International Symposium on Room Acoustics Satellite Symposium of the 19 th International Congress on Acoustics Seville, September 2007

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1 Itatioal Sypoiu o Roo Acoutic Satllit Sypoiu of th 9 th Itatioal Cog o Acoutic Svill, - Sptb 7 A THEORETICA AAYSIS OF UTI-ODA BASS-TRAPPIG RESOATORS COUPED TO COTRO-ROO ACOUSTICS PACS: C Atu, Joé ; Iácio, Octávio Itituto Itituto Tcológico ucla, Applid Dyaic aboatoy, Etada acioal, 686 Sacavé, Potugal; jatu@it.pt ESAE-Itituto Politécico do Poto, uical Acoutic aboatoy, R. da Algia 53, 4-45 Poto, Potugal; OctavioIacio@a-ipp.pt ABSTRACT Hlholtz oato a oft applid fo th oud qualiatio of cotol oo i codig tudio, though adquat lvllig of th low fqucy acoutic odal oo po. Th ub of cotolld acoutic od dpd o th ctal fqucy ad dapig of oato, a wll a o th odal dity of th cotolld yt withi th oato fqucy ag. I a ct pap w popod to ipov th fficicy of uch dvic by, itad of uig baic Hlholtz oato dvlop hap optiizd ulti-odal oato i od to cop with a lag ub of ituiv oo od. I pit of th poiig ult thu obtaid, futh wo i dd to dotat th faibility of uch appoach. Th pt pap i a futh tp i that dictio by aalyig th acoutic of th fully coupld oo/oato yt. o pcifically, uig a ubtuctu coputatioal appoach w thotically div th coupld acoutical od of cotol oo fittd with val optiizd ulti-od oato. ITRODUCTIO Duig th lat dcad, th ha b o cotovy o which hould b th bt dig picipl fo oud cotol oo, cocig aly thi vbatio ad oud diffuio chaactitic. vthl, th i a gal agt that tog oo oac po hould b avoidd, paticulaly at th low fquci. Idd, th ubalac btw ov-hact of oud at th odal fquci ad th abc of oo po at ati-oac oigiat a dtital lac of uifoity of th oo acoutic po ad udu oud colouatio. Thi ffct i o pooucd fo th fqucy ag wh odal dity ad odal dapig a low. Additioally, th oo diio ay b uch that pac of od occu i ctai fqucy ag, ot oly axiizig th oac ffct but alo catig paatio btw difft pa i th oo fqucy po. Th pobl hav oft b tacld, with o o l fficicy, by th u of Hlholtz oato, ba pal o tub-tap, aog oth. Th ucoupld oac bhaviou of th ba cotol dvic i typically focud o a ctal fqucy of axiu oud aboptio which pad ov a dtid badwidth. Th ub of cotolld acoutic od dpd o val facto aog which a th ctal oac fqucy cho, th odal dity i th cotolld fqucy ag, dapig, ad th atio of th oato to oo volu. Th dg of attuatio of th oac ffct i dpdt ot oly o th ub of uch dvic ud, but alo o thi locatio i th oo, idally clo to pu atiod of th od to cotol. Hlholtz oato hav b paticulaly ud i ay difft applicatio wh a accuat cotol of a igl fqucy i did. Th oato hav b thooughly tudid ic th 9th ctuy bgiig with th wo of Hlholtz. o ctly, val ach bca ittd i th dig ad phyical bhaviou of uch yt [,], o th ffct of baic goty chagig o th oat fqucy [3,4], ad o th acoutical couplig btw th oato ad th oo [5,6], to tio a fw. I a ct pap [7] w uggtd that th fficicy of uch oato ay b igificatly ipovd if, itad of uig baic Hlholtz o dvic with uifo co-ctio, o coplx hap-optiizd oato a ud i od to cop with a lag ub of udiabl acoutic od. Hc w applid optiizatio tchiqu i od to obtai optial hap fo uch dvic o that thy oat at a tagt t of acoutic igvalu, withi ipod phyical ad/o gotical cotait. Howv, a coplt

2 Thotical aalyi of oo/oato coupld acoutic aalyi of thi pobl ha to coid th fqucy hift ad oo odhap ditotio aiig fo th acoutical couplig btw th oo ad th oato, a wll a th vicou bouday lay aboptio ffct which accout fo th dapig at th tac of th oato. Thfo, i pit of th poiig pliiay ult obtaid, futh wo i dd to dotat th faibility of uch appoach. I th pt pap w aaly th acoutic of th fully coupld oo/oato() yt. A thotical thod fo coputig th coplx (diipativ) coupld acoutical od of a cotol oo fittd with o o val ulti-od oato i dvlopd ad th illutatd with a xapl. Such pobl ay b tacld uig but-foc uical tchiqu, fo itac th FE o BE thod, coupld with a uitabl odl fo th acoutical dapig phoa. Howv uch appoach i highly coput itiv, ivolvig thouad of dg of fdo, ad hc i ill adaptd to th xtiv odal coputatio calld by a optiizatio pocdu. Hc w dvlop h a ubtuctu coputatioal thod, th coupld acoutical od big coputd fo th ducd odal bai t of th iolatd oo ad oato(), aud clod at thi itfac(). Such appoach i highly advatagou: (a) It lad to aipulatio of a fw hudd od at ot; (b) Wh optiizig th hap ad locatio of th oato() th odal bai of th baic oo oly ha to b coputd oc. I cotat with th appoach dvlopd i th xcllt pap by Fahy ad Schofild [5], th thod popod h appli to ulti-od oato, a itdd, i th piit of [7]. It ca b viwd a xtio of th palty foulatio ptd by Axia & Atu [8], uitabl fo coupld volu with igificat itfac dapig. COSERVATIVE ODE FOR COUPED ROO / RESOATORS W will fit add th baic covativ acoutical pobl, which will th b xtdd to iclud diipativ phoa. h y S z S Q ( ) t x Figu.- (a) Baic oud cotol oo coupld with ulti-od oato; (b) Itfac odl Thotical foulatio Figu how th tch of a cotol oo of gic hap, coupld with oatig ulti-od dvic. O ay wit a ihoogou wav quatio fo th oo fo itac [,5,8]: Q Q p (, ) (, ) = ρ δ( ) + δ( ) t p t () c t t = t H th xcitatio Q( t ) i a giv volu-vlocity poit-ouc. Th Q ( ) = & t Sξ( t ) a localizd ouc latd to th acoutical flow btw th oato() ad th oo, wh ξ ( t ) a th (co-ctio avagd) acoutical diplact i th aociatd itfac S at locatio. Thfo () i witt: = ρ δ + ξ δ & && && p (, t) c p (, t) c Q ( ) S ( ) = () Ad fo ach coupld oato w hav: = ρ ξ δ p&& (, t) c p (, t) c S && ( ) ; =,,..., (3) Equatio (-3) ut b uppltd with uitabl clou coditio, to iu th copatibility of th acoutical flow at th oato itfac. W will au that th a of fiit dpth h, with both S ad h uch all tha th acoutic wavlgth of itt, o that th flow ay b potulatd icopibl i th all itfac volu v = Sh. Hc th dyaical balac of th fluid iid th: ISRA 7 SEVIA

3 Thotical aalyi of oo/oato coupld acoutic ρ && ξ = && ξ = S ( ) (, ) (, ) ( ) = ρ (, ) ( h t S p t p t t p t p, t) ;,,..., (4) h Thi clod t of quatio ay b futh iplifid by fdig (4) ito (-3). W th obtai th quivalt copact fo: c && p (, ) = δ δ t c p (, t) S p(, t) ( ) p (, t) ( ) h (5) = c && p (, ) = δ δ t c p(, t) S p(, t) ( ) p (, t) ( ) ; =,,..., (6) h I a viw copltly difft fo (-4), quatio (5-6) ay b a a palty foulatio fo th coupld pobl, by taig K cs h a palty paat which foc th pu fild of th two coctd ubyt to b a-idtical at ach itfac. Actually, otic that fo all dpth h th valu of K will b quit lag, a thy hould, o that both th volu-ouc foulatio (-4) ad th quivalt yt (5-6) a coitt with th palty appoach dvlopd i [8]. W will u (-4) i th followig, bcau thi foulatio i o aabl to add th gal diipativ pobl. odal foulatio Equatio (-4) will b ow dictizd uig odal pojctio. A uual, th pu fild a dfid a: = φ = φ ( ) ( ) ( ) ( ) p (, t) ( ) P ad p (, t) ( ) P ; =,,..., (7) = = wh φ ( ) ad φ ( ) a th pu odhap pctivly of th iolatd (clod) oo ad (clod) oato. Rplacig (7) ito () ad pojctig th ultig quatio o th oo od w obtai: ( ) φ && ( ) ( ) ( ) ( ) ( ) φ φ ( ) P ( ) dv c ( ) P φ ( ) dv = V = = V (8) = ρ & ( ) φ δ + && ( ) c Q ( ) ( ) dv Sξ φ ( ) δ( ) dv ; =,,..., V = V ad, accoutig fo th odal othogoality, all co-t i th ight-had id of (8) vaih, xcpt fo th t =, o that: with: ISRA 7 SEVIA 3 ( )&& ( ) ( ) ( ) + = ρ & ( ) φ + && ( ) A P B P c Q ( ) Sξ φ ( ) ; =,,..., (9) = ( ) ( ) ( ) ( ) ( ) ( ) ( ) A = φ ( ) ; = φ dv B c ( ) φ ( ) dv = ( ω ) A ; =,,..., () V V Siilaly, fo (3), w obtai fo ach oato: ( )&& ( ) ( ) ( ) + = ρ && ( ) A P B P c S ξ φ ( ) ; =,,..., ; =,,..., () with: ( ) ( ) ( ) ( ) ( ) ( ) ( ) A = φ = φ ( ) dv ; B c ( ) φ ( ) dv = ( ω ) A ; =,,..., ; =,,..., () V V Fially, placig (7) ito th copatibility quatio (4): && ( ) ( ) ( ) ( ) ξ = φ ( ) P φ ( ) P ; =,,..., ρ h = = + + quatio: = ( ) ( ) ( ) A C φ ( ) C φ ( ) && ( ) ( ) P B ( ) P ( ) φ ( ) O O ( ) ( ) ( ) O A C φ ( ) φ ( ) C && ( ) ( ) P B ( ) P ( ) ( ) t φ ( ) () () () A C φ ( ) && () P B () P O O O () () A Cφ ( ) && () P () B () P O + O = cρ & Q ( ) ( ) A C φ ( ) && ( ) ( ) P B ( ) P O O O ( ) ( ) A Cφ ( ) && ( ) ( ) P B ( ) P ( ) ( ) () () && ξ φ φ φ φ D ( ) D ( ) ( ) ( ) D D ξ O O O O O && ( ) ( ) ( ) ( ) ξ D φ ( ) Dφ ( ) φ ( ) φ ( ) D D ξ ( ) t W ow abl quatio (9-3) ito covit atix fo, coitig o (3) (4)

4 Thotical aalyi of oo/oato coupld acoutic H oly th lvat o-zo ub-atic a highlightd ad w hav itoducd paat ρ D = ρ h. Th coupld yt atix quatio (4) ay b witt i copact fo a: C c S ad ( ) = o: A [ ] [ ] [ ] A [ ] {&& } {&& } [ ] [ ] [ ] [ ] [ ] [ ] ( ) ( ) ( ) ( ) C P ( ) B P ( ) { φ ( )} () () () () C P B () { P } + { } O O = ρ & c Q ( ) ( ) [ ] [ ] { && ( ) ( ) } [ ] [ ] A C P B [ ( ) ] { P } { } ( ) () ( ) [ ] [ ] [ ] [ ] {&& ξ } ( t ) D D [ ] { ξ } { } D [ ]{&& } [ ]{ } { } ρ Th, auig ig-olutio of th fo { } = { } (6) lad to th claic igvalu pobl: wh λ ω ISRA 7 SEVIA 4 { } (5) A V + B V = E c Q& ( t ) (6) V V xp( λ t ), th hoogou quatio tig fo ([ ] + λ [ ]){ } = { } B A V (7) a th cicula fquci of th coupld od ad { } V a th copodig igvcto. Fially th coupld odhap ψ ( c ) a coputd by odal cobiatio (withi th coplt doai c ) of th oigial ub-yt odhap: ψ = Φ Φ ( c) ( ) ( ) ( ) ( ) ( ) ( ) V ( ) V (8) T T { } { } U { } { } c = ( wh { ) ( V } a th igvcto copot ptaiig to th t of oo odhap { ) ( )} ( ) { V } a tho ptaiig to th t of oato odhap { Φ ( ) ( )}. Φ, whil th Illutativ xapl A a ptativ but ipl illutatio of th pcdig foulatio, th followig will b addd, coitig o a ho-box oo with diio x = 5, y = 9 ad z = 4, which i coupld with additioal oato. Th a ipl cylid with lgth = 3 ad diat D =.5, locatd pctivly at poit = [.5..8] ad = [ ]. Th itfac aa, ach with S., a tho of th cylid co-ctio. Th valu of phyical paat a c = 343 / ad ρ =.5 Kg/ 3. Th od of both th ub-yt ca b witt aalytically. Fo th clod oo, odal fquci ad odhap a giv a: / ( ) c i j ( ) iπ x jπ y π z fij = + + ; φ (,, ) = co co co (,, =,,,...) ij x y z i j (9) x y z x y z both big otd i od of icaig fqucy ad th tucatd byod a giv fqucy, typically.5~ ti th axiu fqucy of itt fo th coupld od to b coputd. Th cylid a odlld i t of pla wav, o that fo th clod oato: π ( c ) = ; φ ( ) ( ) = co f ( =,,,...) () Coputatio of th odal cofficit () ad () i taightfowad, ladig to: ( ω ) ( ω ) ( ) P ( i, j, ) ( ) ( ) ( ) ( ) P ( ) ( ) ( ) ( ) A = ; B = A ; A = S ; B = A () ij x y z ij ij ij wh P ( i, j, ) ad P ( ) a th ub of o-zo idx fo ach od (9) ad (). Coputatio of ( th couplig cofficit i ub-atic ) ( C, ) ( C, ) ( ) D ad D of (5) pt o difficulty. Tabl.- odal fquci of th iolatd clod ub-yt od Roo Roato Tabl diplay th fit odal fquci of th idpdt ub-yt, clod at thi itfac. A o xtiv viw of th odal ba i how i Figu (a). Coputatio of th yt coupld od wa pfod uig all od i th ag ~ Hz, aly 8 oo od ad 4 oato od, ladig to atix iz of 8x8 i Equatio (5). Th coupld yt odal fquci a

5 Thotical aalyi of oo/oato coupld acoutic how i Tabl ad Figu (b), coputd uig th pt odl ad th FE, pctivly. Th fiitlt odl, which i ta a a copaig fc, calld fo o tha 5 dg of fdo, patly du to th fid h dd i th oo/oato itfac taitio gio Figu 3. I thi paticula ca th pt appoach wa fat by th od of agitud. Tabl.- odal fquci of th coupld yt od Pt appoach FE Figu.- (a) odal fquci of th iolatd (clod) oo ad oato; (b) odal fquci of th coupld yt Bcau uch dyaical bhaviou i at th oppoit of th bouday coditio i th tatig odal bai of thi foulatio, th paticula od a o itiv to th odal tucatio i th coputatioal odl. Idd, icaig futh th fqucy ag of coputatio ultd i dcaig th o. Futho, th ult obtaid a alot iitiv to th (all) valu of th itfac dpth, alo a xpctd, which wa i th pt coputatio ta a h =.. Figu 3.- h ud i th FE coputatio Th ult i Tabl how gligibl diffc fo all th coputd coupld od, xcpt fo od 3 ad 4 (wh all acoutical activity lay o th oato), which diplay a a-od of pu at th itfac Figu 4. Figu 4 diplay th fit i (o-zo fqucy) coupld odhap coputd by th pt thod, which a aly iditiguihabl fo tho tig fo th FE coputatio. Th plot claly highlight th od wh oo/oato couplig i igificat ad tho wh th ubyt ai dcoupld. DISSIPATIVE ODE FOR COUPED ROO / RESOATORS ac of pac pvt u fo ptig h i dtail th diipativ ca, of paticula lvac fo applicatio, howv th ai ida will b tchd. Diipativ phoa will b odlld: (a) Though odal dapig cofficit i quatio (9) ad (), to cov th gy aboptio i oo ad acoutic volu though th uual diipativ poc (ai vicoity, wall ad fuitu aboptio, tc.). Such dapig cofficit ζ ( ) ad ζ ( ) a typically low, a fw pct at ot, ad ay owhat abuivly b potulatd a th ult of popotioal dapig. I oth wod, th odal bai of th dcoupld ubyt a potulatd to b al, a fa a th phoa a cocd. Hc: ( )&& ( ) ( ) + & ( ) ( ) ( ) + = ρ & ( ) φ + && ( ) A P Z P B P c Q ( ) Sξ φ ( ) ; =,,..., () = ( )&& ( ) ( ) + & ( ) ( ) ( ) + = ρ && ( ) A P Z P B P c S ξ φ ( ) ; =,,..., ; =,,..., (3) (b) At th oo/oato itfac, tog diipatio ay ai du to local vicou phoa, which ay b igificatly icad by th u of dapig poou atial with pcific acoutic itac (p uit aa) η - Figu. Th, at ach itfac, th dyaic balac quatio (4) i placd by: && ξ + η & ξ = S = ρ p(, t) p (, t) ;,,..., (4) h Such togly local dapig lad to coplx (.g. o-al) acoutical od fo th coupld yt. Fially, aft odal pojctio of (-4) w obtai quatio (5), which ay b olvd uig th uual T tafoatio to th fit-od fo { W } = V, V & : ISRA 7 SEVIA 5

6 Thotical aalyi of oo/oato coupld acoutic [ ]{&& } [ ]{& } [ ]{ } { } ρ A V + D V + B V = E c Q& ( t ) (5) Figu 4.- odhap of th fit coupld od coputd uig th popod thod FORCED RESPOSES To copa th fqucy-po fuctio of th oigial oo with tho of th coupld oo/oato(), th volu-ouc at xcitatio locatio i aud haoic, Q ( t ) = Q xp( iωt ), with a wpig fqucy i th ag ω ω ax. I tady-tat gi, po will alo b haoic at th xcitatio fqucy, o that th po of th coupld yt ay b foally witt a: ( ) { } { } = [ ] + [ ] [ ] V( ω) B iω D ω A E iωc ρ Q ( ω) (6) fo which th odal ad phyical po at ay giv locatio o ay b coputd. Fo th oigial ucoupld oo, th pcdig quatio iplify datically, a thy oly cotai th oigial oo ucoupld od. COCUSIOS I thi pap w hav addd th pobl of coputig th od of a oo wh coupld to a t of ulti-od oato, accoutig fo th vicou diipativ phoa at th oo/oato itfac. A ipl but ptativ xapl wa ptd ad th ult copad, fo th covativ ca, with FE coputatio. Th odal/ub-tuctuig tchiqu ud h i uch l coput-itiv tha th fiit-lt appoach. Futho, icopoatio of diipativ ffct i lativly taightfowad, th hot aalyi tchd h big xpadd lwh. I fothcoig pap w will u th pt odl fo coputig th focd po of coupld oo/oato ad optiiz th hap of t of ulti-odal oato (a wll a thi locatio) to obtai optial oo qualizatio. Rfc. K.Igad: O th Thoy ad Dig of Acoutic Roato. Joual of th Acoutical Socity of Aica 5 (953) P.o, K.Igad: Thotical Acoutic. Picto (968). 3. R.Chaaud: Effct of Goty o th Roac Fqucy of Hlholtz Roato. Joual of Soud ad Vibatio 78 (994) D.i, J.Vippa: O th dig of log T-hapd acoutic oato. Joual of th Acoutical Socity of Aica 6 (4) F.Fahy, C.Schofild: A ot o th Itactio Btw a Hlholtz Roato ad a Acoutic od of a Eclou. Joual of Soud ad Vibatio 7 (98) A.Cuig: Th Effct of a Roato Aay o th Soud Fild i a Cavity. Joual of Soud ad Vibatio 54 (99) O.Iácio,.Hiqu, J.Atu: Dig of Duct Co-ctioal Aa i Ba-Tappig Roato fo Cotol-Roo. oi Cotol Egiig Joual 55 (7) F.Axia, J.Atu: odllig of chaical Syt: Fluid-Stuctu Itactio. Elvi (7). ISRA 7 SEVIA 6