The physics of wedge diffraction: A model in terms of elementary diffracted waves

Transcription

1 coustcs 8 Pas The hyscs of wedge dffacto: model tems of elemetay dffacted waves M. eda Pedo Meguo cece Laboatoy Meguo Meguo-ku Tokyo Jaa ueda-mt@fty.com 6563

2 coustcs 8 Pas to-dow hyscal cle called vtual dscotuty cle of dffacto s aled to waves dffacted by a wedge. I the aalyss dffacted waves ae descbed by a sum of two moe fudametal quattes called elemetay dffacted waves ad the hyscs of wedge dffacto s made clea tems of the elemetally dffacted waves. I addto the smle stuctue the fa feld eables us to eoduce the fa feld goous solutos of waves dffacted by the wedge. Thus the cle s justfed fmly by ths esult. 1 Itoducto We have oosed a ew hyscal cle that s called vtual dscotuty cle of dffacto (abbevated by VP fo aalyzg waves dffacted by a obstacle [1]. ce VP s a to-dow cle t has bee justfed by the fact that the elato fo dffacted waves by VP always satsfes the bouday codto at the suface of the obstacle. Ths oety s ot suoted by othe cles fo aalyzg dffacted waves fo examles Kchhoff s fomula ad Bouday lemet Method. I ths ae VP s aled to waves dffacted by a wedge to make the hyscs of wedge dffacto clea. The ma featue of VP les o the asseto that waves dffacted by the aex of a obstacle ae exessed by a sum of two moe fudametal quattes that ae called elemetally dffacted waves. ce the wedge has oe aex t s suted to study the oetes of elemetally dffacted waves. d the otetal the wedge ca be exessed by geometcal otcs waves ad elemetally dffacted waves so that the ole of elemetally dffacted waves s qute udestadable. Thus the hyscs of wedge dffacto s made clea tems of elemetally dffacted waves. I addto the elemetally dffacted waves have smle stuctue the fa feld ad t eables us to eoduce the fa feld goous solutos of waves dffacted by the wedge. Thus VP s justfed futhe by ths esult. Ths ae s ogazed as follows. I ec.2 a vtual sace s fomulated by cooatg mo mages eflected by edges of wedge to the sace ad the otetals t ae defed usg the oety of mo eflecto. I ec.3 the Gee s theoem s aled to the vtual sace ad the esultg elatos make the hyscs of wedge dffacto udestadable tems of elemetally dffacted waves. I ec.4 elemetally dffacted waves ae deved the fa feld ad the goous solutos fo dffacted waves ae eoduced fom them. shot summay s gve ec.5. 2 omulato of a vtual sace 2.1 Physcal descto of a wedge Let us daw a half le L the 2 sace ad deote the statg ot as Q as show g.1 ad toduce the ola coodate system (θ by secfyg as a dstace measued fom Q ad θ as a agle measued fom L the atclockwse decto. Let W be a wedge of g.1 ofguato of a wedge. agle 2Ф ad defed by W {( θ Φ θ } (1 Φ whee <Ф πad Фπ coesods to a sem-fte lae Ф>π/2 a cocave wedge Ф<π/2 a covex wedge ad Фπ/2 a eflectg lae. The aex of W les o Q ad W s bouded by two edges ad. Let us deote a half le that stats fom Q ad us the θ decto as L(θ. The the edges ca be secfed as L(Ф ad L(-Ф ad let us deote the fome as the atclockwse edge ad the latte as the clockwse edge. 2.2 oud feld The waves oagatg W ae statoay tme ad satsfy the followg elato k δ ( whee stads fo the otetal of waves k the wave umbe δ the delta fucto ( θ the osto vecto of the ot souce ad the elatos < < ad -Ф<θ <Ф hold. I the fee soud feld adates the dect waves ( H ( k /(4 (3 j whee stads fo the dect waves fom H the -th ode Hakel fucto of the secod kd j the magay ut ad the statoay tme fucto ex(jωt s deleted whee ω stads fo the agula fequecy. The obsevato ot lays a motat ole ths ae ad ts osto vecto s deoted by ( θ ad t belogs to W. The otetal s exessed as a fucto of o that s ( o (. I the wedge dffacto the dstace fomato s ofte umotat ad ths case the otetal may be exessed as a fucto of θ o θ. 6564

3 coustcs 8 Pas s to the bouday codto the chlet codto ( / o the Neuma codto ( s set to edges of the wedge whee stads fo a e ut vecto omal to the edges. Let us deote the wedge that satsfes the chet codto as the had wedge ad the Neuma codto as the soft wedge. The edge of the had wedge ca be egaded as a mo of m1 whee m stads fo the eflecto coeffcet of the mo. mlaly the edge of the soft wedge ca be egaded as a mo of m-1. The dffacted waves ca be cosdeed as a devato fom the geometcal otcs waves. The the otetal ca be exessed as G ( ( + ( (4 whee G stads fo the otetal fo the geometcal otcs waves ad that fo the dffacted waves. I ths ae a ew feld quatty that s called elemetay dffacted waves s toduced by ( θ g( k( + l ( / ( l dl L θ (5 whee stads fo the otetal fo the elemetay dffacted waves l the coodate take alog L(θ l the osto vecto of a ot o L(θ a ut vecto the atclockwse decto omal to L(θ g the Gee s fucto that s gve by g ( H ( /(4. (6 j The model fo wedge dffacto descbed ths ae s fomulated tems of ad t ca be calculated usg W as see q.(5. Physcally howeve s cosdeed as a cotbuto of o L(θ to a ot located at (θ+π as show as g.1. Thus f should be ket W t would be ecessay to daw L(θ the aea outsde W. 2.3 xteso of a wedge sace Let us exted a wedge sace beyod the edges by cooatg mo mages eflected by the edges the sace that s the edges ad ae cosdeed as mos ad moed mages ae assumed to be sead out beyod the edges. The wedges W 1 W 2 W 3... ae sead out beyod ad the wedges W -1 W -2 W ae sead out beyod as show g.2. Let us exess these wedges by W whee stads fo a tege ad coesods to the eflecto umbe. The wedge W s bouded by L((2+1Ф ad L((2-1Ф ad let us deote the fome as ad the latte as. If a ot W secfed by (θ* s maged to a ot (θ W by mo eflectos the the followg elato θ ( 1 θ * + 2Φ (7 holds. Let us deote θ* as the ogal agle of θ. The the wedge umbe ad the ogal agle θ* fo ay agle θ ca be assged by the followg elatos g.2 omulato of a vtual sace. d( θ t(( θ / Φ + sg( θ θ* h( θ ( 1 ( θ 2Φ whee t( ad sg( ae fuctos that show the tegal at ad sg of the eal umbe x esectvely. The the otetal at ( θ ca be assged by the followg elato (8 (9 ( θ m ( θ* (1 whee ad θ* ae calculated by qs.(8 ad (9 ad m eflects the amltude evesal case of the soft wedge (m-1. ce the ogal agle s symmetc wth esect to the edge (θ the had wedge s symmetc wth esect to the edge ad atsymmetc the soft wedge. ccodg to the bouday codtos that s / θ fo the had wedge ad fo the soft wedge the cotuty of ad / θ at the edge holds fo had ad soft wedges. mlaly the followg elato holds fo the elemetally dffacted waves ( θ ( m ( θ*. (11 osequetly (θ the had wedge s atsymmetc wth esect to the edge ad symmetc the soft wedge. The cotuty of ad / θ at the edge also holds fo had ad soft wedges sce at the edge s mmedately obtaed fom q.(5 fo the had wedge ad / θ ca be also deved fo the soft wedge. Thus ad ae exteded beyod the edges cotuously. o the sake of late efeeces let us deote the mo mage of W as ad ts osto vecto as ( θ the θ ( 1 θ + Φ (12 2 holds ad the dect waves fom s exessed as ( m H ( k /(4 (13 j whee the case of ths elato s educed to q.(3 sce. 6565

4 coustcs 8 Pas 2.4 omulato of a vtual sace V vtual sace V s defed as a sace that ca be obseved by whee moed mages ae assumed to be sead out beyod the edges. Let us toduce a half le L(θ +π that s a half le that stats fom Q ad us the decto of θ +π as show g.2. If s otated the atclockwse decto utl t touches the cotact takes lace W whee d(θ +π s a oegatve tege. Let us exess ug W as ad a wedge bouded by ad as W ad call t as a atal wedge tucated by. mlaly f s otated the clockwse decto t touches W whee d(θ -π s a oostve tege. Let us exess ug W as ad a wedge bouded by ad as W. The the vtual sace V ca be fomulated by 1 +1 V W W ( W. (14 If ethe o holds the thd tem of q.(14 becomes zeo. The otetals o ad ae dffeet fo most cases ad the otetal V s ot cotuous alog. ccodgly t has bee called as a vtual dscotuty le. 3 Wave aalyss the vtual sace The Gee s theoem s aled to V to deve a ew exesso fo tems of ad the oetes of dffacted waves ae aalyzed usg the ew exesso. 3.1 lcato of Gee s theoem to V Let us daw a closed cuve W fo... as show g.2. ach s comosed by two ccula acs of adus ε (ε<<1 ad γ (γ >>1 esectvely ad two segmets coectg these acs alog the edges. The cetes of cuvatue of acs le o Q ad the ad ε ad δ ae commo fo all. The the followg elato s obtaed by alyg the Gee s theoem to ( ( / l g k l g( k l ( l / dl (15 whee a ccle of vey small adus ceteed at s cluded ad that ceteed at. If s ot cluded sde o ccle s added to. The same stoy holds fo ad. s see g.2 two tegal aths u aallel to the edge that s oe ad the othe +1 ( -1. ce the otetal s cotuous at edges these as of tegals ae cacelled out each othe. d thee ema the tegals alog ad sce s the oly edge g.2 whee the otetal s dscotuous. I ths case les o the exteso of ad the elatos l l + g /. (16 Hold. Thus at the lmts of ε ad γ the tegals alog ad ca be exessed as ( θ +π ad ( θ -π esectvely ad the tegals alog the acs become zeos. osequetly q.(15 ca be ewtte as ( + ( W ( + ( W 1 + ( ( ( θ + π + ( θ π 1 (17 whee the dect waves ae esulted fom the tegals alog the small ccle ceteed at ad ( W takes 1 f є W ad othewse. q.(17 s the exesso of the otetal of waves the wedge tems of the elemetally dffacted waves. The fucto s cluded q.(17 sce the souces ad may ot be cluded the tucated wedges. The sum of the fst thee tems q.(17 comses the geometcal otcs waves. The as see fom qs.(4 ad (17 the dffacted waves s exessed by the sum of the fouth ad ffth tems q.(17 that s ( θ ( θ + π + ( θ π (18 whee s deleted the exesso fo smlcty. The hyscs of wedge dffacto s qute clea g.(17. The fst tem q.(17 chages dscotuously wheeve cosses but ths jum s comesated by the fouth tem so that the otetal chages cotuously. The same stoy holds fo the secod tem ad t s comesated by the ffth tem. The dffacted waves ae exessed by the sum of the fouth ad ffth tems as show g.(18. ccodgly the hyscs wedge dffacto has bee uclea the coveto theoy of dffacto. q.(18 ca be ewtte tems of the ogal agles as ( θ ( m + ( m d ( θ + π d ( θ π (( θ + π * (( θ π * (19 whee qs.(8(9(11 ae used ad case of the soft wedge deeds oly o the ogal agles. 3.2 Poetes of dffacted waves Let us toduce ew vaables xy y b as follows x θ / Φ y ( θ + π * / Φ y b π / 2Φ ( θ + π * / Φ whee agles ae omalzed by Ф ad the aamete b s used to descbe the wedge shae fom ow o. The the followg elatos y ( ( 1 y ( ( 1 ( ( ( x + 2b 2 ( ( x 2b 2( ( t( x / 2 + b + sg( x + 2b ( t( x / 2 b + sg( x 2b (21 ae deved fom q.(8 ad q.(9. d q.(19 ca be ewtte as ( ( ( ( m ( y ( + ( m ( y (. (22 Let take x alog the hozotal axs ad y ad y alog the vetcal axs as show g.3(a ad exame the elato betwee them gahcally. s see fom q.(21 the followg elatos 6566

5 coustcs 8 Pas ( ( y ( y y ( 1 ( ( y (2b 2 ( (23 hold at x. The the eghbohood of x the followg elatos y ( m y ( m x + y x y (24 hold utl ethe y o y becomes geate tha 1 whee m (-1 ( ad m ad y ae show g.3(b as a fucto of b. The two les descbed q.(24 ae aallel to each othe ad the sloe of the le s ethe 1 o -1 as see g.3(b. s a esult of ths veso the two les coss at x1 ad x-1 as show g.3(a. Thus y ad y daw a ectagle that s cled to ethe 45 o 45 ad scbed to a squae of edge legth 2 ad ceteed at the og Bouday codto s metoed the evous secto the two ogal agles coss at θ±ф that s whe les o the edges. The as see fom q.(22 holds at the edges of soft wedge (m-1 that s the Neuma bouday codto s satsfed. I the case of the had wedge (m1 the two tems the ght sde of q.(22 become the same at the edges sce the eflecto umbes ad ae dffeet by 1. Thus the chet bouday codto / θ s satsfed at the edges of had wedge. osequetly t becomes clea that the bouday codtos ae satsfed q.(22. It s the ecessay codto that the exesso fo dffacted waves should satsfy but o covetoal exessos have satsfed t so fa Nodffactve wedge Let q be a atual umbe (q12. If bq holds y s esulted fom q.(23 ad q.(24 becomes y ( x y ( x m x (25 ad the eflecto umbes do ot chage fo x 1. The as see fom q.(22 holds that s the dffacted waves ae detcally zeo fo the soft ad had wedges. Let us deote the wedges that satsfy bq as the odffactve wedges ad b1 coesods to a eflectg lae ad b2 to a cocave wedge of Фπ/4. Ths s the well-kow esult but the covetoal aalyss of dffacted waves ths fact s useless sce thee ae o dffacted waves these wedges [2]. I ths aalyss howeve they ae vey useful sce does exst them ad the otetal them ca be exessed by the geometcal otcs waves ymmety of dffacted waves If b(2q-1/2 that s b1/2.3/2 holds y ±1 s esulted fom q.(23 ad the ectagle chages to a lozege as see g.3(a. The y ad y become symmetc wth esect to x ad the soft wedge g.3 (a two ogal agles fo b (b y ad m fo y as a fucto of b. becomes symmetc wth esect to x ad that the had wedge atsymmetc. s see fom the damod shae g.3(a ( s exessed by the sum of (1 ad (-1 ad they ae the bouday values of as dscussed the secto 2.3. Thus ad / θ hold at x fo the had ad soft wedges esectvely. These ae the ecessay codtos to mata the cotuty of the symmetc. 4 the fa feld I ode to efom the tegal gve by q.(5 to deve the aalytcal exesso of let us assume the odffactve wedge ad exess the otetal as the sum of the dect waves fom the souce that s 2q 1 ( ( (26 whee bq ad s gve by q.(12. Let be the otetal esulted by elacg ( l the tegad of q.(5 by ( l that s ( g( k( + l ( l / dl. (27 θ L( Let us assume that ad ae laced the fa feld of the wedge that s the q.(27 becomes ( ( R / 4cot(( θ θ R ex( jk( + /( j2πk ( 1/ 2 (

6 coustcs 8 Pas whee R s the omalzato facto ad the followg aoxmatos 1/ 2 H ( (2 / π ex( jx + jπ / 4 (29 1/ 2 H1 ( (2 / π ex( jx + j3π / 4. ae used. The the odffactve wedge of bq ca be calculated as 2q 1 θ ( θ ( / 4 R m cot(( θ. ( x (a had.5 x.5 x 1 If q.(12 ad the followg elato [3] -2 q 1 cot( π / q + qcot( qx (31 ae used the q.(3 ca be ewtte as ( θ ( br / 4{cot( π ( θ θ / 4Φ + m ta( π ( θ + θ / 4Φ}. The fo the had wedge (m1 s exessed as ( x (b soft.5 1 x.5 x ( θ ( br cos( πx /{s( πx ad fo the soft wedge (m-1 s( πx } (33-2 g.4 Gahs of omalzed fo x ±.5. ( θ ( br cos( πx /{s( πx s( πx } (34 whee the omalzed agles xθ/ф x θ /Ф ad the followg elato cot ± ta B 2cos( m B /{s( + B + s( B} (35 the soft wedge the followg elato br cos( πx ( { 2 s( πx / 2 + πb s( πx cos( πx + s( πx / 2 πb s( πx } (38 ae used. s see fom qs.(33 ad (34 omalzed by br/2 s exessed as a fucto of x ad x that s t s deedet o b. d t s show g.4(a fo the had wedge ad g.4(b fo the soft wedge. The omalzed shows the dscotuty at xx ad the cuves g.4(a ad (b show the almost same behavo ea xx but dffeet at x±1 as dscussed the secto ce the amltude of the souce s costat ad s sead out ove W t would be easoable to assume that the amltude of s ootoal to the aea of W that s to b as see qs.(33 ad (34. The elatos gve by qs.(33 ad (34 ae oved to hold fo bq that s b12. It would be atual howeve to assume that they also hold fo ay b 1/2. The fo ay b ca be calculated by setg to q.(18 ad q.(18 ca be ewtte tems of the omalzed agle ad b as ( ( x + 2b + ( x 2b. (36 I the case of the had wedge the followg elato br cos( πx / 2 + πb ( { 2 s( πx / 2 + πb s( πx cos( πx / 2 πb + s( πx / 2 πb s( πx } (37 s deved by setg q.(33 to q.(36. I the case of s obtaed ad these elatos agee wth the goous oes lteally [4]. The above statemet about qs.(33 ad (34 s justfed by ths ageemet. I addto the to-dow hyscal cle of VP s justfed by ths calculato sce comlex elatos fo dffacted waves ae eoduced by the smle aalytcal calculato based o the cle. 5 ocluso The hyscs of wedge dffacto s made clea by exessg the wave feld tems of elemetally dffacted waves. The elemetally dffacted waves have smle stuctue ad the fa feld goous solutos of waves dffacted by the wedge ae eoduced wth the hel of ths smle stuctue. By ths esult VP s justfed as the to-dow hyscal cle fo aalyzg waves dffacted by a wedge ad obably by a olygo sce t s comosed by wedges. Refeeces [1] Mtsuho eda J.coust.oc.m (1994. [2] old ommefeld Patal ffeetal quatos Physcs 8 (cademc Pess1949. [3].P.Pudkov Yu..Bychkov.I.Machev Itegals ad ees vol (Godo ad Beach1986 [4]..Joes coustc ad electomagetc Waves 588 (laedo xfod