Acoustic Black Hole Effect in Ducted Geometries for Enhanced Dissipation at Low Frequencies

10 h Con en ion o he Eu opean Acous ics Associa ion Tu in, I aly • 11 h – 15 h Sep embe 2023 • Poli ecnico di To ino ACOUSTIC BLACK HOLE EFFECT IN DUCTED GEOMETRIES FOR ENHANCED DISSIPATION AT LOW FREQUENCIES Te esa B a o1* Ced ic Mau y2 1 Spanish Na ional Resea ch Council (CSIC), Mad id, Spain 2 Aix Ma seille Uni , CNRS, Cen ale Ma seille, Ma seille, F ance ABSTRACT* Imp o emen o acous ic abso p ion and ansmission in he low equency ange when acing cons ain s on o al olume and weigh cons i u es a cons an challenge in he ield o noise con ol. Eme gence o acous ic me ama e ials has allowed he de elopmen o compac sub-wa eleng h abso be s ha can p o ide solu ions in e ms o he equi ed wideband low- equency sound dissipa ion. In a lined wa eguide, he e ec i e eloci y o he incoming sound can be p og essi ely educed o ze o leading o he Acous ic Black Hole (ABH) e ec . In his wo k, his concep o sound apping has been mos ly explo ed o widely-opened ABH silence o he educ ion o he ansmission p ope ies wi hou obs uc ing low ci cula ion. A en ion has been paid o he a enua ion o he acous ic back- e lec ions ha can be o in e es o imp o ing engine combus ion e iciency o ins ance. Pa ame ic s udies a e ca ied ou o s udy he in luence o he physical ABH ac o s on he acous ic pe o mance. We ha e also pe o med a causali y-cons ained op imiza ion o de e mine he minimum bandwid h- o-leng h a io equi ed o a ixed dissipa ion equi emen . Resul s ha e been alida ed wi h expe imen al esul s. Keywo ds: me asilence , acous ic black hole, low equency dissipa ion. ————————— *Co esponding au ho : e esa.b a [email protected]. Copy igh : ©2023 Fi s au ho e al. This is an open-access a icle dis ibu ed unde he e ms o he C ea i e Commons A ibu ion 3.0 Unpo ed License, which pe mi s un es ic ed use, dis ibu ion, and ep oduc ion in any medium, p o ided he o iginal au ho and sou ce a e c edi ed. 1. INTRODUCTION Imp o emen o acous ic abso p ion and sound insula ion a low equencies using s uc u es wi h limi ed size o weigh cons i u es a long-s anding enginee ing challenge in he ield o en i onmen al noise con ol. Po ous and ib ous abso be s show good b oadband pe o mance in he high equency ange, bu op imal solu ions o low equency p oblems a e oo bulky and no sui able o la ge-scale applica ions. Mic o-Pe o a ed Panels backed by a igid ca i y [1, 2] a e sound abso be s ha ha e p o ed o be e icien in his equency ange when p ope ly selec ing hei physical cons i u i e pa ame e s [3, 4]. In spi e o his, he a enua ion alues a e es ic ed wi hin wo o h ee oc a es bandwid h a ound hei Helmhol z- ype esonance. The addi ion o se ial o pa allel a ays o igidly-backed MPP esona o s wi h di e en ca i y dep hs o pe o a ion a ios can b oaden he bandwid h o in e es by me ging he mul iple disc e e na owband esonances o each indi idual mic o-pe o a ed esona o [5, 6]. The d awback o his s a egy is he ade-o ha is no mally equi ed in eal p oblems in ol ing he maximum abso p ion alue, he abso p ion bandwid h and he abso ben o e all hickness [7, 8]. De elopmen o acous ic me ama e ials has allowed he g ow h o sub-wa eleng h abso be s ha can p o ide solu ions in e ms o he equi ed wideband low- equency sound dissipa ion [9, 10]. Me ama e ials de i e hei p ope ies om hei designed s uc u es and geome ies. Unlike adi ional composi es, acous ic me ama e ials can exceed known bounds on con en ional ma e ial p ope ies. Tha is accomplished by exploi ing subwa eleng h mic o s uc u e ha has been ab ica ed om o dina y ma e ials and embedded in a backg ound medium ( o ai - o wa e bo ne acous ic wa es, he backg ound medium is simply he su ounding luid). The no el e ec i e p ope ies o acous ic me ama e ials enable o manipula e 10 h Con en ion o he Eu opean Acous ics Associa ion Tu in, I aly • 11 h – 15 h Sep embe 2023 • Poli ecnico di To ino acous ic wa e ields in ways ha a e impossible o achie e wi h na u ally occu ing ma e ials o adi ional composi e s uc u es. The esponse o p ope ly de ined uni cells can be ansla ed in o a e aged e ec i e pa ame e s, namely an e ec i e densi y and bulk modulus [11, 12]. Fo ins ance, he in oduc ion o pe iodic esonan inclusions induce nega i e e ec i e p ope ies ha c ea e “s op-bands” whe e sound ansmision is o bidden. Tunable esonance sys ems can be concei ed o design selec i e dissipa ion il e s o he p opaga ion o sound wa es in a lined wa eguide. The beha io o acous ic wa es in he p esence o a esonan me ama e ial in a duc ed sys em can be a ec ed by he slow sound gene a ion [13]. The goal is hen he educ ion o he incoming sound wa e eloci y esul ing in a non- e lec ing edge condi ion [14] wi h he sound wa es, e en ually con ined in he me a- s uc u e. I has been coined as an Acous ic Black Hole (ABH) and p esen s g ea po en ial o acous ic ene gy apping. This echnique has been o iginally de eloped o passi e ib a ion con ol. Vib a ional black holes a e made up o sui ably sha pened beams o pla e edges ha slow down and p e en e lec ion o lexu al wa es a hei bounda ies [15]. A p ac ical ealiza ion has been pe o med using a se ies o igid discs ixed on a od inside a ube wi h a ying diame e s acco ding o a pa abolic law. Ene gy abso p ion can be enhanced in a eas o low eloci y by he inse ion o abso bing ma e ials. ABHs cons i u e also e a ding s uc u es induced by a powe law decay o he wa e eloci y, bu wi h a di e en p opaga ion ope a o [16]. Guasch e al. [17] s udied he ABH e ec using he ans e ma ix me hod o p edic he abso p ion pe o mance o linea and quad a ic ABH duc e mina ions as well as he in luence o hei physical pa ame e s. I was ound ha his solu ion ends o he wa e solu ion in a me a luid wi h powe -law a ying densi y using a high numbe o ings. The ABH concep has also been conside ed o open-ended wa eguides, in which he educ ion o bo h abso p ion and ansmission inside he me ama e ial has o be accomplished. This con igu a ion has been shown o be e icien p o ided ha he a io be ween he inle and ou le adii s ays g ea e han 10 [14]. Fu he in es iga ions ha e been ca ied ou analyzing he sensi i i y o ABHs o unca ion e ec s, he di e en damping mechanisms and he powe -law decay a e ha has o be selec ed o maximize he dissipa ion mechanisms. Zhang and Cheng [18] add damping by in oducing mic o-pe o a ed bounda ies ha conside ably imp o ed he b oadband low- equency pe o mance o open-ended ABHs. They p o ed ha he unca ion leng h a he ex emi y inc eased he cu - o equency o he e ec i e abso p ion ange. Conce ning widely-opened mu le s, he ans e ma ix me hod has also been used o pe o mance p edic ions and he esul s ha e been expe imen ally alida ed on 3D p in ed p o o ypes o he no- low case, wi h an inle /ou le adius a io equal o 2 [19]. Cu en ends a e paying a en ion o he p esence o a mean low wi h he de elopmen o me ama e ial windows [20] o allow bo h noise con ol and na u al en ila ion. The same conside a ions ha e been aken in o accoun o he de elopmen o ul a-spa se acous ic en ila ed me a- ba ie s [21]. Sound insula ion and ai low anspo ha e been expe imen ally demons a ed wi h a measu ed wind eloci y a io s agge ingly highe han 90%. In he p esen wo k, we aim a de e mining om op imiza ion s udies he main pa ame e s o ABH duc line s ha limi hei low equency b oadband pe o mance unde a ange o en i onmen al condi ions. We will ocus on he design o a compac ully-opened ABH o ge bo h low e lec ion and ansmission wi hou duc sec ion a ia ion and in he no- low case. Special a en ion will be paid o he causal-based limi a ion pe o mance o he me amu le o op imal dissipa ion in he low equency ange. This subjec has been s udied be o e wi hin he ame o igidly-backed mic ope o a ed pa i ions [7, 8] and will be ex ended he e o he case o open-ended con igu a ions. Sec ion 2 will p esen he model o he ABH duc line analyzing in de ail he educ ion in he eloci y o he inciden sound wa e. Sec ion 3 will p esen pa ame ic s udies on he ABH acous ic pe o mance whose causal- based op imiza ion will be discussed in Sec ion 4 and expe imen ally e i ied in Sec ion 5. The main conclusions and guidelines o u u e wo k will be ou lined in Sec ion 6. 2. THEORETICAL MODEL 2.1 Reduc ion o he wa e eloci y The ABH duc sys em ha is p oposed in his wo k is ou lined in Figu e 1. I is composed o se o ing sec ions sepa a ed by ai ca i ies and dis ibu ed o e an o e all leng h L along he axial dimension. The adii o he ai ca i ies in he ully-opened silence p og essi ely inc ease om 0 o R ollowing a powe law, and i is expanding om he inle si ua ed a L z owa ds he ou le a 0  z. The analy ical exp essions ha go e n he wa e p opaga ion in o he e a ding s uc u e ha e been de ined in [15]. Taking he con en ion  j e, he linea ized mass conse a ion equa ion in a duc ed geome y wi h a wall 10 h Con en ion o he Eu opean Acous ics Associa ion Tu in, I aly • 11 h – 15 h Sep embe 2023 • Poli ecnico di To ino Figu e 1. Schema o he ABH ully-opened con igu a ion. impedance on he bounda y a ying along he duc axis is gi en by [15]  0, j d logd 2 00   zLp c z S z z H n z   , (1) wi h z he axial eloci y componen , Zp n/ he no mal eloci y o e he bounda y  0,  zLR and p he acous ic p essu e. US H/ is he hyd aulic adius wi h S he c oss-sec ional a ea o he duc and U he ci cum e ence o he lining. In oducing he linea ized momen um conse a ion equa ion, zp z 0 j  in o Eq. (1) and aking in o accoun ha  0d/logd zS o a ully-opened mu le wi h cons an sec ion, a plane wa e equa ion is p o ided as  0j1 d d 0 2 0 2 2       p k zy k z p H , (2) whe e 00 ck   is he acous ic wa enumbe and ZZy / 0  is he wall speci ic admi ance no malised by 000 cZ   he luid cha ac e is ic impedance wi h 0  he ai densi y and 0 c he sound speed. The wall admi ance o he ABH cons i u ed o a con inuous dis ibu ion o annula ca i ies can be app oxima ed in he low equency ange 1( 0  Rk ), by    2 2 0 jzDRRkzy  . (3) When in oducing his exp ession in o Eq. (2), one ob ains   0 2 2 1 d d2 2 0 2 2                 p R zD zD R k z p, (4) ha can be educed o a Helmhol z equa ion when he ca i y dep hs inc eases wi h he law    zRzD m   1, wi h     0,/  mLzz m m m  . Wi h his assump ion, he wa enumbe o he acous ic wa e p opaga ing inside he ABH silence akes he exp ession    zkzk mz   2 0, and he co esponding phase speed is gi en by   z c zc m z   20 . (5) I can be no ed ha he acous ic wa e phase speed dec eases p og essi ely om 0 c a he inle owa ds 2/ 0 c a he ABH ou le . This si ua ion is clea ly di e en om he closed ABHs whe e he phase and g oup eloci y dec ease o ze o when app oaching he ABH e mina ion [17]. 2.2 T ans e Ma ix Fo mula ion An analy ical desc ip ion is de eloped using he T ans e Ma ix Me hod (TMM) o a widely-open ABH cons i u ed o a ini e numbe o annula ca i ies and pa ame e ized in disc e e o m as    imi zRD    1 wi h Ni ,...,1 and 0m. Assuming plane wa e p opaga ion in he duc and he adjacen ca i ies, hey a e ep esen ed by he localized sideb anch olume admi ance  0ca ,ca ZzySY ii , wi h   i zy gi en by Eq. (3) and dRS  2 ca  he ca i ies en ance a ea. Applying con inui y o he acous ic p essu e and acous ic low a e ac oss he i h ca i y- ing uni leads o he ela ionship,    T 11 Tp iiiii uup T, be ween he p essu e and olume eloci y ields a he inpu in e ace   T ii up and hose a he ou pu in e ace  T 11  ii up , wi h i T he associa ed ans e ma ix gi en by                                     1 01 cos)sin(j )sin(jcos cos)sin(j )sin(jcos ,ca 00 0 0 0 0 00 0 0 0 0 i i Y dkdk Z S dk S Z dk dkdk Z S dk S Z dk T . (6) The o e all ans e ma ix T be ween inle and ou le sa is ies     T 11 T 11  NN upup T and is exp essed as he p oduc o he ans e ma ices,         2221 1211 1TT TT N i i TT . Assuming a pe ec anechoic downs eam condi ion, he p essu e ansmi ed 1N p is ela ed o he olume eloci y 10 h Con en ion o he Eu opean Acous ics Associa ion Tu in, I aly • 11 h – 15 h Sep embe 2023 • Poli ecnico di To ino 1N u by SuZp NN 101  . The solu ion o he e lec ion and ansmission coe icien s a e gi en by                12011 2202112011 2202112011 1 TzT TzTTzT TzTTzT , (7) wi h 00 ZSz . The powe dissipa ed by he ABH hen eads   22 1 wi h  he abso p ion coe icien and  he ansmission coe icien . The ansmission loss (TL) is de ined as   10 log10(dB)TL  . We can ake in o accoun he isco- he mal losses wi hin he ca i ies and he duc using he Johnson-Champoux- Alla d-La a ge (JCAL) model [22], exp essed as a unc ion o complex acous ic wa enumbe s    cdcd Ck ,,0   and impedances    1 ,,0   cdcd CZ in he ans e ma ices o Eq. (6), wi h    cd, he e ec i e densi y ha con ains he isco-ine ial e ec s and    cd C, he ai comp essibili y ha accoun s o he he mal e ec s. The JCAL pa ame e s ha e been selec ed conside ing he ela ed li e a u e [22]. 3. STUDY OF THE INFLUENCE OF THE PHYSICAL PARAMETERS A compa ison has been ca ied ou o alida ion o he TMM p edic ions using a Fini e Elemen Me hod (FEM) comme cial p og amme. The moViscous Acous ics model in Comsol Mul iphysics conside s he g adien s o eloci y and empe a u e o include he iscous losses and hea conduc ion e ec s. The bounda y condi ions a he duc sec ions ha e been selec ed o simula e an in ini e leng h duc and a oid plane wa es e lec ed om he inle and ou le duc sec ions. The mesh has been de ined o wa an a leas en nodal poin s pe acous ic wa eleng h a he highes equency o analyses ha co esponds o he i s duc cu -on equency  Rc c  284.1 0 . 2D axisymme ic s udies ha e been conside ed using quad a ic elemen s. The ABH pa ame e s simula ed analy ically and nume ically co espond o he adius m047.0Rand he leng h m1.0L, ha p o ide a cu -on equency Hz2142 c . The ABH mu le is cons i u ed o 10  N annula ca i ies o axial wid h m008.0  d sepa a ed by ing walls o hickness m002.0 d, hus leading o a wall po osi y  %80 ddd  o e he silence sec ion. The axial a e a which he ca i y dep hs inc ease is chosen as 4.2  m. Analy ical simula ions ha e been pe o med using hese pa ame e s o he es ima ion o he abso p ion coe icien and he TL o he ABH s uc u e. The esul s a e p esen ed in Figu e 2. 200 400 600 800 1000 1200 1400 1600 1800 2000 0 0.5 1 (a) F equency (Hz) α 500 1000 1500 2000 0 20 40 (b) F equency (Hz) TL (dB) Figu e 2. Simula ion esul s o he abso p ion (a) and he ansmission loss (b) o he ABH mu le ob ained analy ically using TMM (blue) and nume ically wi h FEM ( ed). As i can be app ecia ed, he abso p ion coe icien p esen s good pe o mance alues o e almos he o al equency ange o analysis. In pa icula , in he band be ween 1300 Hz and 1600 Hz, he abso p ion exceeds 0.9. In he same equency ange, he co esponding TL exceeds 30 dB. These esul s show ha he p oposed mu le ac s as an ABH a oiding bo h e lec ion and ansmission wi hin his equency ange. Fo low equencies, he ansmission becomes signi ican and he pe o mance is limi ed. I can be also seen ha he ag eemen be ween he analy ical and nume ical p edic ions is co ec , al hough o he TL esul s he FEM unde es ima es he esul s p o ided by he TMM. This could be due o he simpli ying TMM app oach ha disca ds e anescen in e ac ion be ween neighbo ing ca i ies. A pa ame ic s udy has been done o ind pa ame ic dependencies and ends o ob aining he maximal dissipa ion alues inside he ABH de ice. Figu e 3 p esen s he es ima ed o al dissipa ion  in eg a ed o e a pa icula equency band when a ying he ca i y dep h inc ease a e m and he leng h- o- adius a io RL . 10 h Con en ion o he Eu opean Acous ics Associa ion Tu in, I aly • 11 h – 15 h Sep embe 2023 • Poli ecnico di To ino Di e en equency anges be ween min and c ha e been conside ed o he calcula ion o he o al dissipa ion     c c min d 1 min  , which a e supe imposed in Figu e 3. Maximum alues a e displayed wi h he co esponding ma ke s. 0.5 1 1.5 2 2.5 3 3.5 4 0 0.5 1 (a) m η 23456 0 0.5 1 (b) L/R η Figu e 3. In luence o he ca i y dep h inc ease a e m (a) and he leng h- o- adius a io RL (b) o es ima ing he o al dissipa ed powe simula ed wi h TMM and in eg a ed be ween 100 Hz (cyan), 800 Hz (yellow) and 1200 Hz (pink) up o Hz2142 c . The pe o mance esul s p og essi ely imp o e when conside ed a mo e educed equency ange be ween he lowe and he uppe equency limi s. When conside ing he b oades equency band, he selec ion o 2.2m p o ides op imal dissipa ion and minimiza ion o e lec ion and ansmission. Conce ning he leng h- o- adius a io, he alue 5.1RL can be selec ed when while keeping a ixed numbe o ca i ies 10N. Highe alues de e io a e he es ima ed dissipa ion p og essi ely. In conclusion, op imal selec ion o he physical pa ame e s can ha e an impo an impac on he expec ed esul s. This will be u he explo ed in he nex sec ion. 4. OPTIMIZATION BASED ON THE CAUSALITY CRITERION 4.1 S ochas ic op imiza ion Global op imiza ion o he selec ion o he ABH op imal pa ame e s is a combina o ial op imisa ion p oblem whe e all he physical pa ame e s a e c oss- ela ed and a a ia ion o one o hem may signi ican ly a ec he o he s. The equency-a e aged dissipa ion o e a selec ed band can also p esen many sub-op imal maxima and classical op imiza ion algo i hms ake he isk o being apped in hese non-op imal solu ions. O he op imiza ion echniques such as na u al algo i hms ha e been used classically. Fo compa ison pu poses wi h he p oposed causali y-c i e ion de eloped in he nex sec ion, we ha e used he Pa icle Swa m Op imiza ion (PSO), and adap i e op imiza ion echniques ha mimic he beha io o ce ain species g oup in e ac ions such as bi ds o ishes. Unlike gene ic algo i hms, i does no combine gene ic ma e ials o p e ious indi iduals o p og essi ely imp o e owa ds gene a ions, bu uses coope a ion o he explo a ion o he sea ch space [23]. We ha e used a PSO algo i hm o ind op imal alues       NLm NLm ,,, op op op op maxa g,,,  imposing he cons ain s 43.0   m, 9.01.0   , 4.31  RL and 305   N o a gi en duc adius m047.0R. The es ima ion o he a e aged dissipa ion has been done using TMM and he b oades equency band o in e es , om 20 Hz o he i s cu -on duc equency. The op imal alues ob ained a e p o ided by 2 op m, %7.46 op   , m15.0 op  L and 20 op  N. The co esponding op imal ca i ies wid h is m0035.0 op  d. 4.2 Causal-based op imiza ion A causal-based op imiza ion c i e ion has been o mula ed o he op imal pa ame e s o igidly-backed mic o- pe o a ed abso be s [7, 8]. I co ela es he bandwid h- o- leng h a io o he pa i ion o i s ul ima e abso p ion pe o mance. In he p esen con igu a ion, we will subs i u e he o al e lec ed powe by he o al dissipa ed powe o e all wa eleng hs o ob ain he ABH op imal ca i ies wid h. In he equency domain, i eads            0 22 0d )(-1log 4 c T    , (8) whe e  is he powe dissipa ed by he ABH silence and calcula ed by he TMM. When s udying he a ia ion o  T wi h espec o he ABH ings hickness d, wi h he o he pa ame e s being gi en by hei op imal alues ob ained in Sec . 4.1, i should be no ed ha he op imal ings hickness, m004.0 op ,  d, al eady ob ained in Sec . 4.1 using PSO, is also ob ained om maximizing he sensi i i y o he o al 10 h Con en ion o he Eu opean Acous ics Associa ion Tu in, I aly • 11 h – 15 h Sep embe 2023 • Poli ecnico di To ino in eg a ed dissipa ion wi h espec o d, such as ddTd   maxa g op , . Acco ding o his c i e ion, when op dd , , he o al amoun o ene gy en e ing he ABH silence is ully dissipa ed by isco- he mal losses wi hin he ca i ies, he eby leading o pe ec dissipa ion abo e 1.4 kHz up o he duc cu -on equency. Fo he analysis o ABH mu le s, i is o g ea impo ance o de elop an app oach ha p o ides he maximum pe o mance ha can be ob ained o e a a ge bandwid h wi h cons ain s on he mu le geome y. Gi en ha 2 1   , i esul s ha an uppe bound o               0 2 2 1 2dlog4 T  will also be an uppe bound o  T. Following he same app oach p esen ed in [7], using an ancilla y unc ion ha has no ze os, no poles in he lowe hal complex equency plane and applying Cauchy’s heo em o e a closed con ou in his plane, one ob ains an inequali y o he o al in eg a ed dissipa ion as          0 0d d Im 4 c T   . (9) A i s -o de expansion o )( a ound 0 can be achie ed om he TMM o mula ion. Conside ing o wa d ans e ma ices be ween each cell, one ge s an explici exp ession o 1 i T om Eq. (6), ela ing  T 11, ii up o  T ,ii up . Because he ABH silence is ully-opened a 0z, he ansmi ed p essu e 1N p and olume eloci y 1N u a e linked by SuZp NN 101   in plane wa e egime assuming a pe ec anechoic downs eam condi ion. F om hese equali ies, one inds a causal-based uppe bound o he o al in eg a ed dissipa ion in he ABH silence       D ABH V V L T   2 4, (10) whe e ABH V is he olume occupied by he ABH silence and SLVD is he olume o he lined duc sec ion. Assuming a cons an a ge dissipa ion 0  o e a speci ic bandwid h minmax     , Eq. (10) o mula ed in he wa eleng h domain p o ides an uppe limi on he ul ima e bandwid h- o-leng h a io L/   ha can be achie ed by he ABH silence ,           D ABH V V L   2 1log 0 2. (11) Se ing 0 min   , a cu -o equency min can be deduced om Eq. (11) below which a wideband ABH silence will exhibi poo dissipa ion pe o mance. I is gi en by          D ABH V V L c    2 1log 2 00 min . (12) Al hough he esul s a e no p esen ed he e, i can be deduced ha dec easing min can be achie ed by inc easing he ABH o e all leng h L o he ela i e olume DABH VV be ween he silence and he duc sec ion o by dec easing he wall po osi y  . 5. EXPERIMENTAL VALIDATION The simula ed esul s ha e been e i ied expe imen ally in a s anding wa e acili y o measu e he e lec ed, ansmi ed and dissipa ed powe in plane wa e egime. A pho og aph o he acili y is ske ched in Figu e 4 whe e he le loudspeake is connec ed o he ABH mu le inle h ough he impedance ube. I is made o a hick cylind ical ube o leng h 1000 mm, inne diame e 100mm wi h i s i s cu o equency a 2.1 kHz. A he end o he ube is plugged a whi e sample holde in which he ABH mu le can be inse ed. Figu e 4. Pho og aph o he expe imen al se -up used o he de e mina ion o he dissipa ion o he ABH mu le . Acous ic cha ac e iza ion is based on he de e mina ion o he sca e ing ma ix om he measu emen s o he ans e unc ions be ween le and igh loudspeake s and ou 10 h Con en ion o he Eu opean Acous ics Associa ion Tu in, I aly • 11 h – 15 h Sep embe 2023 • Poli ecnico di To ino lush-moun ed condense mic ophones sepa a ed by a dis ance cm5  . The loudspeake s a e d i en by whi e noise om 50 Hz o 2.5 kHz. Each acquisi ion is achie ed using he OROS (OR38) mul i-channel acquisi ion sys em, igge ed on he gene a ion o he d i e signal, a a sampling a e o 12.8 kHz and wi h a spec al esolu ion o 1.56 Hz. I is ca ied ou in he plane wa e egion be ween 60 Hz and 2000 Hz wi h a signal- o-noise a io (SNR) la ge han 10 dB. An ABH mu le has been ab ica ed wi h he op imized pa ame e s using used deposi ion modelling o ABS polyme on a hea ed p in su ace. The compa ison esul s be ween he TMM, he nume ical FEM and expe imen al abso p ion coe icien and TL a e p esen ed in Fig. 5. 200 400 600 800 1000 1200 1400 1600 1800 2000 0 0.5 1 (a) F equency (Hz) α (dB) 500 1000 1500 2000 0 20 40 (b) F equency (Hz) TL (dB) Figu e 5. Resul s o he abso p ion (a) and he ansmission loss (b) o he ABH mu le p edic ed using TMM (blue), FEM ( ed) and measu ed on he expe imen al se -up (g een). As i can be app ecia ed, he ag eemen o he abso p ion coe icien be ween he p edic ed and he measu ed alues is e y good. The achie ed alues a e qui e ema kable and a e main ained almos cons an o e he whole equency ange o analysis. As o he TL, he es ima ed and expe imen al alues p esen mo e impo an di e ences, bu i can be no ed ha he TL alues exceed 25 dB abo e 1400 Hz. 6. CONCLUSIONS In his wo k, he concep o sound apping has been explo ed o widely-opened ABH silence s aiming a he educ ion o he e lec ion and ansmission p ope ies. The s udy o hei pe o mance has been made analy ically, using TMM wi h JCAL model o isco- he mal losses, and nume ically, wi h Visco- he mal Acous ics FEM Comsol Mul iphysics. Compa ison be ween he app oaches has shown ha al hough TMM o e es ima es sligh ly he esul s p o ided by FEM due o he simpli ying app oach ha disca ds e anescen in e ac ions be ween neighbo ing ca i ies, i cons i u es a cos -e icien es ima ion o he ABH pe o mance. I has been shown ha he p oposed mu le slows down he inciden sound wa e due o p og essi e inc ease o he s i ness-con olled wall admi ance. I is able o p o ide bo h low e lec ed and low ansmi ed powe s in he e iciency ange ha ex ends be ween 1300 Hz and 1600 Hz. Pa ame ic s udies using TMM o mula ion ha e e ealed he impac o he selec ion o he physical pa ame e s on he acous ic esul s. In pa icula , we ha e shown he in luence o he axial a e o inc ease o he ca i y dep hs and he leng h- o- adius a io RL . A causal-based c i e ion has been p oposed conside ing he dissipa ed powe in eg a ed o e a ce ain equency band ha can p o ide an uppe bound o he bandwid h- o-leng h a io in o de o achie e a cons an a ge dissipa ion. The esul s ha e been compa ed o hose ob ained om o he na u al algo i hms, such as PSO global op imiza ion and hey i ex emely well. Finally, he op imized ABH mu le has been 3D p in ed wi h he p oposed pa ame e s and he simula ed esul s ha e been e i ied expe imen ally using a s anding wa e acili y o measu e he e lec ed, ansmi ed and dissipa ed powe s in plane wa e egime. The measu ed esul s closely ollow he p edic ions wi hin he equency egion o 1300 Hz-2000 Hz, wi h almos no back- e lec ed powe o e his bandwid h. 7. ACKNOWLEDGMENTS This wo k is pa o he p ojec TED2021-130103B-I00, unded by MCIN/AEI/10.13039/501100011033 and he Eu opean Union “Nex Gene a ionEU”/PRTR. I has also ecei ed suppo om he F ench go e nmen unde he F ance 2030 in es men plan, as pa o he Ini ia i e d'Excellence d'Aix-Ma seille Uni e si é - A*MIDEX (AMX-19-IET-010). 8. REFERENCES [1] D. Y. Maa, “Po en ial o mic ope o a ed panel abso be s”, Jou nal o he Acous ical Socie y o Ame ica, ol. 104, pp. 2861–2866, 1999. 10 h Con en ion o he Eu opean Acous ics Associa ion Tu in, I aly • 11 h – 15 h Sep embe 2023 • Poli ecnico di To ino [2] D. Y. 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