is eihe he ime inegal, o disappeas by muliplying all fields in (1) by. Sacificing genealiy fo simpliciy, we choose hee simple consiuive ( ) elaions, e.g., in a maeial medium a-es in Γ, ( ) ( ) ( ) ( ) ( ) ( ) D = ε E, B = µ H (11) ( ) ( ) whee in (11) he scalas ε, µ, ae he maeial paamees. Only in fee-space ( ) ( ) (vacuum) we have ε = ε, µ = µ fo all ineial efeence-sysems. In maeial media (9), (11), pescibe he Minkowski consiuive elaions [1], e.g., fo fields measued in Γ, and a medium a-es in Γ, we have D+ v H = E+ v B 2 / c ε ( ) B v E = H v D 2 / c µ ( ) (12) degeneaing in fee-space ino (11) in he fom D= ε E, B= µ H. In geneal, he foms (12) ae much moe difficul o handle [1]. Subsiuion fom (5) ino (12) and noing ha U V = γ I yields diffeenial opeaos D= ε M E v H/ c, B= µ M H+ v E/ c M = I v I = U W γ 2 2 1 Γ, Γ ( ) / (13) displaying he dependence of D on E and H, and similaly he dependence of B on H and E. Again, fo v = he consiuive elaions degeneae o he feespace case in all efeence-sysems. Alenaively, he Minkowski consiuive elaions can be expessed as = = N = V ( I + v I / c ) Γ, Γ Γ, Γ Γ, Γ Γ, Γ N H ε W E, N E µ W H Γ, Γ 1 2 (14)
>5< and once again fo v = he consiuive elaions (14) degeneae o he fee-space case in all efeence-sysems. 3. PLANE WAVE SCENARIOS The following simple examples illusae scenaios invesigaed below. We sa wih a monochomaic plane wave given in a medium a-es in Γ {, } {, ikr E H = Ee Hh } e, K = ( k, iω / c), K R = k ω (15) whee in (15) and houghou he elecical and magneic fields ae wien ogehe, i.e., symbols in baces apply coespondingly; K denoes he specal (popagaionveco and fequency) Minkowski fou-veco. In an isoopic medium a-es waves ae ansvesal, displaying he popeies E H = k, E H = E k = H k = ω / k = ( µε ), e / h = Z = ( µ / ε ) 1/2 1/2 (16) and in fee-space µ = µ, ε = ε. We conside he iniial wave (15) in he half space denoed by {1}. To saisfy he bounday condiions a he pefecly-conducing plane-ineface, a efleced wave mus exis in {1}, given by {, } {, ikr E H = Ee Hh } e, K = ( k, iω / c) (17) whee in an isoopic medium we have v = ω/ k = ω/ k, denoing he phase velociy which in fee-space becomes vph simila o (16). ph = c, and he ansvesaliy popeies ae The oal angenial elecic field mus vanish a all poins N on he bounday, a all imes. This pescibes fo he phases, i.e., wave popagaion vecos and fequencies, and fo he ampliudes, especively, ω ω =, N ( k k) =, N ( E e + E e ) = (18) Fo he case of a magneic wall he condiion on he ampliude is N ( H h + Hh) =. The symmey condiion N ( k k ) = is usually efeed o as Snell s law, obviously i is no a law, bu a esul of applying bounday condiions. Inasmuch as he pai of plane waves (15), (17), subjec o (18), saisfies he bounday condiions, i is feasible o exend he waves ino egion {2}, he halfspace iniially shielded by he pefecly-conducing plane-ineface, and emove he pefecly-conducing plane-ineface, wihou aleing he oiginal waves (15), (17), in egion {1}. The angenial componens of boh he elecic and magneic fields
>6< ae coninuous acoss he ineface, heefoe no equivalen suface chage and cuen densiy souces ae equied. Thusly E can be called he image of E, consiuing he simples example fo he applicaion of he mehod of images. In ode o obseve he fields in a diffeen efeence-sysem, (1) is applied o (15), (17), yielding in Γ E H = W E H = E e H h e Γ, Γ ik R {, } {, } {, } E H = W E H = E e H h e W = V ( I v I ), K = ( k, iω / c), K = ( k, iω / c) Γ, Γ ik R {, } {, } {, } Γ, Γ 1 (19) whee in (19) he phase is obained subjec o he phase-invaiance pinciple discussed below, and v is he velociy of Γ as obseved fom Γ. Once E, H, E, H, ae compued, he associaed fields ae found fom he peinen Maxwell equaions (5) B = E, D = H 1 1 B = E, D = H 1 1 (2) As long as we deal wih fee-space, he opeaions in (2) ae ivial and in Γ we have ansvesal plane wave as well. Inasmuch as K, K, R ae Minkowski fou-vecos, we have fo he inne poducs K R = K R, K R = K R (21) also efeed o as he phase-invaiance pinciple (aleady used in (19)), e.g., see [4]. Subsiuing fom (6), (7), ino (21) and collecing ems yields k = U k v = v k k = U k v = v k 2 ( ω/ c ), ω γ ( ω ) 2 ( ω/ c ), ω γ ( ω ) (22) whee in (22) he ansfomaions k = k [ k], k = k [ k ] ae he fomulas fo he elaivisic Fesnel dag phenomenon, and ω = ω [ ω], ω = ω [ ω] ae usually efeed o as he elaivisic Dopple effec fomulas. Similaly o (6), we symbolize (22) by K = K [ K], K = K [ K ]. The invese ansfomaions K = K[ K ], K = K[ K ] ae eadily deived. In Γ we have he condiions (18). By ansfomaion ino Γ, he fomulas fo eflecion of a plane wave fom a moving mio [5] ae obained, ofen dubbed as he elaivisic abeaion effec. The ineesing aspec of his quie ivial analysis is ha when eveyhing is obseved fom Γ, we have now he plane bounday moving a a velociy v. The effec of he bounday can be eplaced by an image wave, namely E, (19), whose paamees ae deemined by he bounday-value poblem in Γ and he peinen elaivisic ansfomaions, and hus we have defined he image wave in he case of a moving mio.
>7< We also need o conside waves in a efeence-sysem Γ moving wih velociy v when obseved fom Γ. The elevan fomulas ae obained fom (19)- (22) by eplacing he pimes wih double-pimes E H = W E H = E e H h e Γ, Γ ik R {, } {, } {, } Γ, Γ ik R { E, H } = W { E, H} = { E e, H h} e W = V ( I v I ) Γ, Γ 1 B = E, D = H, K = ( k, iω / c) 1 1 1 1 B =,, (, iω E D = H K = k / c) 2 k = U ( k v ω/ c ), ω = γ ( ω v k) 2 k = U ( k v ω/ c ), ω = γ ( ω v k) (23) The wo efeence-sysems Γ, Γ, ae moving symmeically wih espec o he plane-ineface, i.e., N ( v v ) =, nn ( v + v ) = (24) Unlike fee-space scenaios, maeial media equie moe scuiny. The fis scenaio involving maeial media is quie simila o he fee-space case: In Γ in egion {1} we now assume a medium a-es, possessing abiay paamees ε, µ. The same waves (15), (17), exis hee. The bounday condiions a he pefeclyconducing plane ae once again saisfied by (18). Also, by exending he waves and he medium fom he iniial half-space {1} o he half-space {2}, he fields in {1} emain unaleed. When ansfomaions ino Γ ae effeced, popagaion is obseved in a moving medium, yielding he waves (19)-(22), evealing he abeaion effec fo he pesen case of moving maeial media. I is noed ha in Γ he waves ae no ansvesal any moe. The nex scenaio assumes in {1} a medium moving elaive o he pefecly-conducing plane-ineface, e.g., see [1]. I mus be emphasized ha we neglec hee he fluid-dynamical aspec of he poblem, namely he flow coninuiy, allowing an abiay jump in he maeial velociy-field a he bounday. One way of looking a i is o assume he plane-ineface o be poous in a way ha allows he fluid o feely move hough i, a he same ime we assume he poes o be sufficienly small wih espec o he wavelengh, such ha he ineface acs like a faaday cage sceen, elecically pefoming as a pefec conduco. The siuaion is complicaed by he fac ha when we eside in Γ, whee he bounday is a-es, we encoune a moving medium, while fom he efeence-sysem Γ, whee he medium is a-es, he bounday appeas o be in moion. Wih his in mind, conside a pai of ansvesal plane waves (19) in he medium a-es in Γ in {1}, possessing consiuive paamees ε, µ. Accoding o (1), he invese of (19)-(22) is compued. In Γ we have
>8< Γ, Γ ikr { E, H} = W { E, H} = { Ee, Hh} e E H = W E H = E e H h e W = V ( I + v I ) Γ, Γ ikr {, } {, } {, } Γ, Γ 1 2 ( / ), ( ) 2 1 1 1 1 B= E, D= H, B= E, D= H k = U k + v ω c ω = γ ω + v k k = U ( k + v ω / c ), ω = γ ( ω + v k ) (25) In geneal, he waves (25) in Γ ae no simple ansvesal plane waves, because we ae dealing hee wih popagaion in a moving medium. Fo boundaies a-es, bounday condiions ae independen of he maeial composiion of he medium, including is moion, and ae diecly deived fom he Maxwell equaions [11, 12]. In he pesen case he bounday condiions ae evaluaed in Γ, a he planeineface a-es, hence he same bounday condiion (18) ae pescibed hee oo. Howeve, because of he medium moion, an equivalen anisoopic medium is encouneed hee, depending on he velociy-field v. I follows ha hee, unlike he cases of waves in fee-space o in a medium a-es wih espec o he bounday, we have nn ( k k), nn ( E e + E e ) (26) i.e., in geneal he componens of he popagaion vecos and field vecos pependicula o he plane-ineface ae unequal. The values of he unknown vecos in (26) ae pescibed by he ansvesaliy popeies of he waves in he medium s es-fame Γ. We now seek o define condiions, such ha he plane-ineface can be emoved wihou affecing he waves in {1}. To ha end we have o define appopiae waves in Γ in egion {2}. One ivial scenaio is suggesed by exending he same medium, is velociy, and he waves (15), (17), fom egion {1} ino {2}. As fa as emoving he plane-ineface is concened, his povides a valid configuaion. Howeve, i is immediaely ealized ha when waves ae consideed in Γ, Γ, he symmey (24) fo he velociy is violaed. Consequenly, unlike he case of fee space o media a-es in Γ discussed below, i is no feasible o define symmeically moving image scaees in {2}. We heefoe un o anohe alenaive, whee in Γ in {2} a medium aes is defined, maeially idenical o he medium a-es in Γ in {1}. When obseved fom Γ, he medium in Γ in {2} is moving wih a symmeical velociy saisfying (24). Similaly o he above agumen egading he flow disconinuiy a he plane-ineface, when we emove now he plane bounday and assume medium velociy v in {2}, he jump in he diecion of he velociy field is disegaded. Similaly o (19), whee he wave { E, H } is given in Γ in egion {1}, we now define { E, H } accoding o (23) in Γ in {2} in a medium a-es. Thus we have waves in media a-es in egions {1}, {2}, fo Γ, Γ, coespondingly
>9< { E, H } { E e, H h } e, K ( k, i / c) ik{1} R {1} {1} = {1} {1} {1} {1} {1} = {1} ω {1} { E, H } { E e, H h } e, K ( k, i / c) ik{1} R {1} {1} = {1} {1} {1} {1} = {1} ω {1} { E, H } { E, H ik{2} R {2} {2} = {2} e {2} {2} h {2} K {2} = k{2} ω {2} } e, (, i / c) { E, H } { E e, H h } e, K ( k, i / c) ik{2} R {2} {2} = {2} {2} {2} {2} {2} = {2} ω {2} (27) In ode o have symmeical waves we impose ω ω =, N ( k k ) =, nn ( k + k ) = {1} {2} {1} {2} {1} {2} N ( E e + E e ) =, nn ( E e E e ) = {1} {1} {2} {2} {1} {1} {2} {2} N ( H h H h ) =, nn ( H h + H h ) = {1} {1} {2} {2} {1} {1} {2} {2} (28) I follows ha similaly o (18) we now have in Γ on boh sides of he pefecly-conducing plane-ineface, in he coesponding egions {1}, {2}, ω ω ω =, N ( k k ) =, N ( E e + E e ) = {1} {1} {1} {1} {1} {1} {1} {1} ω =, N ( k k ) =, N ( E e + E e ) = {2} {2} {2} {2} {2} {2} {2} {2} (29) and heefoe ω = ω = ω = ω, N ( k k ) =, N ( k k ) = {1} {1} {2} {2} {1} {2} {1} {2} N ( E e E e ) =, N ( E e E e ) = {1} {1} {2} {2} {1} {1} {2} {2} N ( H h H h ) =, N ( H h H h ) = {1} {1} {2} {2} {1} {1} {2} {2} (3) Accoding o (3), in Γ, on boh sides of he pefecly-conducing planeineface, in he coesponding egions {1}, {2}, he fequencies ae equal, and he angenial componens of he popagaion vecos and fields ae coninuous, implying ha he plane-ineface can be emoved wihou affecing he oiginal fields in Γ in {1}. Adheing o ou pevious convenion, fields wih, wihou, uppe ba ae consideed as he inciden, he image, waves especively. I is ineesing o noe ha due o he diffeen velociies v, v, in effec we have now wo diffeen anisoopic media in egions {1}, {2}. Hence he changes of he popagaion vecos k{1}, k {2}, and k{1}, k {2}, acoss he bounday, display how he waves, popagaing fom one medium ino he ohe, ae efaced a he ineface. Wih his he discussion fo individual plane waves is compleed, faciliaing he applicaion of he esuls o moving scaees. 4. IMAGES IN VELOCITY-INDEPENDENT SCATTERING
>1< The pesen secion deals wih a simple case of wave scaeing fom an infinie cylinde, in he pesence of a pefecly-conducing plane-ineface. The choice of a wo-dimensional geomey seves o inoduce basic conceps, ools, and elevan noaion in a simple way. The coesponding hee-dimensional analog follows along he same lines, bu he mahemaical deails become moe complicaed, due o he veco spheical waves and hei associaed special funcions. In Γ, whee he plane-ineface is a-es, in fee-space, an inciden planewave (15) is assumed in he half-space {1}. Fo simpliciy, conside E = z o be polaized along he cylindical axis. I follows ha H, k, ae in he pependicula xy -plane, in which a adius-veco is defined. The plane-ineface is defined by he diecion of is nomal n, wih n z = fo he cylindical case, and by he oigin = locaed on i. Define fo he scaee a local igh-handed cylindical coodinae sysem ρ= ρ( ρ, ψ ), ρ ψ = z (31) wih pola coodinaes ρ, ψ. In he coesponding hee-dimensional case we would have ρ= ρ( ρ, θ, ψ), ρ θ = ψ. The local coodinae sysem is locaed elaive o he iniial one by means of = + ρ (32) The inciden wave (15) is anslaed o he local coodinae sysem by subsiuing (32) ino he phase K R, yielding K R = k ω = k + k ρ ω (33) The geomey of he cylindical scaee is defined in he coodinae sysem (31) elaive o he local oigin ρ =. This geomey, as well as he consiuive paamees ae unifom along he cylindical z -axis, such ha no coss-polaizaion occus. We wish o esic he pesen analysis o ougoing waves only. I has been shown by Twesky [13-15] ha he scaeed wave is given in ems of ougoing waves a leas ouside he cicumscibing cicle (o cicumscibing sphee fo he coesponding hee-dimensional case) of adius ρ max. Hence fo he pesen case we have o ensue ha he plane-ineface is ouside his egion, i.e., ρ max < n. Accodingly he scaeed wave, due o he exciaion (15), is given by E ρ z k ik iω m imψ (, ) = ee e Σmi am( ) Hm( κ) e k ik ee H( κ) gk (, ρ), κ= kρ = kρ ρ= k ρ ρ 1/2 i i ( ) (2/ ), (, ) ( im κ = πκ gk ρ = zσm m k) (34) H i e a e κ ω ψ and he associaed H -field is compued by applicaion of he Maxwell equaions (4). In (34) H m denoes he Hankel funcions of he fis kind, which ogehe wih
>11< i he ime faco e ω guaanee ougoing waves, and i e k is he exa phase faco esuling fom he anslaion of he inciden wave ino he local coodinae sysem ρ. The coefficiens a ( ) m k, fo he specific geomey of he cylinde a-hand, depend on he diecion of incidence of he exciaion wave (15). In he fa field E k ends asympoically o an ougoing symmeical cylindical wave imes he scaeing ampliude gk (, ρ ), wih k, ρ ( ψ ), indicaing diecions of incidence and obsevaion, especively. The noaion k ρ ρ seves o emphasize he quasi planewave behavio of he scaeed wave in he fa field. Using (32), he fa field (34) is anslaed back o he iniial coodinae sysem, yielding E (, ) (2/ ) g( k, ρ ) (35) i( k kρ ) 1/2 ikρ iω ee iπκ e k and in view of he consans and he slowly vaying ems in (35), E (, ) k esembles a plane wave wih he phase kρ ω. Similaly o (31), conside now a igh-handed local coodinae sysem defined in {2} ρ = ρ( ρ, ψ), ρ ψ = z (36) Fo (31), (36), o be symmeical and have mio azimuhal angles, implies = + ρ, N ( ) =, nn ( + ) = N ( ρ ρ) =, nn ( ρ + ρ) = ψ= ψ ψ= ψ N ( ψ + ψ) =, nn ( ψ ψ) = ψ = ψ ψ= ψ (37) Coesponding o he iniial scaee defined in {1}, a geomeically symmeic mio-scaee is now defined in {2}. While he iniial scaee is excied by (15) in {1}, he image objec is excied by he image plane-wave (17) consideed in {2}. These waves ae symmeical accoding o (18) and nn ( k + k ) =. The wave scaeed fom he image cylinde is (cf. (34)) is E ρ = z e e e Σ i a k H κ e ik iω m imψ (, ) ( ) ( ) k m m m ik ee H( κ) gk (, ρ), κ= kρ = kρ ρ = k ρ ρ, e = e (38) H i e a e 1/2 iκ iω imψ ( κ) = (2/ πκ), gk (, ρ) = zσm m( k) The symmeies pescibe fo (34), (38) gk (, ρ ) = g( k, ρ ), a ( k ) = a ( k ) (39) ψ= ψ m m
>12< Similaly o (35), we now have E g k ρ (4) i( k kρ ) 1/2 ikρ iω (, ) ee (2/ iπκ ) e (, ), e = e k Remove now he pefecly-conducing plane-ineface and conside he vaious symmeies ogehe wih he expessions (35), (4), a poins on he planeineface. Clealy he oiginal bounday condiions (18) ae mainained on boh side of he plane-ineface, and he fields in {1} emain unaleed, fo boh he inciden and efleced waves, and he scaeed waves oiginaing in he half-spaces {1} and {2}, especively. Plane waves ae vey convenien fo discussing image-mehod siuaions in velociy-dependen sysems. Accodingly, abiay wave funcions, fo abiay disances fom he scaee, can be ecas in ems of plane-wave inegal epesenaions. Fo cylindical geomeies in paicula, he Sommefeld inegal epesenaions [2, 4, 8] ae exploied. Accodingly (34), (38) ae ecas in ems of a supeposiion of inhomogeneous plane waves, in he fom E ρ gk p /2 i i i p iω τ= ψ+ π k k ρ (, ) = e e e (, ) dτ / π, = k ψ, τ ψ, τ τ= ψ π /2+ i imτ g= zg( k, p) = zσ a ( k) e, k ρ = kp ρ = kρcos( ψ τ) m m p E ρ gk p /2 i i i p iω τ = ψ+ π k k ρ (, ) = ee e (, ) dτ / π, = k ψ, τ ψ, τ τ = ψ π /2+ i imτ g = zg( k, p) = zσ a ( k) e, k ρ = kp ρ, kρcos( ψ τ ) m m p (41) Alhough in he inegand (41) he plane waves popagae in complex diecions indicaed by p, p, in all ohe especs hey ae eaed like plane waves wih eal popagaion vecos. The waves (41) ae now anslaed back o hei iniial coodinae sysems, yielding i( k kp) ikp iω E (, ) = e e e g( k, p ) dτ / π k ψ, τ i( k kp) ikp iω E (, ) = e e e g( k, p ) dτ / π k ψ, τ (42) Similaly o he agumen following (4), in he inegands in (42), a he bounday, pais of plane waves wih symmeical diecions p, p, can be idenified, combining o saisfy he bounday condiions a he plane-ineface. In eospec, he same conclusions could have been obained by diecly using he exac seies in (34), (38), howeve he concep of plane-wave epesenaions, which applies o hee-dimensional siuaions as well, is moe geneal [2, 4, 13-15], applicable o abiay wave funcions. This appoach will also be needed hee in he sequel. Finally, we have o pay aenion o he muliple-scaeing aspecs of he poblems a hand. The configuaion of a scaee and a plane bounday, o alenaively, a scaee and is image, consiues a many-body sysem in which
>13< successive scaeing of waves bouncing fom one objec o anohe akes place. Thus in addiion o he scaeing pocesses discussed above, he waves efleced by he bounday (o scaeed by he image scaee), also excie he oiginal scaee, ec. A closed self-consisen muliple-scaeing fomalism has been devised by Twesky [13-15], bu in velociy-dependen sysems, in geneal, only a successivescaeing fomalism is feasible, enumeaing one scaeing pocess afe is pedecesso. This is due o he fac ha new fequencies ae ceaed wih each successive-scaeing pocess [8], and he new fequencies ae unknown befoe he peceding modes ae compued. The poblem of dealing wih highe-ode modes becomes inceasingly complicaed [8, 16]. Subsequenly only a limied numbe of ineacions will be consideed, namely, he ineacions of he exciaion planewave wih he scaee, and he efleced plane-wave wih he image scaee, as demonsaed above fo he velociy-independen case. 5. VELOCITY-DEPENDENT IMAGES IN FREE-SPACE The elaively simple case of fee-space scaeing is consideed fis. As in he pevious secion, fo simpliciy we deal wih cylindical scaees, oiened along he cylindical z -axis, and pependicula velociies in he xy -plane. Conside a scaee a-es in he efeence-sysem Γ, defined elaive o (*) Γ accoding o (6) wih v = v. In Γ an inciden wave (15) is given, wih E = z, coesponding in Γ o he wave given by (19). By inspecing (34), (41), (42), and judiciously modifying he peinen noaion, he scaeed wave in Γ is E ρ z e e e i a k H e = ik iω m imψ (, ) = Σm m( ) m( κ ) k (, ) τ / π ik i p i ee k ρ ω e g k p d ψ, τ i( k k p ) ik p iω E (, ) = e e e g ( k, p ) dτ / π k ψ, τ (43) whee in (43) i is assumed ha g ( k, ), hence also a ( ) m k, ae aleady available, e.g., by solving he bounday-value scaeing poblem in Γ. The las expession (43) esuls fom he anslaion back ino he iniial,, coodinae sysem in Γ. The plane waves in he inegand (43) individually saisfy (21), (22), hence he ansfomaion fom Γ ino Γ accoding o (25), (cf. (42)), yields i( k k p ) ikp iω p E = W E (, ) = e e e g( k, p ) dτ / π Γ, Γ k k ψ, τ k ω = k ω g k p = W g k p p p p Γ, Γ, (, ) (, ) W = V ( I + v I ) = V ( I v k I / c) Γ, Γ 1 p = z γ (1+ v k / c) p (44) whee in (44) in fee space k = ω / c, k = ω / c. The plane-wave phase is p p p epesened in ems of he Γ paamees, bu eveyhing else is lef in ems of Γ
>14< paamees fo convenience. Only when explici compuaions ae pefomed, fuhe subsiuions may be advised. Noe ha in Γ a single exciaion fequency ω is pesen, wih a eal value ω. Accoding o (21), and he invese of (22) K = K[ K ], he phase k ω p p, in ems of Γ coodinaes,, is obained fom k p ω, which is in ems of Γ coodinaes. Fo he fequency in paicula we have p p Inasmuch as (45) involves boh eal ω and complex k p fequencies ω p. Thus p e ω ω = γ ( ω + v k ) (45), we now have complex manifess he ime dependen vaiaion of he wave ampliudes due o he eceding o appoaching scaee, as obseved in Γ. In (44) we have a supeposiion (inegal) of plane waves, fo which we seek a coesponding se of efleced waves, o hei images. This can be achieved by assuming a symmeical image-objec moving in he half-space egion {2}, a-es in he image efeence-sysem Γ. This efeence-sysem is elaed o Γ accoding (*) o (6), wih he velociy v = v. The wo efeence-fames Γ, Γ, consideed wih espec o he bounday a-es in Γ, ae moving symmeically accoding o (24). By inspecion of (38), (41), (43), he wave scaeed fom he imagecylinde, a-es in Γ, is given by E ρ z k = ik iω m imψ (, ) = ee e Σmi am( ) Hm( κ ) e k (, ) τ / π ik i p i ee k ρ ω e g k p d ψ, τ (, ) ( imτ g = zg k p = zσ a k ) e, ρ = ρ ( ρ, ψ ) k ρ = k p ρ = k ρ cos( ψ τ ) p m m (46) and similaly o (43), anslaion o he iniial coodinaes Γ yields,, of efeence-fame i( p ) i p iω (, ) = e k k k E e e g ( k, p ) dτ / π (47) k ψ, τ As in (44), he scaeed wave (46) is now ansfomed ino Γ i( k k p ) ikp iω p E = W E (, ) = e e e g( k, p ) dτ / π Γ, Γ k k ψ, τ k ω = k iω g k p = W g k p p p p Γ, Γ, (, ) (, ) Γ, Γ 1 W = V ( I + v ) ( I = V I v k p I / c) = z γ (1 + v k / c), k = ω / c, k = ω / c p p p p (48)
>15< Once again (48) displays he phenomenon encouneed in (45), namely ha in Γ he fequency ω fo each complex diecion indicaed by p is complex, and p e ω p manifess he ime dependence of he ampliude due o he moion. Inasmuch as we ae now dealing wih plane waves in Γ, we can ake pais of waves fom he inegands of (44), (48), and impose on hem he bounday condiions a he plane-ineface in Γ. Thus fom he symmeies (24), (39), see also (43), (46), we have ( k k p ) = ( k k p ) ψ = ψ ωp = ωp ψ = ψ, N ( k p k p ) = ψ = ψ (49) gk (, p ) = gk (, p ) ψ = ψ which finally allows us o emove he pefecly-conducing plane-ineface, and conside he image poblem as solved. Obsevaion fom Γ, whee he plane bounday is moving wih velociy v follows upon ansfoming (46) fom Γ o Γ, by applying he diffeenial opeao aken fom (19) E W E W V I v I (5) Γ, Γ Γ, Γ 1 =, = ( ) k k consiuing a vey complicaed compuaion, whose deail should be lef o a numeical simulaion pojec. In pinciple, each of he plane waves in he inegand, which in Γ saisfy he condiions of equal fequencies and he Snell law in (49), now will behave accoding o Einsein s abeaion fomula [5], as emaked above afe (22), fo he single plane wave. 6. VELOCITY-DEPENDENT IMAGES IN MATERIAL MEDIA Once again we sa wih a maeial medium a-es in Γ in egion {1}, possessing abiay paamees ε, µ. The waves (15), (17), ae assumed, saisfying he bounday condiions pescibed by (18) a he pefecly-conducing plane. The waves ansfom ino Γ accoding o (19)-(22). Similaly, accoding o (23), he waves can be ansfomed ino efeence-sysem Γ, moving wih velociy v when obseved fom Γ, wih Γ, Γ, elaed by he symmey condiion (24). I mus be bon in mind ha in Γ, Γ, we pesenly deal wih waves in moving media. The wave scaeed by an objec a-es in he moving medium in Γ, excied by E in (19), is in geneal given by he inegal epesenaion in (43). Howeve, since we ae dealing hee wih a moving medium having a pefeed diecion pescibed by he velociy, he medium is effecively anisoopic. Consequenly he scaeed wave will no submi o a Hankel-Fouie seies epesenaion as in (43). The Fesnel dag effec k = k [ k ] given in (22) mus be aken ino accoun, pescibing ha fo he pesen case k p in he las wo lines of (43) is velociy-dependen. Theefoe, fo each plane wave in he inegand, we ae dealing wih an individual effecive phase-velociy v ph. The evaluaion of specific
>16< scaeing poblems of his kind is complicaed. Some elevan wok fo scaees moving elaive o he ambien medium have been discussed befoe, e.g., see [11, 12, 17]. Tansfoming he plane waves in he inegand (43) ino Γ yields (44). Similaly, we have he plane waves in he inegand (46), (47), scaeed by an image-objec a-es in he moving medium in Γ, subjec o (24), excied by E, (23). Similaly o (44), he scaeed wave (46), (47), is ansfomed ino Γ, yielding (48). Obviously he pesen scenaio follows closely along he lines of he feespace and velociy-independen cases. Thus in (42), pais of plane waves wih symmeical diecions wee sough, each pai saisfying he bounday condiions a he plane-ineface. Hee he same siuaion applies o he plane waves in he inegands of (44) and (48). Hence he bounday condiions ae saisfied, and he pefecly-conducing plane-ineface may be emoved wihou aleing he fields in egion {1}. By ansfoming all waves ino Γ, we ae dealing wih a scaee a-es embedded in a moving maeial medium, in he pesence of a plane-ineface a-es in Γ, obseved fom Γ o be moving a velociy v. This is he same siuaion which fo he single plane-wave is descibed by (19)-(22), associaed wih he abeaion phenomenon. We ae now eady o discuss he scenaio involving a maeial medium and a scaee, boh a-es in Γ in {1}, in he pesence of a pefecly-conducing plane-ineface a-es in Γ. In Γ a medium moving elaive o he bounday is encouneed, heefoe he above emaks egading he flow coninuiy apply hee oo. We seek o eplace his configuaion by an image medium and an image scaee a-es in Γ in {2}. The evaluaion of he bounday-value poblems fo he scaees a-es wih espec o he media in Γ, Γ, is classical and need no be fuhe discussed. The exciaion wave, and is eflecion which is equivalen o an image wave, have been discussed above, (27)-(3). As shown in (27), we need o epesen he waves in boh egions {1} and {2}, because in Γ we encoune wo diffeen effecive anisoopic media in hose egions, whose popeies ae govened by he moion. This is he key o undesanding and solving he poblem. Coesponding o he fis line in (27), we have he scaeed wave (43), in Γ in {1} given by E ρ z k ik{1} iω{1} m imψ (, ) = e {1} e e Σmi am{1} ( {1}) Hm ( κ {1}) e k {1} = e e e g ( k, p ) dτ / π ik{1} ik p {1} ρ iω{1} {1} ψ, τ {1} {1} i( k{1} k p {1} ) ik p {1} iω{1} E (, ) = e e e k {1} {1} ψ, τ g ( k, p ) τ / π {1} {1} d (51) and accoding o (44), he wave (51) is ansfomed ino Γ in {1}, yielding
>17< E = W E (, ) Γ, Γ k {1} k {1} = e e e g ( k, p ) dτ / π i( k{1} k p {1} ) ikp {1} iω p {1} {1}, {1} {1} ψ τ k ω k ω g k p W g k p Γ, Γ p {1} p {1} = p {1} {1}, {1}( {1}, ) = {1}( {1}, ) Γ, Γ = V 1 ( I + v ) ( I = V I v k p{1} I / c) W = z γ (1 + v k / c) p{1} (52) Each plane wave in (52) mus be associaed wih a efleced wave, such ha ogehe, accoding o he fis line (29), he bounday condiions ae saisfied ω ω =, N ( k k ) = p {1} p {1} p {1} p {1} N ( g ( k, p ) e + g ( k, p ) e ) = {1} {1} {1} {1} {1} {1} (53) by Theefoe in Γ in {1}, he wave efleced fom he plane-ineface is given E = e ( e e g k, p ) dτ / π (54) i( k{1} k p {1} ) ikp {1} iω p {1} k {1} {1}, {1} {1} ψ τ Once again, in ode o emove he pefecly-conducing ineface, we need o define a medium a es in Γ in {2}, maeially idenical o he medium a-es in Γ in {1}. Obseved in Γ, he medium is moving wih a symmeical velociy v saisfying (24). Consisenly, we mus have waves in Γ in {2}, such ha he second line (29) is saisfied, i.e., similaly o (53) we now have ω ω =, N ( k k ) = p {2} p {2} p {2} p {2} N ( g ( k, p ) e + g ( k, p ) e ) = {2} {2} {2} {2} {2} {2} (55) accoding o he subsequen expessions. Similaly o (46), (47), he wave scaeed by he image objec in Γ in {2} (cf. (51)) is given by E ρ z k ik{2} iω{2} m imψ (, ) = e {2} e e Σmi am{2} ( {2}) Hm ( κ {2}) e k {2} = e e e g ( k, p ) dτ / π ik{2} ik p {2} ρ iω{2} {2}, {2} {2} ψ τ i( k{2} k p {2} ) ik p {2} iω{2} E (, ) = e e e g ( k, p ) dτ / π k {2} {2} ψ, τ {2} {2} (56) Tansfoming (56) ino Γ in {2}, accoding o (48) (cf. (52)), yields
>18< E W E (, ) Γ, Γ = k {2} k {2} = e e e g ( k, p ) dτ / π i( k{2} k p {2} ) ikp {2} iω p {2} {2}, {2} {2} ψ τ k ω k ω g k p W Γ, Γ p {2} p {2} = p {2} i {2}, {2}( {2}, ) = W = V ( I + v I ) = V ( I v k I / c) Γ, Γ 1 p {2} = z γ (1 + v k / c) p{2} g ( k, p )(57) {2} {2} In Γ in {2} in ode o saisfy he bounday condiions (55), efleced waves ae needed, consiuing he analog of (54) E = e e e (, ) dτ / π {2} g k p (58) i( k{2} k p {2} ) ikp {2} iω p {2} k {2}, {2} {2} ψ τ Wih all ha accomplished, he pefecly-conducing plane-ineface can be emoved wihou affecing he iniial waves in {1}. I is noed ha on he wo sides of he ineface, in he diffeen media as defined by he moion, he fields wihou he uppe-ba and hose endowed wih an uppe-ba ae diffeen. As fo he single plane waves, his phenomenon can be ascibed o efacion a he ineface sepaaing he wo media, see discussion afe (3). 7. CONCLUDING REMARKS The mehod of images in velociy-dependen wave sysems is invesigaed. Appaenly i woks only fo a limied class of poblems, namely scaeing in he pesence of magneic o elecic pefec-mios, wih plane-ineface geomey. Poblems involving maeial half-spaces, e.g., [18], mus be excluded, bu hei limiing cases, when he ineface becomes a pefec mio, could be esed agains he pesen mehod. A heoeical discussion of he poblem is pesened hee. Fuhe numeical simulaions ae needed fo depicing he new physical phenomena. This subjec bings ogehe scaeing poblems in media a-es and in-moion, muliple scaeing, and special elaiviy, hus poviding an exension of classical mehod of images ideas. Plane wave inegal epesenaions and he use of diffeenial opeaos fo field ansfomaions faciliae compac noaion, allowing fo he descipion of vaious scenaios. Fee space siuaions ae simple o analyze, poviding he basis fo moe elaboae cases involving maeial media, boh a-es and in-moion wih espec o he pefecly-conducing plane-ineface. When appopiae moving image scaees and media ae povided, his bounday can be emoved wihou affecing he fields in he half-space whee he iniial poblem is saed. Maeial media pose he poblem of evaluaing scaeing poblems in he pesence of moving media: In he scenaio whee he plane-ineface is a-es wih espec o he maeial medium, he scaee is embedded in a moving medium. The ohe case involves a scaee a-es wih espec o he embedding medium, bu
>19< he plane-ineface is immesed in a moving media. These cases ae no simple o analyze, and equie sacificing he mechanical flow-coninuaion in favo of a manageable definiion of elecomagneic poblems. REFERENCES 1. Sommefeld, A., Elecodynamics, Academic Pess, 1964. 2. Saon, J.A., Elecomagneic Theoy, McGaw-Hill, 1941. 3. Censo, D., Applicaion-oiened elaivisic elecodynamics (2), PIER, Pogess In Elecomagneics Reseach, Vol. 29, 17 168, 2. 4. Censo, D., "The mahemaical elemens of elaivisic fee-space scaeing", JEMWA Jounal of Elecomagneic Waves and Applicaions, Vol. 19, pp. 97-923, 25. 5. Einsein, A., Zu Elekodynamik bewege Köpe, Ann. Phys. (Lpz.), Vol. 17, 891-921, 195; English anslaion: On he Elecodynamics of moving bodies, The Pinciple of Relaiviy, Dove. 6. Pauli, W., Theoy of Relaiviy, Pegamon, 1958, also Dove Publicaions. 7. Censo, D., I. Anaoudov, and G. Venkov, Diffeenial-opeaos fo cicula and ellipical wave-funcions in fee-space elaivisic scaeing, JEMWA Jounal of Elecomagneic Waves and Applicaions, Vol. 19, pp. 1251-1266, 25. 8. Censo, D., Fee-space muliple scaeing by moving objecs", JEMWA Jounal of Elecomagneic Waves and Applicaions, Vol. 19, pp. 1157-117, 25. 9. Censo, D., Boadband spaioempoal diffeenial-opeao epesenaions fo velociy-dependen scaeing, PIER Pogess In Elecomagneic Reseach, Vol. 58, pp. 51-7, 26. 1. Censo, D., Scaeing of a plane wave a a plane ineface sepaaing wo moving media, Radio Science, Vol. 4, pp. 179-188, 1969. 11. Censo, D., "Non-elaivisic elecomagneic scaeing: "Revese engineeing" using he Loenz foce fomulas", PIER-Pogess In Elecomagneic Reseach, Vol. 38, pp. 199-221, 22. 12. D. Censo, "Non-elaivisic bounday condiions and scaeing in he pesence of abiaily moving media and objecs: cylindical poblems", PIER- Pogess In Elecomagneic Reseach, Vol. 45, pp. 153-18, 24. 13. Twesky, V., "Scaeing of waves by wo objecs", pp. 361-389, Elecomagneic Waves, Ed. R.E. Lange, Poc. Symp. in Univ. of Wisconsin, Madison, Apil 1-12, 1961, The Univesiy of Wisconsin Pess, 1962. 14. Twesky, V., "Muliple scaeing by abiay configuaions in hee dimensions", Jounal of Mahemaical Physics, Vol. 3, pp. 83-91, 1962. 15. Twesky, V., "Muliple scaeing of elecomagneic waves by abiay configuaions", Jounal of Mahemaical Physics, Vol. 8, pp. 589-61, 1967. 16. Censo, D., Velociy dependen muliple scaeing by wo hin cylindes, Radio Science, Vol. 7, pp. 949-954, 1972. 17. Censo, D., Non-elaivisic scaeing in he pesence moving objecs: he Mie poblem fo a moving sphee, PIER-Pogess In Elecomagneic Reseach, Vol. 46, pp. 1-32, 24.
>2< 18. De Cupis, P., Radiaion by a moving wie-anenna in he pesence of ineface, JEMWA Jounal of Elecomagneic Waves and Applicaions, Vol. 14, pp. 1197 123, 2.