An analytical solution versus half space BEM formulation for acoustic radiation and scattering from a rigid sphere

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1 Journl of Phyc: Conference Sere PAPER OPEN ACCESS An nlycl oluon veru hlf pce forulon for couc rdon nd cerng fro rgd phere To ce h rcle: B. Soenrko nd D. Sedkrun 06 J. Phy.: Conf. Ser Reled conen - Acouc Rdon Induced by Ineny Moduled Ion Be Kenj Kur, Kenj Nknh, Akr Nhur e l. - Meureen of vcoy by he ubhronc preure hrehold for bubble n vcou lqud P W Vughn - Applcon of he boundry eleen ehod o he udy ofboundry poenl n elecrcl pednce oogrphy Y Sh, P M Mrl, W W D e l. Vew he rcle onlne for upde nd enhnceen. Th conen w downloded fro IP ddre on /0/08 3:5

2 Journl of Phyc: Conference Sere 776 (06) 0065 do:0.088/ /776//0065 An nlycl oluon veru hlf pce forulon for couc rdon nd cerng fro rgd phere B. Soenrko nd D. Sedkrun Elecrcl Engneerng Depren, Unver Kren Mrnh, Jl. Prof.Drg.Sury Sunr 65, Bndung 4064, INDONESIA E-l: Abrc. A hlf pce proble n couc decrbed by nroducng n nfne plne boundry h reflec he wve cong no he plne. A nuercl oluon ung Boundry Eleen Mehod () h been known whch foruled ung odfed Green funcon n he Helholz Inegrl Forulon, whch elne he dcrezon over he nfne plne. Hence, he dcrezon re confned o he body or obcle n queon only. Th feure conue he n dvnge of he forulon for hlf pce proble. However, no generl nlycl oluon vlble o verfy he reul for hlf pce proble. Th pper ed o propoe n nlycl oluon for he o copre wh, hence o verfy he clculon. Th nlycl pproch currenly developed for hlf pce proble nvolvng rdon nd cerng of couc wve fro rgd phere. The ge of phere well he ge of he feld pon re defned wh repec o he nfne plne. Then, n d hoc oluon ued nvolvng conn nd he dnce fro he cener of he phere o he feld pon nd he dnce fro he cener of he ge of he phere o he feld pon. The conn deerned by pong he boundry condon. Te ce were run wh everl confguron nvolvng he locon of feld pon nd he phere. Copron of he nlycl oluon wh clculon how good greeen beween he wo reul...inroducon The Boundry Eleen Mehod () one of he nuercl ehod h h been known o reercher n ny re of reerch. The n dvnge offer by he h requre only he dcrezon of he boundry of he body n queon. Therefore, n h chee he denon of he proble reduced by one, e.g. hree denonl proble olved ung wo denonl clculon n he. In couc, he pplcon of he nclude ny cle of proble nvolvng rdon nd cerng of couc wve fro rbrry hpe bode. Inenve developen n he hve been propoed by que nuber of reercher [-9]. The couc edu y be confned n fne or nfne don. Fne don of couc ed found n o clled neror proble n couc, whle couc proble wh nfne don known exeror proble n couc. When he nfne don bounded by n nfne plne, he proble becoe hlf pce proble. The forulon for hlf pce proble ue odfed Green funcon n he Helholz Inegrl Forulon nd nclude he ge of he vbrng body n he rdon proble or he obcle n he cerng proble. Th forulon vod he dcrezon of he nfne plne whch plfe he nuercl clculon nce nvolve only he dcrezon of he rdng body n he rdon proble or he obcle n he cerng proble. However, no Conen fro h work y be ued under he er of he Creve Coon Arbuon 3.0 lcence. Any furher drbuon of h work u nn rbuon o he uhor() nd he le of he work, journl con nd DOI. Publhed under lcence by Ld

3 Journl of Phyc: Conference Sere 776 (06) 0065 do:0.088/ /776//0065 nlycl oluon vlble for hlf pce proble o verfy he reul. I he of h pper o propoe n nlycl oluon o verfy he clculon, pecfclly for rdon nd cerng proble n hlf pce for rgd phere. Te ce were run for everl confguron wh repec o he feld pon locon nd he dnce beween he cener of he phere nd he nfne plne. Copron of he reul obned fro he nlycl oluon wh hoe fro he clculon how very good greeen. Hence he reul re well verfed..the Hlf Spce Forulon..Rdon The forulon for couc rdon n hlf pce gven by he followng equon [6,0] : H ( ( P. H ( P, ( ds( C( P) ( P), () S0 where S 0 he urfce of he rdng objec, ψ H he hlf pce Green funcon defned by kr( P, / R( P, R exp kr( P, / R( P, ) ( P, exp Q, () H p where Q ny pon on S 0, P y on S 0 or oude of S 0, k he wve nuber /c where c he peed of ound nd ω he frequency, nd he ouwrd un norl on S 0, R p he reflecon coeffcen of he plne, C( P) 4 ds( R( P,, (3) S0..Scerng For cerng proble n hlf pce, he forulon ke he for [6,0]: H ( ( P. H ( P, ( ds( 4 ( P) C( P) ( P), (4) S0 where he ol velocy poenl, he cered velocy poenl, he velocy poenl of he ncong wve, (P) he u of he velocy poenl of he ncong wve nd he velocy poenl P. 3. Anlycl oluon for rdon fro rgd phere n hlf pce Conder unforly pulng phere of rdu loced wh cener O dnce B fro n nfne rgd plne hown n Fgure. The orgn of he coordne ye defned he cener of he phere. Le O be he cener of he ge of he phere wh repec o he plne. The velocy on he urfce of he phere U e ω. Now ue n d hoc oluon: ( r, ) A ( r) e ( r ) e (5) ( kr) ( kr ) where r he dnce fro O o ny feld pon P, r he dnce fro O o P, nd A conn.

4 Journl of Phyc: Conference Sere 776 (06) 0065 do:0.088/ /776//0065 Fgure Rdon fro pulng phere n hlf pce On : r n r r z z ( x y z ) z r x y ( z B) z B r n r z z where n he un norl on. Furherore r=r nd z=b on. Therefore r n B r o h r n B r nd r n r n (6) Then kr ) kr r k r e r n r k r e r n 0 n A (7) SH ( by vrue of Eq.(6). Thu he d hoc oluon (Eq.(5)) fe he boundry condon on. The conn A cn be deerned fro he boundry condon on he phere,.e.: SH or r r r U r U (8) where he e dependen fcor e uppreed for brevy. Expre r n er of r nd θ: r 4 r B 4Br co (9) where he ngle beween r nd he z-x. Then Ung Eq.(5) n Eq.(8) n lgh of Eq.(0) yeld: r Bco r 4B 4Br r r co (0) 3

5 Journl of Phyc: Conference Sere 776 (06) 0065 do:0.088/ /776//0065 A U k ke kq Q B co Q e kq () co where Q 4B 4B () Incorporng Eq.() nd Eq.() n Eq.(5) gve he fnl oluon follow: ( r, ) 3 kr krq 3 k ru Q e e Qe kq e kq B co where Q 4B r 4B r co e kq (3) (4) Eq.(3) nd Eq.(4) yeld he nlycl oluon of he rdon of pulng rgd phere n hlf pce wh rgd nfne plne. I y be noed h A no pure conn ued n Eq.(5), bu A A(co ). However, kng no ccoun h co co n he vcny of he plne, cn be hown h he boundry condon on he rgd plne fed. 4. Anlycl oluon for cerng fro rgd phere n hlf pce Suppoe he phere decrbed n Fg. no ovng nd pnged by plne wve kx 0 e hown n Fg.. The ncong wve cn be expnded no ere of phercl wve [0]: 0 0e ( ) P (co ) j ( kr) (5) where P (co) re Legendre polynol of order, j (kr) re phercl Beel funcon of order nd he ngle beween r nd he ncong wve. The ge of he phere h cener O, r he dnce fro O o ny feld pon P of nere. Fgure Scerng fro rgd phere n hlf pce Aue he cered poenl o be gven by 0 P ) h ( kr) b P (co ) h ( kr e (co ) (6) 4

6 Journl of Phyc: Conference Sere 776 (06) 0065 do:0.088/ /776//0065 where he ngle beween r nd he drecon of he ncong wve, h (kr) re he Hnkel funcon of he econd knd. Eq.(6) decrbe he cered preure ere of phercl wve fro he orgn O nd fro O. Ech er n Eq.(6) oced wh he correpondng er n Eq.(5). The ol poenl gven by (7) Snce he plne rgd, he boundry condon on he plne z 0. Thu, z( ) 0 Drop he e dependen fcor e ω kx for brevy, z z( 0 e ) 0, o SH z 0 (8) Ung Eq.(6) n Eq.(8) nd nong h for feld pon on he plne: z=b, = nd r=r, cn be hown h = b. Hence, Eq.(6) cn be rewren: P ) h ( kr) P (co ) h ( kr e (co ) 0 (9) The boundry condon on he phere (he phere rgd) : r 0 or r r 0 (0) r The -h er of I nd u fy Eq.(0). Subung Eq.(5) nd Eq.(9) n Eq.(0) wh lle hecl book keepng gve: r r where ( ) kp (co ) D ( k)n ( k) Ter Ter 3 0 Ter () Ter ( k) P (co ) kh ( k) kp (co ) D ( k) e () Ter Ter P (co )( kd ( kr 0 0 ) e ( kr0) ) ( B )co Q h ( ) 3 co P (co ) P (co ) co co 4 B B co Q kr0 (3) (4) δ (k) he vlue of δ (kr) for r = n whch δ (kr) defned uch h nd D (kr) = h (kr) (kr) = D (kr)e {δ (kr)+ π } + [{j (kr) ( + )j +(kr)} + {n (kr) ( + )n + (kr)} ] j (kr) re he phercl Beel funcon wh rguen kr 5

7 Journl of Phyc: Conference Sere 776 (06) 0065 do:0.088/ /776//0065 n (kr) re he phercl Neunn funcon wh rguen kr r 0 = [4B + -4Bcoθ] / Mulplyng he nueror nd he denonor by, Eq.() becoe: where k ( ) P (co ) D ( k)n ( k) D D D (5) o D = Ter, D = Ter, D 3 = Ter 3 I y be noed h doe no look lke conn ued n Eq.(6), bu = (coθ). However, cn be hown h z 0 whch ply h z 0. Furherore, cn lo be hown h z 0. Thu (Eq.(5)), when ued n Eq.(9) wll fy he boundry condon on,.e. z 0. Hence, he oluon for he cered poenl cn be expreed : k ( ) P (co ) D ( k)n ( k). P (co ) h ( kr) P (co ) h ( kr D D D 0 e ) 0 Equon (6) he nlycl oluon of he cerng fro rgd phere n hlf pce wh rgd nfne plne. 5. Reul nd dcuon A nuber of e ce were run ung he propoed nlycl oluon nvolvng rdon nd cerng of couc wve n hlf pce where copron wh he clculon were exned. The nfne plne ued o be rgd. The objec nvolved rgd phere loced vrey of dnce fro he nfne plne. The fr e ce rdon fro pulng rgd phere. The cener of he phere loced dnce B=3 fro he nfne plne. The phere vbrng wh velocy on he urfce equl o U. The boundry condon on he phere unfor U (pulng phere), nd he norlzed frequency of vbron of he phere k =. Fgure 3 nd 4 how he norlzed couc preure (p=kz 0 ) dnce of 3 nd fro he cener of he phere, repecvely funcon of polr ngle ploed on plne png hrough he cener of he phere nd perpendculr o he nfne plne. The plo of he wo confguron how good greeen beween he nlycl oluon nd he reul. 3 3 (6) 6

8 Journl of Phyc: Conference Sere 776 (06) 0065 do:0.088/ /776//0065 Norlzed Rded Preure v Polr Angle y=0 vercl plne Anlycl Fgure 3 Norlzed rded preure dnce r = 3, when k =, B = 3 Norlzed Rded Preure v Polr Angle y=0 vercl plne Anlycl Fgure 4 Norlzed rded preure dnce r =, when k =, B = 3 Nex we loce he phere cloer o he plne uch h B= for he e frequency he prevou ce,.e. k =. Fgure 5 how he norlzed rded preure ploed veru polr ngle dnce r=. The gree wh he nlycl reul o whn 5% for h confguron. Norlzed Rded Preure v Polr Angle y=0 vercl plne Anlycl Fgure 5 Norlzed rded preure dnce r =, when k =, B = The nex e ce for cerng proble condered he cerng fro rgd phere of rdu loced dnce B=3 fro he nfne rgd plne. The ncden wve plne wve wh frequency of k=, propgng prllel o he rgd plne. Fgure 6 nd 7 re he polr plo of he cered velocy poenl, norlzed by he velocy poenl of he ncden wve, ploed dnce of 3 nd repecvely fro he cener of he phere. Boh Fgure 6 nd 7 how good greeen beween he nd he nlycl reul 7

9 Journl of Phyc: Conference Sere 776 (06) 0065 do:0.088/ /776//0065 / v Polr Angle for Rgd Sphere y=0 vercl plne Anlycl 0.05 Fgure 6 Scered velocy poenl dnce r = 3, when k =, B = Anlycl Fgure 7 Scered velocy poenl dnce r =, when k =, B = 3 The followng e ce deonre noher confguron when he phere loced cloer o he rgd plne,.e. B=. Fgure 8 depc he drecvy pern dnce r=. I cn be oberved h he nlycl oluon nd he reul gree very well. / v Polr Angle for Rgd Sphere y=0 vercl plne Anlycl Fgure 8 Scered velocy poenl dnce r =, when k =, B = In concluon, n nlycl oluon h been derved for hlf pce proble nvolvng rdon nd cerng of couc wve fro rgd phere bounded by n nfne rgd plne. Verfcon h been deonred for vrou confguron kng vree of he dnce of he phere fro he nfne plne nd he dnce of he feld pon fro he cener of he phere where he couc feld ncped. Copron wh he clculon how good greeen beween he wo reul. A he e e he nlycl oluon y be ued verfcon ool for nuercl ehod clculon uch he.. 8

10 Journl of Phyc: Conference Sere 776 (06) 0065 do:0.088/ /776//0065 Reference [] Cherock G 964 J. Acou. Soc. A [] Schenck H.A. 968 J.Acou.Soc.A., [3] Engblo J.J nd Nelon R.B. 975 J.Appl.Mech [4] Meyer W.L. Bell W.A., Znn B.T., nd M.P.Sllybr M.P 978 J.Sound nd Vb [5] Koopn G.H. nd Benner H 98 J.Acou.Soc.A [6] Seyber A.F., Soenrko B, Rzzo F.J., nd Shppy D.J. 985 J.Acou.Soc.A, [7] Seyber A.F., Soenrko B.,Rzzo F.J., nd Shppy D.J. 986 J.Acou.Soc.A [8] Lch M.A. nd Aky A. 986 J.Vb.Acou.Sre Relb De [9] Jng J.K. nd Prd M.G., 986 J.Vb.Acou.Sre.Relb De [0] Soenrko B, PhD The, Unvery of Kenucky, 983 9

Introduction. Section 9: HIGHER ORDER TWO DIMENSIONAL SHAPE FUNCTIONS

Introduction. Section 9: HIGHER ORDER TWO DIMENSIONAL SHAPE FUNCTIONS Secon 9: HIGHER ORDER TWO DIMESIO SHPE FUCTIOS Inroducon We ne conder hpe funcon for hgher order eleen. To do h n n orderl fhon we nroduce he concep of re coordne. Conder ere of rngulr eleen depced n he