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Enhancing rigid frame porous layer absorption
withthree-dimensional periodic irregularities
J.-P. Groby,a) B. Brouard, and O. DazelLaboratoire d’Acoustique
de l’Universit�e du Maine, UMR6613 CNRS/Univ. du Maine,Avenue
Olivier Messiaen, F-72085 Le Mans Cedex 9, France
B. NennigLaboratoire d’Ing�enierie des Syst�emes M�ecaniques et
des Mat�eriaux, Supm�eca, 3 Rue Fernand Hainaut,F-93407 St Ouen
Cedex, France
L. KeldersLaboratorium voor Akoestiek en Thermische Fysica,
KULeuven, Celestijnenlaan 200D,B-3001 Heverlee, Belgium
(Received 10 April 2012; revised 27 November 2012; accepted 10
December 2012)
This papers reports a three-dimensional (3D) extension of the
model proposed by Groby et al.[J. Acoust. Soc. Am. 127, 2865–2874
(2010)]. The acoustic properties of a porous layerbacked by a rigid
plate with periodic rectangular irregularities are investigated.
The Johnson–
Champoux–Allard model is used to predict the complex bulk
modulus and density of the equivalent
fluid in the porous material. The method of variable separation
is used together with the radiation
conditions and Floquet theorem to derive the analytical
expression for the acoustic reflection coeffi-
cient from the porous layer with 3D inhomogeneities. Finite
element method is also used to validate
the proposed analytical solution. The theoretical and numerical
predictions agree well with the
experimental data obtained from an impedance tube experiment. It
is shown that the measured
acoustic absorption coefficient spectrum exhibits a quasi-total
absorption peak at the predicted
frequency of the mode trapped in the porous layer. When more
than one irregularity per spatial
period is considered, additional absorption peaks are
observed.VC 2013 Acoustical Society of America.
[http://dx.doi.org/10.1121/1.4773276]
PACS number(s): 43.55.Ev, 43.20.El, 43.20.Ks, 43.20.Gp [KVH]
Pages: 821–831
I. INTRODUCTION
This work was initially motivated by a design problem
connected to the determination of an optimal profile of a
dis-
continuous spatial distribution of porous materials and of
the
geometric properties for the absorption of sound. Acoustic
porous materials (foam) suffer from a lack of absorption at
low frequencies, when compared to the absorption values at
higher frequencies. The usual way to solve this problem is
by multi-layering,1 while trying to keep the thickness of
the treatment relatively small compared to the incident
wavelength that has to be absorbed. The purpose of the pres-
ent article is to investigate an alternative to
multi-layering
by considering periodic irregularities of the rigid plate on
which a porous sheet is attached, thus creating a
diffraction
grating and therefore extending previous works2 already
conducted in two-dimensional configurations to three-
dimensional ones.
The influence of rigid backing irregularities on the
absorption of a porous sheet was previously investigated by
use of the multi-modal method in Ref. 2 by considering peri-
odic rectangular air-filled irregularities of the rigid plate
on
which porous sheets are often attached in two-dimensional
configurations. This leads, in the case of one irregularity
per
spatial period, to a total absorption peak associated with
the
excitation of the fundamental modified mode of the backed
layer. This mode is excited due to the presence of the
surface
grating and traps the energy inside the porous sheet. Such
configurations have been widely studied in room acoustics
whereby irregularities are introduced to the walls in a
space
to enhance the diffusion and absorption effects,3 but the
con-
sidered phenomenon is mostly related to the resonance of
the irregularities. Other works related to surface
irregular-
ities were carried out, notably related to local resonances
associated with fractal irregularities4,5 or to porous
material
surface roughness.6
One-dimensional surface gratings consisting of periodic
rectangular irregularities of infinite length in one
direction
have been extensively studied in optics and electromagnet-
ism,7–9 in geophysics in relation to the city-site effect,10,11
in
urban acoustics to study wave propagation in streets.12
Three-dimensional configurations for acoustic waves seem
to have been studied only recently, mainly for urban acous-
tics purposes.13
The phenomena associated with surface irregularities of
the rigid backing were also coupled with those associated
with the embedment of a volumic heterogeneities in a porous
layer in order to increase the absorption properties of the
configuration. It was investigated by use of the multipole
method or a mode matching technique, by embedding a peri-
odic set of high-contrast inclusions, whose size is not
small
a)Author to whom correspondence should be addressed. Electronic
mail:
[email protected]
J. Acoust. Soc. Am. 133 (2), February 2013 VC 2013 Acoustical
Society of America 8210001-4966/2013/133(2)/821/11/$30.00
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mailto:[email protected]
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compared with the wavelength, in a macroscopically-
homogeneous porous layer backed by a rigid flat backing in
Refs. 14 and 15 or by a periodic irregular rigid backing in
Ref. 16 leading to a structure whose thickness and weight
are
relatively small. It was found, that the structure possesses
almost total absorption peaks, below the so-called quarter
wavelength resonance of the layer, when the irregularities
and
heterogeneities are correctly designed. These peaks are
asso-
ciated with trapped modes that trap the energy between the
heterogeneities and the rigid plate, and associated with the
resonances of the irregularity that trap the energy inside it,
to-
gether with the modified mode of the backed layer
excitation.
In this paper, the effect of a three-dimensional periodic
irregularity of the rigid backing on which a porous plate is
attached is investigated theoretically, numerically, and
experimentally.
II. FORMULATION OF THE PROBLEM
Rather than to solve directly for the pressure �pðx; tÞ[with x ¼
ðx1; x2; x3Þ], we prefer to deal with pðx;xÞ, relatedto �pðx; tÞ by
the Fourier transform �pðx; tÞ ¼
Ð1�1 pðx;xÞ
� e�ixtdx. Henceforth, we drop the x in pðx;xÞ so as todenote
the latter by pðxÞ.
A. Description of the configuration
A unit cell of the 3D scattering problem is shown in
Fig. 1. The layer is a rigid frame porous material saturated
by air (e.g., a foam) which is modeled as a macroscopically
homogeneous equivalent fluid M½1�. The upper and lowerflat and
mutually parallel boundaries of the layer, whose x3coordinates are
L and 0, are designated by CL and C0, respec-tively. The upper
semi-infinite material M½0�, i.e., the ambient
fluid that occupies X½0�, and M½1� are in a firm contact at
theboundary CL, i.e., the pressure and normal velocity are
con-tinuous across CL:½pðxÞ� ¼ 0 and ½q�1@npðxÞ� ¼ 0, wherein@n
designates the normal derivative operator. The rigidbacking with
period d ¼ ðd1; d2Þ, i.e., along the x1 and x2
axis, respectively, contains parallelepipedic irregularities
that create a diffraction grating. The Jth irregularity of
the
unit cell occupies the domain X½2ðJÞ� of height bðJÞ and
widths
wðJÞ1 , w
ðJÞ2 along the x1 and x2 axis. The x1 and x2 coordinate
of the center of the base segment of X½2ðJÞ� are d
ðJÞ1 and d
ðJÞ2 .
This irregularity is occupied by a fluid M½2ðJÞ�. The
boundary
of X½2ðJÞ� is composed of the rigid portion CNðJÞ [Neumann
type boundary conditions, @npðxÞ ¼ 0] and of CðJÞ throughwhich
media M½2
ðJÞ� and M½1� are in firm contact (continuityof the pressure and
normal velocity). C0 is also composed ofa rigid portion CN (Neumann
type boundary conditions), i.e.,C0 ¼ CN[J2JCNðJÞ .
We denote the total pressure, wavenumber and wave
speed by the generic symbols p, k, and c, respectively, withp ¼
p½0�; k ¼ k½0� ¼ x=c½0� in X½0�, p ¼ p½1�; k ¼ k½1� ¼ x=c½1�in
X½1�, and p ¼ p½2ðJÞ�; k ¼ k½2ðJÞ� ¼ x=c½2ðJÞ� in X½2ðJÞ�.
The azimuth wi of the incident wavevector ki ismeasured
counterclockwise from the positive x1 axis, whileits elevation hi
is measured counterclockwise from theðx1; x2Þ plane. The incident
wave propagates in X½0� andis expressed by piðxÞ ¼
Aieiðki1x1þki2x2�k
½0�i3ðx3�LÞÞ, wherein ki1
¼ �k½0� cos hi cos wi, ki2¼�k½0� coshi sinwi, k½0�i3 ¼ k½0�
sinh
i
and Ai¼AiðxÞ is the signal spectrum.The plane wave nature of the
incident wave and the
periodic nature of [J2JX½2ðJÞ� imply the Floquet relation
pðx1þnd1;x2þmd2;x3Þ¼pðx1;x2;x3Þeiki1nd1þiki2md2 ;
8x2R3; 8ðn;mÞ2Z2: (1)
Consequently, it suffices to examine the field in the unit
cell
of the plate which includes the parallelepipeds X½2ðJÞ�, J 2
J
in order to obtain the fields, via the Floquet relation, in
the
other cells.
The uniqueness of the solution of the forward-scattering
problem is assured by the radiation conditions
p½0�R ðxÞ ¼ p½0�ðxÞ � piðxÞ � outgoing waves;
jxj ! 1; x3 > L: (2)
B. Material modeling
The rigid frame porous material M is modeled using
theJohnson–Champoux–Allard model. The compressibility and
density, linked to the sound speed through c
¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1=ðKqÞ
pare17–19
1
K¼ cP0
/ c� ðc� 1Þ 1þ i x0c
Pr xGðPr xÞ
� ��1 ! ;
q ¼qf a1
/1þ i xc
xFðxÞ
� �; (3)
wherein xc ¼ r/=qf a1 is the Biot frequency, x0c ¼ r0/=qf a1, c
the specific heat ratio, P0 the atmospheric pressure,Pr the Prandtl
number, qf the density of the fluid in the(interconnected) pores, /
the porosity, a1 the tortuosity, rthe flow resistivity, and r0 the
thermal resistivity. The
FIG. 1. Example of a d-periodic fluid-like porous sheet backed
by a rigidwall that contains periodically arranged macroscopic
cubic irregularities
excited by a plane incident wave: (a) in plane dimensions in
case of a one
irregularity unit cell and (b) out of plane dimensions in case
of a two irregu-
larity unit cell.
822 J. Acoust. Soc. Am., Vol. 133, No. 2, February 2013 Groby et
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correction functions GðPr xÞ (Ref. 18) and FðxÞ (Ref. 17)are
given by
GðPr xÞ ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�
igqf Pr x
2a1r0/K0
� �2s;
FðxÞ ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�
igqf x
2a1r/K
� �2s; (4)
where g is the viscosity of the fluid, K0 the thermal
character-istic length, and K the viscous characteristic length.
Thethermal resistivity is related to the thermal characteristic
length18 through r0 ¼ 8a1g=/K02.The configuration is more
complex than the one already
studied in Ref. 2 as the structured backing is composed of a
three-dimensional grating consisting of a two-dimensional
peri-
odic set of parallelepipeds. The method of solution, which
is
quite similar to the one used in Ref. 2 is also briefly
summarized.
C. Field representations in X½0�, X½1�, and X½2ðJÞ �
Separation of variables, radiation conditions, and Flo-
quet theorem lead to the representations:
p½0�ðxÞ ¼X
ðn;mÞ2Z2
hAie�ik
½0�3nmðx3�LÞd0nd0m þ Rnmeik
½0�3nmðx3�LÞ
i
� eik1nx1þik2mx2 ; 8x 2 X½0�;
p½1�ðxÞ ¼X
ðn;mÞ2Z2
�f ½1��nm e
�ik½1�3nmx3 þ f ½1�þnm eik
½1�3nmx3
�
� eik2nx1þik2mx2 ; 8x 2 X½1�; (5)
wherein d0n is the Kronecker symbol, k1n ¼ ki1 þ 2np=d1,
k2m ¼ ki2 þ 2mp=d2, and k½j�3nm ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðk½j�Þ2
� ðk1nÞ2 � ðk2mÞ2
q,
with Reðk½j�3nmÞ � 0 and Imðk½j�3nmÞ � 0, j ¼ 0; 1. The
reflection
coefficient of the plane wave denoted by the subscripts n
and
m is Rnm, while f½1�6nm are the coefficients of the
diffracted
waves inside the slab associated with the plane wave also
denoted by the subscripts n and m.According to Ref. 2 the
pressure field p½2
ðJÞ�, admits thepseudo-modal representation, that already
accounts for the
boundary conditions on CNðJÞ :
p½2ðJÞ�ðxÞ¼
XðN ;MÞ2N2
D½2ðJÞ�NM
�cos�
k½2ðJÞ�1N
�x1�dðJÞ1 þ w
ðJÞ1 =2
���cos
�k½2ðJÞ�2M
�x2�dðJÞ2 þ w
ðJÞ2 =2
���cos
�k½2ðJÞ�3NM
�x3þbðJÞ
��; 8x2X½2ðJÞ�;8J2J ;
(6)
wherein k½2ðJÞ�1N ¼ Np=w
ðJÞ1 , k
½2ðJÞ�2M ¼Mp=w
ðJÞ2 , k
½2ðJÞ�3NM
¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðk½2ðJÞ�Þ2
� ðk½2
ðJÞ�1N Þ
2 � ðk½2ðJÞ�
2M Þ2
q, with Reðk½2
ðJÞ�3NMÞ � 0 and
Imðk½2ðJÞ�
3NMÞ � 0, 8J 2 J and D½2ðJÞ�NM are the coefficients of the
pseudo modal representation.
III. DETERMINATION OF THE ACOUSTICPROPERTIES OF THE
STRUCTURE
A. Application of the continuity conditions across CLand C0
We apply successively the following:
•Ð d1=2�d1=2
Ð d2=2�d2=2 e
ik1lx1þik2gx2 dx1dx2 with ðl; gÞ 2 Z2, to the con-tinuity of (i)
the pressure field and (ii) the normal compo-
nent of the velocity across CL
ðd1=2�d1=2
ðd2=2�d2=2½p½0�ðx3 ¼ LÞ � p½1�ðx3 ¼ LÞ�eik1lx1þik2gx2 dx1dx2 ¼
0;
ðd1=2�d1=2
ðd2=2�d2=2
1
q½0�@p½0�
@x3
�����x3¼L
� 1q½1�
@p½1�
@x3
�����x3¼L
24
35eik1lx1þik2gx2 dx1dx2 ¼ 0; (7)
and to the continuity of (iii) the normal component of the
velocity across CN[J2NCðJÞ (this implicitly includes the
Neumanntype boundary conditions along CN),
ðd1=2�d1=2
ðd2=2�d2=2
1
q½1�@p½1�
@x3
�����x3¼0
�Xj2J
1
q½2ðjÞ�@p½2
ðjÞ�
@x3
�����x3¼0
Yðx1;x2Þ2CðjÞ
24
35eik1lx1þik2gx2 dx1dx2 ¼ 0; (8)
whereinQðx1;x2Þ2CðjÞ is 1 when ðx1;x2Þ2CðjÞ and 0 otherwise.
•Ð dðJÞ
1þwðJÞ
1=2
dðJÞ1�wðJÞ
1=2
Ð dðJÞ2þwðJÞ
2=2
dðJÞ2�wðJÞ
2=2
cosðk½2ðJÞ�
1P x1Þcosðk½2ðJÞ�2Q x2Þdx1dx2 with
ðP;QÞ 2N2, to the continuity of the pressure fieldacross
CðJÞ:
ðdðJÞ1þwðJÞ
1=2
dðJÞ1�wðJÞ
1=2
ðdðJÞ2þwðJÞ
2=2
dðJÞ2�wðJÞ
2=2
hp½1�ðx3¼ 0Þ�p½2
ðJÞ�ðx3¼ 0Þi
�cos�
k½2ðJÞ�1P x1
�cos�
k½2ðJÞ�2Q x2
�dx1dx2¼ 0; 8J 2J :
(9)
J. Acoust. Soc. Am., Vol. 133, No. 2, February 2013 Groby et
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Introducing the appropriate field representation therein,
Eqs. (5), and (6), and making use of the orthogonality rela-
tionsÐ di=2�di=2 e
iðkin�kilÞxi dxi ¼ didnl; 8ðl; nÞ 2 Z2, i ¼ 1; 2 andÐ wðJÞi0
cosðk
½2ðJÞ�iN xiÞcosðk
½2ðJÞ�iP xiÞdxi ¼w
ðJÞi dNP=�N , wherein �0 ¼
1 and �N ¼ 2;8N 2N?, gives rise to a linear set of
equations.After some algebra and rearrangements, this reduces to a
linear
system of equations for the solution of D½2ðJÞ�NM which may
be
written in the matrix form, when the infinite vector of
compo-
nents D½2ðJÞ�NM is denoted by D:
ðA� CÞD ¼ F; (10)
where F is the column matrix of elements FðtÞNM and A and C
are two square matrices of elements Aðn;tÞNM;NM, and C
ðn;tÞNM;NM,
respectively. These elements are
FðtÞNM ¼
Xðn;mÞ2Z2
Ai2a½0�nmDnm
IþðtÞ1nN I
þðtÞ2mMe
ik1n ðdðtÞ1 �wðtÞ1=2Þþik2mðdðtÞ2 �w
ðtÞ1=2Þd0nd0m;
Aðn;tÞNM;NM ¼
1
�N �Mcos�
k½2ðtÞ�3NMb
ðtÞ�dNNdMMdtn;
Cðn;tÞNM;NM ¼
Xðn;mÞ2Z2
iwðnÞ1 w
ðnÞ2 a
½2ðnÞ�NM
�a½1�nm cosðk
½1�3nmLÞ� ia½0�nm sinðk
½1�3nmLÞ
�d1d2Dnma
½1�nm
sin�
k½2ðnÞ�3N Mb
ðnÞ�
� I�ðnÞ1nM I�ðnÞ2mMI
þðtÞ1nN I
þðtÞ2mMe
ik1n ððdðtÞ1 �dðnÞ1Þ�ðwðtÞ
1�wðnÞ
1Þ=2Þþik2mððdðtÞ2 �d
ðnÞ2Þ�ðwðtÞ
2�wðnÞ
2Þ=2Þ;
Dnm¼ a½0�nmcosðk½1�3nmLÞ� ia½1�nm sinðk
½1�3nmLÞ;
I6ðJÞinN ¼
e6ikin�
wðJÞi =2
2eik½2ðJÞ��
wðJÞi
=2
iN sinc
�k½2ðJÞ�iN 6kin
�wðJÞi2
!þeik
½2ðJÞ ��
wðJÞi
=2
iN sinc
�k½2ðJÞ�iN 7kin
�wðJÞi2
!0@1A; i¼ 1;2; (11)
where sincðvÞ ¼ sinðvÞ=v and a½j�nm ¼ k½j�3nm=q
½j�, j ¼ 0; 1; 2ðJÞ. The components FðtÞNM account for the
excitation of the irregularity tby a wave that is previously
diffracted by the layer, the components A
ðn;tÞNM;NM account for the irregularity t while the
components
of Cðn;tÞNM;NM account for the coupling, between the
irregularities t and n, due to the waves that are traveling inside
the porous plate.
B. Evaluation of the fields
Once Eq. (10) is solved for D½2ðnÞ�NM, Rnm, f
½1�þnm , and f
½1��nm in terms of D
½2ðJÞ�NM can be evaluated and, in particular,
Rnm ¼XJ2N
XðN ;MÞ2N2
iwðJÞ1 w
ðJÞ2 a
½2ðJÞ�nm
d1d2Dnma½1�nm
D½2ðJÞ�NMsin
�k½2ðJÞ�3NMb
ðJÞ�
I�ðJÞ1nN I
�ðJÞ2mMe
�ik1p�
dðJÞ1�wðJÞ
1=2
��ik2p
�dðJÞ2�wðJÞ
2=2
�
þd0nd0mAia½0�nmcosðk
½1�3mnLÞ þ ia½1�nmsinðk
½1�3nmLÞ
Dnm: (12)
Introduced in the appropriate field expression, this gives
p½0�R ðxÞ ¼
Xðn;mÞ2Z2
XJ2J
iwðJÞ1 w
ðJÞ2 e�ik1n ðdðJÞ1 �w
ðJÞ1=2Þ�ik2mðdðJÞ2 �w
ðJÞ2=2Þ
d1d2Dnma½1�nm
�X
ðN ;MÞ2N2D½2ðJÞ�NMa
½2ðJÞ�NMsin
�k½2ðJÞ�NMb
ðJÞ�
I�ðJÞ1nN I
�ðJÞ2mMe
ik1nx1þik2mx2þik½0�3nmðx3�LÞ
þAi a½0�icosðk½1�i3 LÞ þ ia½1�isinðk
½1�i3 LÞ
Dieik
i1x1þiki2x2þik½0�i3 ðx2�LÞ;
p½1�ðxÞ ¼X
ðn;mÞ2Z2
XJ2J
iwðJÞ1 w
ðJÞ2 e�ik1n ðdðJÞ1 �w
ðJÞ1=2Þ�ik2mðdðJÞ2 �w
ðJÞ2=2Þ
d1d2Dnma½1�nm
�X
ðN ;MÞ2N2D½2ðJÞ�NMa
½2ðJÞ�NMsin
�k½2ðJÞ�NMb
ðJÞ�
I�ðJÞ1nN I
�ðJÞ2mMe
ik1nx1þik2mx2
� a½1�nmcos�
k½1�3nmðx3 � LÞ
�þ ia½0�nmsin
�k½1�3nmðx3 � LÞ
�þAi 2a
½0�icosðk½1�i3 x3ÞDi
eiki1x1þiki2x2 ; (13)
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wherein Di ¼ D00, a½j�i ¼ a½j�00, j ¼ 0; 1. These fields
areexpressed as a sum of (i) the field due to the irregularities
of
the multi-component grating with (ii) the field in the
absence
of irregularity.
C. Evaluation of the reflection and absorptioncoefficients
In case of an incident plane wave with spectrum Ai,
con-servation of energy leads to a hemispherical reflection
Rexpressed by
R ¼X
ðn;mÞ2Z2
Reðk½0�3nmÞk½0�i3
kRnmk2kAik2 ; (14)
wherein the expressions of Rnm are given by Eq. (12).
Theabsorption coefficient A takes the form A ¼ AD þAS,wherein AD is
the inner absorption of the domains X½1� andX½2
ðJÞ�, 8J 2 J , and AS is the surface absorption induced
byviscosity and related to the interfaces CL and CðJÞ, 8J 2 J .In
our calculations, the irregularities are filled with air. Any
absorption phenomenon is associated to air, and thus the
inner absorption reduces to the one of domain X½1�, and
thesurface absorption related to CðJÞ simplifies.
Nevertheless, A will be simply calculated through theenergy
conservation relation A ¼ 1�R.
IV. MODE OF THE CONFIGURATION
Similarly to the analysis which was performed for two-
dimensional configurations,2,16 the modes of the present
con-
figuration consist of a complex combination between the
trapped mode of the irregularities (TMI) related to the geo-
metric and material properties of the irregularities, the
so-
called modified mode of the backed layer (MMBL), related
to the geometric and material properties of the porous layer
and to the lattice characteristics d1 and d2, and the mode ofthe
grating (MG) when bounded by a semi-infinite homoge-
neous half space, to some extent.
The trapped mode associated with the Jth irregularity,which
concentrates the energy inside the irregularity and
which can be obtained11 by the use of an iterative scheme of
resolution of Eq. (10), satisfies a relation close to
cosðk½2ðJÞ�
3NMbðJÞÞ ¼ 0. The frequencies of excitation of these
modes are also close to
�TMIðJÞ
NNM¼c½2ðJÞ�
2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið2Nþ1Þ
2b½2ðJÞ�
� �2þ 2N
w½2ðJÞ�1
!2þ 2M
w½2ðJÞ�2
!2vuut :(15)
Of course, this trapped mode is coupled with the porous
plate and is not excited at this exact frequency.
Effectively,
this trapped mode does not correspond to a Dirichlet condi-
tion on CðJÞ, but rather to a continuity condition. The
effec-tive height of the irregularity is also larger than the
actual
height of the irregularity itself, because of the pressure
field radiation inside the layer. The main difference com-
pared with the two-dimensional configuration is the three
dimensions of the irregularity, which introduce a third pa-
rameter useful in design of irregularity with specific reso-
nance. The mode density can be higher than in the two-
dimensional configuration and the �ðTMIÞðJÞNNM are not
necessar-
ily equally spaced in frequency.
In order to point out and to get a grip on the modified
mode of the plate, it is useful to consider a unit cell
composed
of only one irregularity, excited at low frequencies, i.e.,
below the fundamental TMI resonance which only depends
on the irregularity height and occurs at �TMI000 ¼ c½2�=4b.
Inthis case, the dispersion relation, i.e., detðA� CÞ ¼ 0,
whosevariables are horizontal wavenumbers-frequency, reduces to
1�X
ðn;mÞ2Z2
iw1w2d1d2
a½2�tanðk½2�bÞsinc2 k1nw12
� �sinc2 k2m
w22
� �a½1�nmDnm
a½1�nmcosðk½1�3nmLÞ� ia½0�nmsinðk½1�3nmLÞ
¼0;(16)
wherein a½2� ¼ a½2�00. By referring to the notion of the
Cutlermode8 and extending it to three-dimensional
configurations,
this relation is satisfied (in the non-dissipative case)
when
the denominator of Eq. (16) is purely imaginary and van-
ishes. It is also useful to introduce the wavenumber
kqnm
¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik21n þ
k22m
p. These conditions are achieved when
jkqnmj 2 ½k½0�;Reðk½1�Þ� and when either Dnm ¼ 0 or a½1�nm ¼
0(i.e., k
½1�3nm ¼ 0), which correspond to MMBL and MG,
respectively. MMBLs depend on the characteristics of the
surrounding material and of the porous layer, on the
thickness
of the latter and on the spatial periodicity, while MGs only
depend on the characteristics of the porous layer and the
spa-
tial periodicity. Both of them are determined by the
intersec-
tion of cnm ¼ x=kqnm, respectively, with Reðc?ðpÞðxÞÞ,wherein
c?ðpÞðxÞ is the pth root of the dispersion relation of aporous
layer backed by a rigid plate Di ¼ 0, and withReðc½1�Þ. The MMBL
are shown by the dots in Fig. 2 for theporous material S1 (see
Table I), when the spatial periodicity
is d1 � d2 ¼ 12 cm� 8 cm. Similarly to the TMI, the fre-quencies
of the modified modes of the plate are controlled in
three-dimensions by two geometric parameters that define the
lattice periodicity. The mode density can also be higher
than
FIG. 2. Real and imaginary parts of the dispersion relation
roots in the ab-
sence of irregularities c?ð1Þ. Real part of the modified modes
of the layer c?nm,
for d1 � d2 ¼ 12 cm� 8 cm are shown by dots.
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in the two-dimensional configuration and the �MMBLpnm are
notnecessarily equally spaced in frequency.
The associated attenuation of each mode can then be
determined by the values of Imðc?ðnÞÞ and Imðc½1�Þ at themode
excitation frequencies. The attenuation associated with
MG is also higher than the one associated with MMBL for
all frequencies. Moreover, MG corresponds to the highest
boundary of jk1qj for Eq. (16) to be true. This implies thatMG
could be difficult to excite. The latter type of mode can
only be poorly excited by a plane incident wave,
particularly
at low frequencies. Because rigorously MG corresponds to a
configuration with a semi-infinite domain directly above the
grating, this phenomenon can be understood as follows. On
one hand, when the thickness of the layer is smaller than or
of the same order as the wavelength in the layer, MG can
hardly be excited because waves associated with it can
hardly stand at the lower bound of the layer, and so modes
of
the configuration are close to MMBL. On the other hand,
when the thickness of the layer is larger than the
wavelength
in the layer, MG can be excited (if the waves could travel
through the layer towards the grating), and so modes of the
configuration are close to MG. This latter case corresponds
to the asymptotic high-frequency regime of MMBL.
V. NUMERICAL RESULTS, EXPERIMENTALVALIDATION AND DISCUSSION
The infinite sumP
n2Z over the indices of the kjn,j ¼ 1; 2 is found to depend on
the frequency and on theperiod of the grating. An empirical rule is
employed,
inspired by Ref. 2 and determined by performing a large
number of numerical experiments. This sum is
truncatedPNþjn¼�N�j
such that N7j ¼ intðdj=2pð3Reðk½1�Þ6kijÞÞ þ 10,j ¼ 1; 2. In
these equations, intðaÞ represents the integerpart of a. In a
similar way, the infinite sum
PN2N over
the indices of k½2ðJÞ�jN is truncated
PNþjN¼0, such that
Nþj ¼ intð3wðJÞj Reðk½2
ðJÞ�Þ=pÞ þ 10.Numerical calculations have been performed for
various
geometrical parameters [ðd1; d2Þ, wðJÞ1 � wðJÞ2 � bðJÞ, and
ðdðJÞ1 ; dðJÞ2 Þ] and within the frequency range of audible
sound,
particularly at low frequencies. One of the main constraints
in designing acoustically absorbing materials is the size
and
weight of the configuration. In this sense, the low
frequency
improvement implies good absorption for wavelength larger
than the thickness of the structure. A 1 cm thick low
resistiv-
ity foam (Fireflex) sheet S1 and a 2 cm thick low
resistivityfoam sheet S2 were used. The parameters of these
twoporous materials are reported in Table I. These parameters
have been evaluated using the traditional methods
(Flowmeter for the resistivity and ultrasonic methods for
the
four other parameters, together with a cross-validation by
impedance tube measurement) described in Ref. 20.
The irregularities are occupied by air, i.e., M½0�, M½2ðJÞ�
and
porous saturating fluid is air (q½0� ¼q½2ðJÞ� ¼ qf ¼
1:213kgm�3,c½0� ¼ c½2ðJÞ� ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffifficP0=qf
q, with P0¼ 1:01325 �105 Pa, c¼ 1:4,
Pr¼ 0:71, and g¼ 1:839 �10�5 kgm�1 s�1).The geometrical
parameters of the configurations stud-
ied therein are reported in Table II. All configurations
have
parallelepipedic irregularities with non-equal dimensions in
order to deal with non-symmetric configurations.
A. One irregularity per spatial period
Two cases can be discussed depending whether the
frequency of the fundamental TMI is lower or higher than
the frequency of the first MMBL. In the first case, the
MMBL would be largely excited, while in the second one,
the TMI would be largely excited as already noticed in
Ref. 2.
1. Absorption coefficient
Figure 3(a) depicts the absorption coefficient of the po-
rous layer with characteristics S1 (see Table I) when backedby a
flat rigid backing and when backed by a rigid grating of
geometry C1 (see Table II). As shown in Fig. 2, the
modifiedmodes of the plate have frequencies �MMBL110 � 2775
Hz,�MMBL101 � 4100 Hz, �MMBL111 � 6500 Hz; etc:; while the modeof
the irregularity are �TMI000 � 2800 Hz, �TMI010 � 4000 Hz,�TMI001 �
5100 Hz; etc: Several remarks should be made onthe solid curve of
Fig. 3(a). First, �TMI000 appears at a lower fre-quency than the
one calculated as �TMI000 ¼ c½2�=4b and isexcited around 1900 Hz.
This phenomenon, already encoun-
tered in Ref. 14, is related to the boundary condition at
Cð1Þ,which is not a Dirichlet condition but rather a continuity
condition, leading to a larger effective height of the
irregu-
larity. The thinner the porous layer, the closer is the
funda-
mental TMI to 2800 Hz. The fundamental TMI being lower
than the first MMBL, the energy is advected by this mode
and the associated absorption coefficient is close to unity
at
this frequency. Additional sharper peaks of absorption are
noticed at higher frequencies and are associated with the
ex-
citation of MMBL, around 2800 Hz, 4100 Hz; etc:Figure 3(b)
depicts the absorption coefficient of the po-
rous sheet with characteristics S1 (see Table I) when backedby a
flat rigid backing and when backed by a rigid grating of
geometry C2. The modified modes the plate stand are�MMBL110 �
1700 Hz, �MMBL120 � 3300 Hz, �MMBL101 � 4100 Hz,
TABLE I. Acoustical parameters of the porous material
constituting the
sheet of thickness L.
/ a1 K (lm) K0 (lm) r (N s m�4)
S1 0.95 1.42 180 360 8900
S2 0.99 1 70 210 7900
TABLE II. Geometry of the configuration. All dimensions are in
cm.
d1 � d2 L dðnÞ1 � dðnÞ2 w
ðnÞ1 � w
ðnÞ2 b
ðnÞ
C1 12� 8 1 6� 4 6� 4 3C2 20� 8 1 10� 4 12� 6 2C3 60� 60 2 30� 30
30� 45 10C4 20� 8 1 fð6� 3Þ; ð16� 4Þg fð12� 6Þ; ð4� 4Þg f2; 8g
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�MMBL111 � 5000 Hz; etc:; while the fundamental TMI is4275 Hz.
In practice, this trapped mode appears around
3000 Hz. The first MMBL being lower than the fundamental
TMI, the energy is advected by this mode and the associated
absorption coefficient is close to unity at this frequency.
An
additional sharper peak of absorption is noticed around
3100 Hz and is associated to the excitation of the second
MMBL. The excitation of the fundamental TMI leads to a
smooth peak leading to an increase of the absorption between
1500 and 5000 Hz.
Finite element method (FEM) computations were also
performed to validate the present calculations (see the Ap-
pendix for details). As illustrated in Fig. 3, both methods
are
in a very good agreement which validates the proposed
approach. The relative error is less than one percent.
Although the FEM offers almost unlimited flexibility, it was
found that for such configurations and dimensions, the pres-
ent semi-analytical method is faster than FEM in terms of
model preparation and computation time, especially when
the frequency or the periodic cell dimension increase.
In both Figs. 3(a) and 3(b), the absorption coefficient
calculated with MAINE3A (Ref. 21) is represented in case of
a
Lþ bð1Þ thick porous sheet of characteristic S1 and in case ofa
L ¼ 1 cm porous sheet of characteristic S1 with a bð1Þ thick
air layer between the porous sheet and the flat rigid
backing.
Any of these two configurations exhibit a quasi-total, i.e.,
close to unity, absorption peak and the geometry C2
enablesabsorption at a lower frequency than in these two cases,
proving the usefulness of a three-dimensional grating back-
ing. Moreover, irregular gratings allow using a smaller
amount of porous material and occupy less space. A particu-
lar feature is that the frequency of the fundamental TMI can
be correctly evaluated through the simplified problem con-
sisting of an air layer of the same thickness as the height
of
the irregularity between the porous layer and a rigid flat
backing, when excited at normal incidence. The amplitude
of the absorption peaks associated with the fundamental
TMI and MMBL excitation are nevertheless higher in case
of the irregular grating.
2. Field analysis
A different type of waves corresponds to each kind of
mode related to the grating (MG and MMBL): evanescent
waves in X½1� (and also in X½0�) for the MG, and evanescentwaves
in X½0� and propagative ones in X½1� for the MMBL. Inorder to
determine the type of modes excited by the plane
incident wave, the transfer function calculated as TF
¼ pðx;xÞ=p½0�iðx;xÞ on Cð0Þ at ðx1; x2Þ ¼ ðd1=2; d2=2Þ forthe
porous layer of characteristics S1 when backed by a rigidbacking of
geometry C1 and C2, excited at normal incidence
FIG. 4. Configuration C1 and C2—Transfer function on C0 atðx1;
x2Þ ¼ ðd1=2; d2=2Þ, and its different contributions, when the
configura-tions are excited at normal incidence: TFð�Þ (solid
line), TF1ð�Þ (dashedline), TF2ð�Þ (dash dotted line), and TF3ð�Þ
(dotted line).
FIG. 3. Absorption coefficient of a porous layer of
characteristic S1 backedby a flat rigid backing (dashed line), of
the same porous material but of
thickness Lþ bð1Þ backed by a flat rigid backing (dash dotted
line), of char-acteristic S1 backed by a rigid flat backing with an
air layer of thickness bð1Þ
in between (dotted line), (a) backed by an irregular rigid
backing of geome-
try C1 (solid line) and (b) backed by an irregular rigid backing
of geometryC2 (solid line) at normal incidence. Finite-element
results are also plottedand pointed out by squares.
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is plotted in Fig. 4. The transfer function is separated by
the
different intervals corresponding to the different types of
waves that are involved in the total pressure calculation:
TFð�Þ is the total transfer function, TF1ð�Þ is the
contribu-tion of the propagative waves in both X½0� and X½1�,
TF2ð�Þ isthe contribution of the evanescent waves in X½0� and
propa-gative ones in X½1� and TF3ð�Þ is the contribution of
theevanescent waves in both X½0� and X½1�.
At the location of the MMBL peaks, the transfer func-
tions also possess large peaks. These peaks are mainly asso-
ciated with the evanescent waves in X½0� and propagativewaves in
X½1�. This also proves that MMBL are the mostexcited modes related
to the grating, at least at low frequen-
cies. These peaks result from a continuous drop between
evanescent waves in both material to evanescent waves in
the air medium. This also means that these peaks are neither
a MMBL nor a MG, but result from a complex combination
of these two types of modes, with a structure closer to the
one of the MMBLs. Because of this structure, the energy is
trapped in the layer, leading to an increase in the
absorption
of the configuration.
At the location of the TMI associated peaks, the transfer
functions are mainly composed of propagative waves in both
domains. This proves that these peaks are associated with
in-
terference phenomena and with a trapped mode. Of particu-
lar interest is the fact that the transfer functions possess
minima at the location of the TMI. This phenomenon is also
completely different from the one associated with the
MMBL for which the transfer functions possess maxima: the
first relates to MMBL that trap the energy inside the porous
layer leading to maxima in the transfer functions, while the
other relate to TMI that trap the energy in the cavity
leading
to minima in the transfer functions calculated in the layer.
In
particular, at the location of the TMI, the pressure field
at
x3 ¼ 0 cm at the location of the cavity possess a minimaclearly
noticeable from the field representation, because
cosðk½2ðJÞ�
3NMðx3 þ bðJÞÞÞ is close to zero.
3. Angular dependence of the absorption peaks
While the excitation frequency of the MMBL depends
on the angle of incidence, the frequency of excitation of
the
TMI is almost independent on it. The frequency of excitation
of the MMBL becomes smaller when close to grazing inci-
dence. Figure 5(a) depicts the absorption coefficient of the
porous layer of characteristics S1 when backed by a rigidbacking
of geometry C1, when excited at hi ¼ p=2(�MMBL110 � 2775Hz), hi¼p=3
(�MMBL110 � 1900Hz), hi¼ p=4(�MMBL110 � 1700Hz), and hi¼ p=6
(�MMBL110 � 1500Hz), forwi¼ 0. The frequency of excitation of the
MMBL becomeslower than the frequency of excitation of the
fundamental
TMI. The absorption coefficient possesses a non-symmetric
peak which is characteristic for the excitation of a mode of
the configuration, i.e., the excitation of the MMBL. The TMI
being excited at higher frequency than the MMBL, the am-
plitude of the associated peak falls down. Here, we focus on
the variation of hi for wi¼ 0, because this leads to
variationsof ki1, i.e., the projection of the incident wave
numberalong the larger dimension of periodicity. Similar trends
are also observed for �MMBL101 , when wi¼/=2, i.e.,
variation
of ki2. In practice, the MMBL cannot be excited
belowminð�MMBL110 ;�MMBL101 Þ calculated at a grazing incidence. In
thepresent case, i.e., porous layer of characteristics S1 backedby
a grating of geometry C1, the MMBL cannot be excitedbelow 1420Hz,
Fig. 5(b).
4. Discussion
These results lead to several conclusions. First, below
the so-called quarter wavelength resonance, the structured
backing leads to a large modification and to an increase of
the absorption coefficient. The absorption coefficient is
always larger with a structured backing than with a flat
rigid
backing. The design of the rigid backing should be subjected
to the dimension of the application. When the fundamental
TMI is lower than the first MMBL, the thickness and width
of the required irregularity should be larger in order to be
correctly excited, than when the first MMBL is lower than
the fundamental TMI.
Moreover, this design rule should be adapted depending
on the material use, and in particular on its
flow-resistivity
value. For medium resistivity as considered in the current
ar-
ticle, or even lower resistivity material, use of the lateral
pe-
riodicity, i.e., the fundamental MMBL constitute a good
solution, because the latter can be efficiently excited. The
fundamental TMI can also be used for such material.
FIG. 5. (a) Absorption coefficient of a porous layer of
characteristic S1backed by an irregular rigid backing of geometry
C1 when the configura-tions are excited at hi ¼ p=2 (solid line),
hi ¼ p=3 (dashed line), hi ¼ p=4(dash dotted line), and hi ¼ p=6
(dotted line) for wi ¼ 0 and (b) evolution of�MMBL110 as a function
of h
i for wi ¼ 0.
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Materials with higher resistivity value can hardly support
the
waves associated to the fundamental MMBL can hardly
stand, because they propagate transversally inside the mate-
rial layer. In this case, assuming that the layer thickness
is
less than lateral periodicity, it is preferable to use the
TMI
excitation.
B. Experimental validation
Remarkable absorption is obtained in case of a periodic
irregular rigid backing, while the response of the structure
without irregularities is quite well known or at least much
more common. The experimental validation also focused on
the periodic structure, its effect having been emphasized in
the previous numerical section by comparison with the flat
rigid backing.
Usually, experiments related to 1D, 2D, or 3D gratings
are carried out in a free field (anechoic room) and/or at
higher frequencies for a finite size sample.15,22
Here, we follow the idea already exploited in Ref. 2,
where the experimental validation has been carried out using
an impedance tube with a square cross section. The square
cross-section impedance tube available in LAUM,
30 cm� 30 cm, with cut-off frequency around 570 Hz, wasused in
the present study. This cut-off frequency corresponds
to a wavelength of 60 cm.
The phenomenon related to the MMBL occurs when the
wavelength is of the order of the spatial period of the
grating.
We also make use of the boundary conditions of the imped-
ance tube, which are perfect mirrors below the cut-off fre-
quency, in order to design the sample. Because of the
impedance tube dimensions, the spatial periodicity along
both x1 and x2 axis should be a multiple of 30 cm. If theprofile
of the unit cell is symmetric with regards to the axis
x1 ¼ d1=2 and x2 ¼ d2=2, the modeled spatial period isðd1 � d2Þ
¼ ð60 cm� 60 cmÞ, as depicted in Fig. 6(a).
The infinitely rigid portion of the sample, where Neu-
mann type boundary conditions are applied, was made of
four 1 cm thick aluminum plates, which were screwed (the
head of the screw was then filled with hard plastic silicone
for the surface to be perfectly flat) in order to create a
step
with 10 cm height and 15 cm width along the x1-axis and22:5 cm
width along the x2-axis, Fig. 6(b). This configurationresults in
the grating of the geometry C3. A L ¼ 2 cm thick
porous foam layer S2, with the characteristics reported inTable
I, was glued to the upper part of the step. In order to
keep the porous layer flat along the step area, a screw of
small diameter (3 mm) was added at the edge of the lower
part of the step and two nylon wires were tightened between
this screw and the upper part of the step, such that the
free
part of the foam layer rests on it.
A comparison between the measured absorption coeffi-
cient experimentally and the one calculated with the present
method is presented in Fig. 7. The so-called measured
absorption coefficient is the averaged value of the
absorption
coefficient measured when the step of the sample lays on the
bottom edge of the impedance tube, Fig. 6(c), and when
rotated by p=2. A rotation of the sample by p and 3p=2
isimpossible in practice because the sample does not lay on
the step and hence not stable in the tube.
The two curves match well and therefore validate exper-
imentally the present method.
C. More complicated unit cell
As pointed out in Sec. IV, the determination of the
modes of the global configuration is even more difficult to
carry out in case of multiple irregularities per spatial
period.
An infinite number of geometries and combinations are
possible. For instance a multi-component irregularities in a
FIG. 6. (Color online) Cross-sectional view of the experimental
set-up and sample design (a), picture of the sample (b), and sample
layer in the impedance
tube (c).
FIG. 7. Comparison between the absorption coefficient of a
porous layer of
characteristic S2, backed by a flat rigid backing (dotted line),
and backed byan irregular rigid backing of geometry C3 as
calculated with the presentmethod (solid line), and measured
experimentally (�)
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unit cell can be used to create a “doubly” periodic or better
a
“fourthly” periodic structure, i.e., a unit cell composed of
a
periodic arrangement of different size irregularities
leading
to a period d ¼ ðd1; d2Þ and a subperiod d0 ¼ ðd01; d02Þ
whichcombines the possibility to excite MMBL associated with
both periodicity: An irregularity with fundamental TMI at
either higher frequency or lower frequency than the previ-
ously used one can also be considered. Here, we will focus
in this last example by adding a wð2Þ1 � w
ð2Þ2 � bð2Þ
¼ 4 cm� 4 cm� 8 cm to the geometry C2. The TMI of thisadditional
irregularity is around 1000 Hz. Figure 8, depicts
the absorption coefficient of this configuration, namely,
ge-
ometry C4, together with the absorption coefficient of
thegeometry C2. The frequencies of excitation of the new TMIsare
�TMI000 � 1000 Hz, �TMI010 � 3200 Hz. Two additional peaksof
absorption can be noticed around 850 Hz and around
2750 Hz. The first is attributed to the excitation of the
funda-
mental TMI. The second peak is attributed to a coupled
mode between the second TMI of the second irregularity and
the fundamental TMI of the first irregularity.
Comparison between the absorption coefficient of the
geometry C4 calculated with the present method and withthe
finite element method also validates the present method
of calculation for a multiple irregularities unit-cell.
VI. CONCLUSION
We studied, theoretically, numerically and experimen-
tally, the acoustic properties of a low resistivity porous
layer
backed by a rigid plate with periodic irregularities in the
form of a three-dimensional grating. This work demonstrates
the possibility to design a three-dimensional porous
material
based system with resonances lower than the usual quarter-
wavelength resonance of the backed layer.
It has been shown that the grating leads to excitation of
modes, whose frequencies depend on both the characteristics
of the surrounding medium and of the porous layer and on
the spatial period of the configuration d1 � d2. These
modes,whose structures are close to the one of the modes of the
layer, can lead to a total absorption peak. This absorption
peak occurs at the frequency of the fundamental modified
mode of the layer and seems to be always a quasi-total
absorption peak. The trapped mode of the irregularity also
leads to quasi-total absorption peaks when excited below the
modified mode of the backed layer. These results are first
validated by comparison with the finite element
calculations.
Experiments were performed in a square cross-section
impedance tube. The boundary conditions of this tube are
perfect mirrors and allow us, thanks to the image theory, to
model diffraction of a plane wave at normal incidence at
fre-
quencies below the cut off of the tube. Experimental results
are in agreement with the theory and particularly exhibit a
total absorption peak at the frequency of the fundamental
modified mode of the layer.
Adding more irregularities per spatial period leads to a
modification of the modes of the configuration, which
become coupled and so are associated with a larger entrap-
ment of the energy than the one encountered in the case of
only one irregularity. When the fundamental frequency of
the irregularity is lower than the fundamental frequency
of the modified mode of the layer, i.e., large high
irregular-
ity, a total absorption peak is obtained at the fundamental
frequency of the irregularity. An infinite number of
combina-
tions are possible, but this type of configuration offers
good
opportunities in the design of three-dimensional structured
acoustic panels.
ACKNOWLEDGMENTS
The authors would like to thank R. Pommier for provid-
ing Solidworks pictures.
APPENDIX: FINITE ELEMENT METHOD
The FEM computations are carried out using Lagrange
quadratic tetrahedral finite elements. The radiation
condition
of the scattered field in the upper air domain X½0�, is
imple-mented with a Dirichlet to Neuman (DtN) map based on the
Floquet decomposition given in Eq. (5). This approach was
favored here as the use of the PML technique is not
efficient
for “low-frequency” applications, i.e., when the wavelength
is large compared to the size of the computational domain.
However, DtN leads to prohibitive computation time when a
significant Floquet mode is cut-on. The highest Floquet
mode index taken into account in each direction is chosen as
the number of cut-on Floquet modeþ 2.To easily apply the Floquet
relation recalled in Eq. (1),
coincident meshes on each opposite lateral boundary of the
periodic cell are used.23,24 Unstructured meshes are
employed in the remainder of the computational domain.
The characteristic element length is fixed in all cases to
ensure two elements in the thickness L of the porous
materialdomain X½1�. This yields approximately to 100 000 degreesof
freedom FEM models.
1O. Tanneau, J. Casimir, and P. Lamary, “Optimization of
multilayered
panels with poroelastic components for an acoustical
transmission
objective,” J. Acoust. Soc. Am. 120, 1227–1238 (2006).2J.-P.
Groby, W. Lauriks, and T. Vigran, “Total absorption peak by use of
a
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FIG. 8. Absorption coefficient of a porous layer of
characteristic S1 backedby a flat rigid backing (dotted line),
backed by an irregular rigid backing of
geometry C2 (solid line) and of geometry C4 (dash dotted line)
as calculatedwith the present method and as calculated by finite
element method (squares).
830 J. Acoust. Soc. Am., Vol. 133, No. 2, February 2013 Groby et
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