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
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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
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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
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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 .
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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

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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
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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).
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Tu in, I aly • 11 h – 15 h Sep embe 2023 • Poli ecnico di To ino
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