THE PRINCIPLE OF THE ACTIVE JMC SCATTERER. Seppo Uosukainen


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1 THE PRINCIPLE OF THE ACTIVE JC SCATTERER Seppo Uoukaie VTT Buildig ad Tapot Ai Hadlig Techology ad Acoutic P. O. Bo 1803, FIN VTT, Filad ABSTRACT The piciple of fomulatig the JC method to poduce ecoday ouce that fuctio a active cattee o a hypothetical catteig uface i etablihed, to be applied, e.g., i cocet hall. The eamiatio i baed o the modified JC method, to eue that the logic doe ot lead to the eed of chagig the pimay ouce. The actively eflectig plae eve a a eample of the JC fomulatio fo the active cattee. The olutio fo the actively eflectig plae wok o the local cotol piciple: each eflectig ubaea eed ifomatio of the pimay field oly at that ubaea. The olutio ca alo apply to piecemeal plaa uface ad to mooth cove uface, appoimately. 1. INTRODUCTION The JC method i uitable fo fomulatig the poblem of active oie cotol with the geeal ytem theoy. oe geeally, the method applie to the ehapig of acoutic o ay othe field [1], [2], [3], to wave ecotuctio [3], ad to wave popagatio poblem [4], [5]. It ame oigiate fom the fit thee pioee of the method: Jeel, agiate, ad Caévet [2] (the JC goup). Uoukaie peeted the modified JC method [6]. The modified JC method diffe fom the oigial oe o that i the fome the pimay ouce ae ot chaged i ay cae. The pupoe of thi pape i to etablih, at a geeal level ad epecially applied to acoutic field, the piciple of fomulatig the JC method to poduce ecoday ouce that fuctio a active cattee o a hypothetical catteig uface. A a eample, a actively eflectig plae i itoduced. The eamiatio i baed o the modified JC method, to eue that the logic doe ot lead to the eed of chagig the pimay ouce. A moe detailed peetatio ha bee give by Uoukaie [7]. 2. ODIFIED JC ETHOD I the oigial ituatio thee i a detemiitic field (of ay type) i which liea opeato L (typically a diffeetial opeato) coect ouce S ad field F via LF = S. (1) Itead of field F, field F i deied, which ca be obtaied fom the oigial field uig opeato a F = F. (2) I the oigial JC method, opeato alo weight the oigial ouce to ouce S. I the modified JC method, the oigial ouce alway emai uchaged. Both i the oigial ad modified JC method, thee i a eed fo additioal ouce S uch that field equatio (1) fo the modified field i valid. The field equatio of deied field F with the oigial ouce uchaged i Joit BalticNodic Acoutic eetig 2004, 810 Jue 2004, aieham, Ålad BNA20041
2 L F = S + S. (3) The epeio above, togethe with equatio (1) ad (2), yield fo the ecoday ouce i the modified JC method whee S = L F S = LF LF = F, (4) = L( I), (5) whee I i the idetity opeato Geeal fomulatio 3. JC FORULATION OF THE ACTIVE SCATTERER A hypothetical catteig obtacle with it bouday uface A i defied accodig to Figue 1. e 1 1 = 0 1 = 10 A Figue 1. A hypothetical catteig obtacle. The modified field i aumed to be the um of oigial field F ad ome eta field F (catteed field) F = F + F F ( ) = F ( ), whee i a patial coodiate vecto, ad ad ae opeato. It i uppoed that opeato map vecto o the othe ide of uface A, i.e., i iide i outide A if i outide A A if i iide A =, if i at A, ee Figue 1. It i futhe uppoed that eta field F vaihe iide A ad obey the homogeeou field equatio outide A, F = 0 iide A LF = 0 outide A. The latte fomula, togethe with equatio (1) ad (3), implicate that the oly poible place fo the ecoday ouce ae o uface A. (6) (7) (8) Joit BalticNodic Acoutic eetig 2004, 810 Jue 2004, aieham, Ålad BNA20042
3 The fact that the modified field geeally obey equatio (2) yield = I +, (9) whee opeato opeate o that F ( ) = F ( ). (10) The ecoday ouce, accodig to the modified JC method, ae ow a tated i equatio (4), whee, accodig to equatio (5) ad (9), = L( I) = L, (11) o S ( ) = L F( ) = L F( ). (12) Accodig to equatio (6) ad (8), opeato ha to be of the fom =, (13) 0ε( whee 0 i a cotiuou fuctio of patial coodiate, ε( 1 10 ) i a tep fuctio, ad whee it i uppoed that bouday A i fomed of a cotat 1 uface 1 = 10, ee Figue 1. The ecoday ouce o A ae due to the dicotiuity of at 1 = 10. Equatio (8) ca be witte outide A, utilizig equatio (6) ad (8), a L F ( ) = L F( ) = L 0 F( ) = 0 outide A. (14) Due to the cotiuity of 0, thi mut hold alo at A. Becaue F vaihe iide A, accodig to equatio (8), the equatio above i valid eveywhee, i.e., L F ( ) 0. (15) 0 = Now the ecoday ouce ae, accodig to equatio (12), (13), ad (15), S ( ) = L = L ( 0ε( F ( ) ) = L( 0F ( ) ) ε( + L( ε( ) ( ε( )) F ( ) at A F( ) The fial geeal olutio above deped o the oigial field at A, opeato 0, ad the field opeato opeatig o the tep fuctio at A Applicatio to acoutic field I acoutic field i flowle ad homogeou ideal fluid field F, ouce S, ad opeato L coectig them ae (16) Q0 L = t p F = u q S =, f ρ0 t (17) Joit BalticNodic Acoutic eetig 2004, 810 Jue 2004, aieham, Ålad BNA20043
4 whee t i time, Q 0 ad ρ 0 ae the compeibility ad the deity of the upetubed fluid, p ad u ae the oud peue ad the paticle velocity of the acoutic field, ad q ad f ae the moopole ad dipole ditibutio pe uit volume. Opeato L opeatig o the tep fuctio yield ow 0 ε( 0 e L ( ε( ) = = δ(, (18) ε( 0 ) 0 e 0 whee δ( 1 10 ) i the Diac delta fuctio ad e i a uit outwad omal vecto o uface A, ee Figue 1. The ecoday ouce ae ow, accodig to equatio (16), (17), ad (18), q S = = δ f 0 e p u e u u ( e e 0 = δ = δ u ( 1 10) p pe ( 1 10) 0 p pi, (19) whee opeato 0 ha bee divided ito two opeato, p opeatig o the oud peue, ad u opeatig o the paticle velocity p p p 0 = u u, (20) u ad whee I i idetic dyadic (I a = a I = a). Itegatig epeio (19) with epect to 1 yield uface ecoday ouce ditibutio S o A a q S = = f u u e p pe = u p u e pi u at A. (21) The olutio above fo the acoutic field deped o the oigial oud peue ad the omal compoet of the oigial paticle velocity at A, ad opeato Reflectig plae Dyadic K poducig the eflectio tafomatio of the oigial field with epect to the plae = 0 ca be peeted a [8] K = I 2e e = e e + e e + e e, (22) y y z z whee e i a uit vecto i diectio (omal to the eflectig plae), ee Figue 2. The dyadic of the eflectio tafomatio ivet the omal compoet (with epect to the eflectig plae) of the vecto a oppoite without chagig the othe compoet i ay way. The eflectio tafomatio opeate o both the actual field vecto ad coodiate vecto, ee Figue 2. The tafomed field may be itepeted to be caued by a mio image of the oigial ouce with epect to the uface. The tegth of the mio image ad it ditace fom the eflectig uface ae equal to thoe of the oigial ouce. Joit BalticNodic Acoutic eetig 2004, 810 Jue 2004, aieham, Ålad BNA20044
5 eflectig plae F () F ( K ) K F ( K ) K image d d ouce 0 Figue 2. The effect of the eflectio tafomatio to field vecto F ad coodiate vecto. If the eflectig uface i ot ideal, the amplitude of the eflected field i malle tha that of the oigial field o the eflectig uface. The eflectio may alo chage the phae of the field. Thi ca be take ito accout with comple eflectio coefficiet R. The eflectio coefficiet mut be popely choe to eue that the eflected field atifie the homogeeou field equatio i the half pace > 0. Oe poibility i to ue a eflectio coefficiet idepedet of the agle of icidece. With a eflectio coefficiet choe a tated above, the eflected acoutic field (ubcipt ) obey ( ) = Rp( K ) ( ) = RK u( K ). p (23) u Accodig to the peetatio of the patial vaiable i the eflectio tafomatio, the popagatio diectio with epect to the omal of the plate i chaged ito the oppoite, emaiig oigial i lateal diectio. Opeato ad, defied i equatio (6), (7), (13), ad (20), ae ow 0 = 0 = = K. ε( ) p u 1 = R K The ecoday ouce deitie, accodig to equatio (21), (22), ad (24), ae ow q uu K u = u e S = = e R e = R at = 0. (25) f p pi pi pe The olutio above wok o the local cotol piciple: the ecoday ouce tegth at ay poit o A deped o the oigial field oly at the ame poit. The plaa ecoday ouce epeio i the equatio above appoimately applie to piecemeal plaa uface ad eve to mooth cove uface. I thoe cae the uit vecto i diectio ha to be eplaced with a uit vecto omal to the uface. (24) Joit BalticNodic Acoutic eetig 2004, 810 Jue 2004, aieham, Ålad BNA20045
6 4. CONCLUSIONS The piciple of fomulatig the JC method to poduce ecoday ouce that fuctio a active cattee o a hypothetical catteig uface wa etablihed. The piciple ca be applied, e.g., i the active cotol of the acoutical popetie of cocet hall. The eamiatio wa baed o the modified JC method, to eue that the logic doe ot lead to the eed of chagig the pimay ouce. A a eample of the cattee, a actively eflectig plae wa itoduced. The olutio wok o the local cotol piciple: each eflectig ubaea eed ifomatio of the pimay field oly at that ubaea. The olutio ca alo apply to piecemeal plaa uface ad to mooth cove uface. 5. REFERENCES [1] Jeel,. J.., ad Agevie, O. L. "Active acoutic atteuatio of a comple oie ouce," IteNoie 80, Poceedig, iami, Floida, , [2] Jeel,. J.. "Active oie eductio a a epeimetal applicatio of the geeal ytem theoy," Ite Noie 83, Poceedig, Edibugh, , [3] Illéyi, A., ad Jeel,. "Decodig/ecodig the ouce ifomatio fom/ito oud field: aothe way of udetadig active oie cotol," IteNoie 88, Poceedig, Avigo, , [4] Caévet, G. "Acoutic Popagatio i Apeiodic Taitio Laye ad Waveguide," J. Acout. Soc. Am. 67: , [5] agiate, G., ad Chale, S. "Abobig bouday coditio fo acoutic wave ad Huyge piciple," 16th Iteatioal Coge o Acoutic, Poceedig, Wahigto, , [6] Uoukaie, S. "odified JC ethod i Active Cotol of Soud," Acutica Uited with Acta Acutica 83: , [7] Uoukaie, S. "Active Soud Scattee Baed o the JC ethod," J. Soud Vib. 267: , [8] Lidell, I., ethod fo Electomagetic Field Aalyi, Claedo Pe, Ofod, Joit BalticNodic Acoutic eetig 2004, 810 Jue 2004, aieham, Ålad BNA20046
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