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:

Jean-Philippe.Groby@univ-lemans.fr

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:Jean-Philippe.Groby@univ-lemans.fr

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 al.: Porous layer backed by rigid grating

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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 al.: Porous layer backed by rigid grating 823

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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)

824 J. Acoust. Soc. Am., Vol. 133, No. 2, February 2013 Groby et al.: Porous layer backed by rigid grating

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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.

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J. Acoust. Soc. Am. 127, 2865–2874 (2010).

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).

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