19 th INTERNATIONAL CONGRESS ON ACOUSTICS MADRID, 2-7 SEPTEMBER 2007

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1 9 h IERAIOAL COGRESS O ACOUSICS MADRID, -7 SEPEMBER 7 AALYICAL AD EXPERIMEAL SUDIES OF HE HERSCHEL- QUICKE UBES BY IS ACOUSIC SCAERIG MARIX PACS: 43.5.Cs Poiie, Benjamin ; Fouca, Fex ; Mauy, Cédic ; Ville, Jean-Michel Laboaoie Robeval FRE UC-CRS o. 833, Univesié de echnogie de Comiègne, BP 59 F65, Comiègne Cedex, Fance ; boiie@uc.f ABSRAC he conce of HQ ubes which is known since he beginning of he h cenuy has been shown o be of inees o educe noise fom ubofan engine. o hel o undesand he hysical henomenon undelying he HQ conce, we chose o base ou wok on he chaaceizaion of he HQ sysem by is scaeing maix [S]. Resuls of some yical coefficiens of [S] deduced fom a D and 3D model ae comaed wih exeimenal esuls. he 3D modeng echnique consides he ubes-inle inefaces as finie ison souces ha coule he acousic field inside a had-walled duc wih he acousic field wihin he HQ ubes. he exeimenal echnique equies measuemen of he modal amudes of essue waves. When he numbe of ubes inceases, he analyical esuls edic an incease of he esonance s amude and a shif of he esonance fequency o he high fequency which ae confimed by he exeience. eveheless, he analyical model does no edic he cazaion of HQ esonances a he same fequencies han exeimenally. IRODUCIO As ai affic coninues o gow u, ubofan engines eesen an inceasingly significan souce of noise olluion, aiculaly a aios aound he communiies. he conce of Heschel- Quincke (HQ ube which was discoveed in he 8 h has shown is oenial o educe he noise a some secific fequencies. he objecive of his sudy is he undesanding of he hysical henomenon which govens he conce. ha s why we chose o base ou wok on he chaaceizaion of he HQ sysem by is scaeing maix [S]. Pevious woks on he HQ conce has consideed he conol of simle one-dimensional lanes waves in ducs [6, 8], bu also he highe-ode modal convesion oeies induced by such a sysem []. In his sudy we oose he scaeing maix fomasm based on a 3D model o chaaceize he acousical oeies of HQ sysem and we confon his o exeimenal esuls. I AALYICAL EXPRESSIO OF HQ UBES SCAERIG MARIX he choice of he scaeing maix [S] was guided by he folwing moivaions. Fis, [S] is a ool which alws a clea ineeaion of he hysical henomena concened: i conains he mode e mode eflecion and ansmission coefficiens as well as he modal convesion coefficiens []. Also, he scaeing maix alws o sudy a sysem excied by a wave coming useam o downseam fom secion HQ simulaing he cases of an insallaion in ai inake o duc oule. Finally, he scaeing maix is useful fo he design of comlex sysems by assembly of simle elemens. An exeimenal faciy and a mehodogy of measuemen of he scaeing maix has been deveed wihin UC [9].

2 DEFO OF HE SCAERIG MARIX OF A DISCOIUIY I a I ( z P ( z P q I L ( z L q R ( z R a z Figue.- Scaeing maix definiion he scaeing maix [ S] of a disconinuiy shown in Figue and caed beween he axial coodinaes zl on side I and zr on side (L and R mean, esecively, he lef side and igh side of he disconinuiy and nked o wo waveguides is a nea elaionshi beween he P q q incoming essue wave veco { } I (......; I veco { } (......; P = P P : = and he ougoing essue wave,, I P S S q = I [ S ] [ ] P S = (,, q S S his maix[ S ], which deends only on he disconinuiy, is indeenden of he useam and downseam acousic condiions and is filled by (² coefficiens whee is he numbe of modes in boh coss secions: S kl, kl, =, and q,,, =,,...,. q, he hysical meaning of each elemenay maix is as folws [7]:,, eflecion of he wave coming ino he elemen fom he lef side;, q, ansmission of he wave coming ino he elemen fom he lef side;, q, eflecion of he wave coming ino he elemen fom he igh side;, q ansmission of he wave coming ino he elemen fom he igh side. Modeng echnique q descibes he descibes he descibes he descibes he he oblem consiss of an infinie cicula igid-walled duc of adius a modelled wih one o seveal cicumfeenial aays of HQ ubes caed aound he duc. his duc eesens he engine inle. A fw of Mach numbe M is assumed o avel in he duc in he osiive z-diecion and he disubance noise field is assumed o oagae wihin he duc in he osiive z- diecion. In he case of ubofan engine, he fw is oagaing in he negaive z-diecion, heefoe, he Mach numbe M is negaive. he essue due o he disubance a any oin in he duc (,?,z, can be exessed as he sum of a se of modes of cicumfeenial ode m and adial ode n given by: Mg g Ι,ΙΙ I, χ imθ ikz (, θ, z = J e e ( m= Mg n= a whee χ / a coesonds o he n h exemum of he Bessel funcion of he fis kind of ode m. is he comlex amude of mode (m,n and m e imθ Disconinuiy J is he fis kind Bessel funcion of ode m. he em indicaes ha he modes ae sinning in he osiive m-diecion. he oagaion chaaceisics fo oagaing (cu-on modes ae given by he axial wave numbes in negaive and osiive diecions esecively k and k exessed as: 9 h IERAIOAL COGRESS O ACOUSICS ICA7MADRID

3 χ and χ k = Mk k ( M² /( M² k = Mk k ( M² /( M² (3 a a he modelng of he HQ ube-duc sysem is caied ou by fis seaaing he HQ ubes fom he duc a he inefaces. he models of he sound fields in he HQ ubes and in he main duc ae hen deveed indeendenly. Finally, hese models ae fully couled by maching he acousic essue and aicle veciy a he ube-duc inefaces. he effec of he ubes on he duc is modelled by consideing he ube-duc inefaces as finie ison souces adiaing ino he duc []. Each ison souce is assumed o have an unknown veciy v, which eesens he aicle veciy a he ends of he ubes (Figue., v, v, v, v Duc modelng he acousic field inside he duc is obained fom he sueosiion of he sound essue due o he disubance and he finie ison souces a he HQ ube ends. In aicula, he aveage essue ove each souce can be exessed in maix foms as folws: { i } [ Z ] { v } { } s i i = (4 i= : i= : i= : whee is he numbe of HQ ubes, v and v ae he aveage inu and ouu veciies of h he l ube esecively; is he aveage essue ove he ison souce due o he disubance, Z is he imedance maix ha elaes he ison souce veciy o he aveage s essue ove anohe ison obseve souce []. he elemens of his maix can be exessed in em of he infinie igid-walled duc Geen s funcion as: u u l Zs = [ jωρ g ( / s ds ] ds (5 S S s whee Figue.- (a Model of he duc wih finie ison souces associaed wih ube ends. (b Model of he ubes. S S is he aea of he obseve souce and g is he Geen s funcion. Fo a oin souce ( a, θ, z osiioned on he duc a s, he Geen s funcion is exessed as: χ imθ Mg J g e i a ik ( z z g (, θ, z, θ, z = e (6 π a² ( M²( k k m= Mg n= whee m is he ohogonazaion consan, M he mach numbe, azimual and adial modes ube modelng M and g g he numbe of In acice, he HQ ubes ae hemi-ooidal. Howeve, fo modelng uoses hey ae consideed as saigh ubes wih unifom coss-secion. he sound fields inside he ubes ae assumed o consis of lanes waves only, a vad assumion well bew he fis cu-on fequency of he ubes. I is also assumed ha hee is no fw inside he ubes. he sound field inside a ube is exessed in ems of osiive and negaive avelng lane waves. he acousic essue and aicle veciy a he ends of he ube can be exessed in he folwing maix fom: 9 h IERAIOAL COGRESS O ACOUSICS ICA7MADRID 3

4 l l cos( sin( kl i kl = and so Zii Zio v = (7 l l ρcv isin( kl cos( kl ρcv l o Zoi Zoo v whee Z, Z, Z, Z ae he ems of he maix of mobiy, L is he lengh of he ube, he l ii l io l oi l oo indices i and o efe esecively o he eny and he exi of he ube. Afe assembly of he maices of ubes HQ, one obains he oal maix of he essues and veciy a he eny and exi of each ube: { } i Z HQ { vi } ube duc coung = (8 i= : i= : he model fo he couled ube-duc sysem is obained by maching he aveage essue on he suface of he souce o he essue in he ube, i.e = i i and he souce veciy in he ube, i.e = v i v i ; oal veciy wih he ineface accoding o he oal essue is deduced fom (4 and he esul subsiued in (8 makes i ossible o obain he folwing elaion (9 which gives he veciy a he inefaces ube-duc as a funcion of he maix of imedance of he sysem couled and he incidenal essues calculaed on he inefaces [] : { v } Z Z l = HQ s { l } (9 l=.. l=.. Analyical exession of HQ scaeing maix he modal inciden essue can be exessed as a veco:, imθ,, ikz = β L β M whee β ( = J k e e ( Fom equaion (9 and ( We can deduce he exession of he veciy accoding o he modal inciden essue (: he modal essue adiaed in side a acousic veciy v ( z is deduced: I ( P,, β L β v,, β β [ Y] [ Y L M = = M M ] v M,, β L β R 9 h IERAIOAL COGRESS O ACOUSICS ICA7MADRID 4 ( z in osiive z-diecion by he ison souce wih, α v ( M = O M = α. [ Y ] β M, I ( P αm n v imθ whee, ( k ρc Jm χ ik ( sin( sin( zd z aα mα d kd e α = e πa² ( M²( k k mα kd We sum hen on all he souces o have he oal ansmied modal essue: I, I I,, I ( P = α [ Y]. β. ( = αm n αm n.[ Y]. β. { i } L (3 = We can hen wie he veco of he oal modal amudes ansmied as a funcion of he veco of he incidenal essues:,, α I L α ( P whee,, I ( P S α L α. [ Y]. β [ I] = =. M M M,, α L α I ( P is he ansmission maix of he wave coming in he elemen fom he lef side. he same wok is made fo,,, and,. i= : (4

9 h IERAIOAL COGRESS O ACOUSICS MADRID, -7 SEPEMBER 7. HE EXPERIMEAL SEUP AD DAA PROCESSIG he measuemen of [S] was caied ou wihou fw wih he duc faciy buil unde a gan wih E.E.C. duing he Duca ogam.

he essue disibuions measued in wo csed coss secions befoe and afe he duc unde es by wo ais of micohones ae Fouie Lommel s ansfomed hen he inciden and efleced modes ae seaaed.

5 9 h IERAIOAL COGRESS O ACOUSICS MADRID, -7 SEPEMBER 7. HE EXPERIMEAL SEUP AD DAA PROCESSIG he measuemen of [S] was caied ou wihou fw wih he duc faciy buil unde a gan wih E.E.C. duing he Duca ogam. I was aleady used o measue eflecion and ansmission maices[9].he essue disibuions measued in wo csed coss secions befoe and afe he duc unde es by wo ais of micohones ae Fouie Lommel s ansfomed hen he inciden and efleced modes ae seaaed. If is now he numbe of cu-on modes, a selecive comuaion of he coefficiens of S is hen efomed afe a ocess based on [ ] he geneaion of indeenden inciden fields fo wo- o six-ad cases deending whehe he diec o ovedeeminaion ocedue is caied ou o no. he ad mehod was chosen insead of he souce [7] one because of acical consideaions. he duc has a diamee of a=75 Figue 3.- Exeimenal seu mm. he ubes have a diamee of d= mm, a lengh of L =.5 cm and a disance beween he inefaces of L= cm. he figue 4 shows an angula sacing of θ = fo he 3 ubes configuaion and θ = 8 fo he ubes configuaion. θ θ (b (a I. EXPERIMEAL RESULS Figue 4- Angula sacing fo wo ubes (a and 3 hee ubes (b configuaions.5.5 R Figue 5.- Scaeing coefficien of eflecion ubes 3 ubes S (a and ansmission S, (b ube,, Figue 5-a,b gives he modulus of esecively he eflecion coefficien of he mode (, fom souce s side S and he ansmission coefficien S fom he lef side o he igh side fo,, (a one ube, wo ubes and hee ubes configuaions. Figue 6-a gives he 3D analyical soluion fo he same configuaions. 3D analyical calculaions wee caied ou on he basis of,,, (b

6 modes in he calculaion of Geen s funcion wihou fw. In figue 5, he ansmission coefficien and he eflecion coefficien oin ou esecively an incease and a decease a =.8 whee he modulus of he coefficien S is equal o.5 fo one ube and.4 and.45 esecively fo wo ubes and hee ubes. I coesonds o he execed effec of ubes HQ [4]. Figue 6-a shows ha when he numbe of ubes inceases, he analyical esuls edic an incease of he esonance s amude and a shif of he esonance fequency o he high fequency which ae confimed by he exeience. eveheless Figue 6-a shows ha analyical model does no edic he cazaion of HQ esonances a he same fequency (=.6 han exeimenally (=.8. Figue 6-b shows he esuls deduced fom he D model deveed by Selame and Dickey [8] fo he same configuaions. he HQ esonance is caed a he same fequency han he aenuaion deduced fom he Hallez s model. his esul confims ha D as 3D models equie an imovemen in he eesenaion of he ube s inefaces.,,.5 (a.5 (b Figue 6.- Scaeing coefficien of ansmission S,, fo ube ubes 3 ubes (a Analyical 3D (b Analyical D COCLUSIOS A heoeical model based on he chaaceizaion of a HQ sysem by is scaeing maix was deveed fo highe ode mode oagaion condiions. One ube, wo and hee ubes configuaions wee esed exeimenally and comaed wih heoy. he analysis of he esuls showed a diffeence a he HQ fequencies. his diffeence is confimed by he esuls fom he D model. he devemen of an analyical mehod which akes ino accoun a bee eesenaion of he ube s ineface could eesen an imovemen of he modeng echnique. he auhos would ke o acknowledge he financial suo fom he BDI-CRS and SECMA gan. Refeences: [] R. F. Hallez, J. P. Smih, and R. A. Budisso, Conol of Highe-Ode Modes in Ducs using Aays of Heschel-Quincke Waveguides, ASME Conol of Vibaion and oise (: -8. [] A.Siel, J.M.Ville and F.Fouca. «Muad exeimenal ocedue fo measuemen of acousic scaeing maix of a duc disconinuiy fo highe ode modes oagaion condiions». JASA, vol n 5, , ovembe 6 [3] J. P. Smih and R. A. Budisso, Acive Conol of Inle oise Fom a ubofan Engine Using Inle Wavenumbe Sensos, CEAS/AIAA (999. [4] J. F. W. Heschel, On he Absoion of Ligh by Coed Media, viewed in connecion wih he Undulaoy heoy, Phisohical Magazine and Jounal of Science. 3 (833: 4-4. [5] G. Quincke, Uebe inefeenzaaae fü schallwellen, Ann. Phys. Chem. 8 (866: [6] G. W. Sewa, he heoy of he Heschel-Quincke ube, Physical Review 3.4 (98: [7] A Siel Méhodes de mesue des maices acousiques des disconinuiés à un ou deux os en ésence des modes élevés hèse UC 5 décembe 5 [8] A. Selame,. S. Dickey, and J. M. ovak, he Heschel-Quincke ube: a heoeical, comuaional, and exeimenal invesigaion, J. Acous. Soc. Am. 96 (5 (994: [9] A. Siel, J. M. Ville, and F. Fouca, An exeimenal faciy fo measuemen of acousic ansmission maix and acousic owe dissiaion of duc disconinuiy in highe ode modes oagaion condiions, Acus. Aca Acus. 89, (3. 9 h IERAIOAL COGRESS O ACOUSICS ICA7MADRID

7 9 h IERAIOAL COGRESS O ACOUSICS ICA7MADRID 7

The Production of Polarization

The Production of Polarization Physics 36: Waves Lecue 13 3/31/211 The Poducion of Polaizaion Today we will alk abou he poducion of polaized ligh. We aleady inoduced he concep of he polaizaion of ligh, a ansvese EM wave. To biefly eview