A wave-fluid saturated poroelastic structure interaction model based on the modified Biot theory (MBT) and plane-wave decomposition using orthogonal cylindrical functions is developed. The model is employed to recover from real data acquired in an anechoic chamber, the poromechanical properties of a soft cellular melamine cylinder submitted to an audible acoustic radiation. The inverse problem of acoustic diffraction is solved by constructing the objective functional given by the total square of the difference between predictions from the MBT interaction model and diffracted field data from experiment. The faculty of retrieval of the intrinsic poromechanical parameters from the diffracted acoustic fields, indicate that a wave initially propagating in a light fluid (air) medium, is able to carry in the absence of mechanical excitation of the specimen, information on the macroscopic mechanical properties which depend on the microstructural and intrinsic properties of the solid phase.

The investigation focuses on the study of the interaction of acoustic waves with a soft, open-cell poroelastic cylinder immersed in a light fluid (air). The objective is to develop a method for the characterization of open-cell poroelastic materials such as foams in the audible frequency range where these materials are often designed to reduce noise and vibration in mechanical fields as the automobile, airplane and construction industries.

In previous studies,1,2 it was shown that the macroscopic structural, porosity and resistivity (related to permeability) of a cellular foam, could be recovered from the diffracted field at audible frequencies using the equivalent fluid model (EFM). In the EFM, the skeleton of the porous material is considered motionless (rigid-frame) in the absence of mechanical excitation and the waves inside the porous material to propagate only inside the fluid phase.3–6 However, it has been shown that it was possible to retrieve both the mechanical and the structural micro-geometric parameters of the soft, open-pore porous material, thanks to waves transmitted by a poroelastic foam panel at ultrasonic frequencies.7 In that study the waves in the soft cellular foam were considered to propagate, both in the fluid and the frame notwithstanding the supposed absence of mechanical excitation. The forward acoustic wave transmission problem was modeled using a modified Biot8,9 theory (biphasic theory) whereby some parameters have physical interpretation closely related to the pore structure at the microscale. The dynamic density function (that describes the viscous effects) is associated with the narrow sections of the pores and the dynamic bulk modulus (describing the thermal effects) is associated with the wider sections of the pores.10–13 The advantage of using the biphasic interaction model over the EFM is that it provides the possibility of retrieval of both the mechanical and the structural micro-geometric parameters of the open-pore porous material.

The aim of this study is to recover poromechanical parameters (density, Young's modulus and Poisson ratio) from the acoustic field, diffracted by a poroelastic specimen in form of a cylinder. We employ the biphasic interaction theory in the low frequency regime. The direct problem developed in this study is for a free-field configuration of the poroelastic cylinder compared to that of Hasheminejad et al.14 who have studied the diffraction of sound waves by an infinite cylinder near an impedance plane. Real data pertaining to the diffracted acoustic field is acquired in the smaller of the two anechoic chambers of the Laboratoire de Mécanique et d’Acoustique using the experimental method for acquiring the diffracted field described in reference 1. The diffracted field is obtained in two steps. First the incident field pinc is acquired without the cylinder, then the total pressured field ptot is acquired in the presence of the cylinder. The diffracted pressure field psc is then computed from the difference,

\begin{equation}p_{sc}=p_{tot}-p_{inc}\end{equation}
psc=ptotpinc
(1)

The model equations of wave propagation in the host medium and in the poroelastic cylinder are detailed in section II B. The inverse problem is formulated in section III.

The forward diffraction problem (notably for simulating measured data) is formulated as follows. Given: (i) the geometry of the diffracting poroelastic cylinder body and its composition (material properties); (ii) the material properties of the host medium; (iii) the incident wavefield, determine: The pressure field measured at points circumscribing a circle around the object.

A monochromatic incident plane-wave (pressure) field with angular frequency ω, propagating in the host medium (air) Ω0, impinges on a cylindrical, linear, macroscopically homogeneous, isotropic poroelastic target whose boundary is ∂Ω described by a circle of radius rc. The pressure field is measured at a point M circumscribing the circle Γ around the object. The problem configuration is depicted on Fig. (1).

FIG. 1.

The problem configuration (Ω0 and Ω1 are the host and poroelastic medium respectively). The incident wave impinging on the target is denoted pinc. The acoustic pressure field is measured by the microphone at point M circumscribing a circle Γ.

FIG. 1.

The problem configuration (Ω0 and Ω1 are the host and poroelastic medium respectively). The incident wave impinging on the target is denoted pinc. The acoustic pressure field is measured by the microphone at point M circumscribing a circle Γ.

Close modal

1. The acoustic wave propagation equations in the host medium (Ω0)

The field equations of state for the hydrostatic stress in the fluid can be expressed more conveniently in terms of a scalar velocity potential which is related to the acoustic particle displacement (d) and pressure (p) as follows

\begin{equation}\dot{d}=\nabla \Phi ,\,\, p=-\rho _{0}\dot{\Phi },\,\,\nabla ^{2}\Phi +k_{0}\Phi =0,\end{equation}
ḋ=Φ,p=ρ0Φ̇,2Φ+k0Φ=0,
(2)

where

$\dot{d}=\partial d/\partial t$
ḋ=d/t is the particle velocity, Φ is the scalar velocity potential,
$\dot{\Phi }=\partial \Phi /\partial t$
Φ̇=Φ/t
, ρ0 the fluid density,
$\displaystyle k_{0}=\omega /c_{0}$
k0=ω/c0
is the wave number, ω is the angular frequency and c0 is the wave velocity in the fluid.

Consider the poroelastic cylinder of radius rc and its cylindrical coordinate system in Fig. (1). The plane wave solution of the incident wave propagating in the host medium (Ω0) is first transformed into a superposition of cylindrical waves satisfying the Helmholtz equation in cylindrical coordinates using the integral representation for the Bessel function and employing orthogonality relations15,16 (assuming harmonic time t dependency, the factor exp ( − iωt) with the angular frequency ω, is implicit),

\begin{equation}\varphi _{inc}=\sum _{m=-\infty }^{\infty }i^{m}J_{m}(k_{0}r)\exp (im\psi ),\quad \vec{r}\in \Omega _0,\end{equation}
ϕinc=m=imJm(k0r)exp(imψ),rΩ0,
(3)

in which Jm is the cylindrical Bessel function of the first kind15 and

$i=\sqrt{-1}$
i=1⁠.

The field scattered by the poroelastic cylinder,

\begin{equation}\varphi _{sc}=\sum _{m=-\infty }^{\infty }A_{m}H_{m}(k_{0}r)\exp (im\psi ),\quad \vec{r}\in \Omega _0,\end{equation}
ϕsc=m=AmHm(k0r)exp(imψ),rΩ0,
(4)

where Am are unknown scattering coefficients and Hm() is the m-th order cylindrical Hankel function which indicates that the scattered field satisfies the Sommerfeld radiation condition at infinity. This condition, which manifests the characteristic of the wave problem (the decay of waves) in far field, is commonly imposed to ensure the uniqueness of the solution in exterior acoustic scattering problems

The total acoustic velocity potential17 is expressed as

\begin{equation}\Phi =\varphi _{inc}+\varphi _{sc}=\sum _{m=-\infty }^{\infty }i^{m}J_{m}(k_{0}r)\exp (im\psi )+\sum _{m=-\infty }^{\infty }A_{m}H_{m}(k_{0}r)\exp (im\psi )\end{equation}
Φ=ϕinc+ϕsc=m=imJm(k0r)exp(imψ)+m=AmHm(k0r)exp(imψ)
(5)

2. The wave propagation equations in the poroelastic media

The equations of motion for a fluid-saturated poroelastic media were formulated by Biot.8,9 This motion is described by the macroscopic displacement of the solid and fluid phases represented by the vectors

$\mathbf u$
u and
$\mathbf U$
U
respectively. The harmonic equation of motion can be written in the form

\begin{eqnarray}\nonumber P {\mathbf \nabla }({\mathbf \nabla }\cdot {\mathbf u})-N {\mathbf \nabla }\times ({\mathbf \nabla }\times {\mathbf u})+Q {\mathbf \nabla }({\mathbf \nabla }\cdot {\mathbf U})+\omega ^{2}(\tilde{\rho }_{11} {\mathbf u}+\tilde{\rho }_{12}{\mathbf U}) &=& 0, \\Q {\mathbf \nabla }({\mathbf \nabla }\cdot {\mathbf u})+R {\mathbf \nabla }({\mathbf \nabla }\cdot {\mathbf U})+\omega ^{2}(\tilde{\rho }_{12} {\mathbf u}+\tilde{\rho }_{22} {\mathbf U}) &=& 0,\end{eqnarray}
P(·u)N×(×u)+Q(·U)+ω2(ρ̃11u+ρ̃12U)=0,Q(·u)+R(·U)+ω2(ρ̃12u+ρ̃22U)=0,
(6)

where P, Q, R are generalized elastic constants which are related, via gedanken experiments, to other measurable quantities, namely ϕ, Kf (bulk modulus of the fluid), Ks (bulk modulus of the elastic solid), Kb (bulk modulus of the porous skeletal frame) and N (the shear modulus of the frame in vacuum).

The equations which explicitly relate P, Q and R to ϕ, Kf, Ks, Kb and N are given by3 

\begin{eqnarray*}P&=&\frac{(1-\phi )(1-\phi -K_{b}/K_{s})K_{s}+\phi (K_{s}/K_{f})K_{b}}{1-\phi -K_{b}/K_{s}+\phi K_{s}/K_{f}}+\frac{4}{3}N \\&&Q=\frac{(1-\phi -K_{b}/K_{s})\phi K_{s}}{1-\phi -K_{b}/K_{s}+\phi K_{s}/K_{f}} \\&&R=\frac{\phi ^{2}K_{s}}{1-\phi -K_{b}/K_{s}+\phi K_{s}/K_{f}}\end{eqnarray*}
P=(1φ)(1φKb/Ks)Ks+φ(Ks/Kf)Kb1φKb/Ks+φKs/Kf+43NQ=(1φKb/Ks)φKs1φKb/Ks+φKs/KfR=φ2Ks1φKb/Ks+φKs/Kf

in which the bulk modulus of the frame can be evaluated from the relation,3 Kb = 2Nb + 1)/[3(1 − 2νb)] (νb is the Poisson ratio of the frame).

Expressions for the complex density and the bulk modulus considering different frequency regimes (frequency-dependent viscous and thermal effects respectively) that constitutes the modification of the Biot model are given in Appendix  A. Consequently the equivalent mass density coefficients with their corrections (in the frequency domain)

$\tilde{\rho }_{11}(\omega )$
ρ̃11(ω)⁠,
$\tilde{\rho }_{12}(\omega )$
ρ̃12(ω)
and
$\tilde{\rho }_{22}(\omega )$
ρ̃22(ω)
; which account for inertial and viscous coupling between both the solid and fluid phases, are given as

\begin{eqnarray}\nonumber \tilde{\rho }_{11}(\omega ) &=& \rho _{11}+i\sigma \phi ^{2}G_{j}(\omega )/\omega , \\\nonumber \tilde{\rho }_{12}(\omega ) &=& \rho _{12}-i\sigma \phi ^{2}G_{j}(\omega )/\omega , \\\tilde{\rho }_{22}(\omega ) &=& \rho _{22}+i\sigma \phi ^{2}G_{j}(\omega )/\omega ,\end{eqnarray}
ρ̃11(ω)=ρ11+iσφ2Gj(ω)/ω,ρ̃12(ω)=ρ12iσφ2Gj(ω)/ω,ρ̃22(ω)=ρ22+iσφ2Gj(ω)/ω,
(7)

where ρ11 = ρ1 + ρa, ρ12 = −ρa, ρ22 = ϕρ0 + ρa, ρ1 = (1 − ϕ)ρs + ϕρ0, ρa = ϕρ0 − 1) and ρs the density of the porous skeletal frame.

The displacement fields can be resolved into a superposition of longitudinal scalar potential φ and transverse vector

${\bm \Psi}$
Ψ components using the Helmholtz decomposition theorem,18 

\begin{equation}{\mathbf u}={\mathbf \nabla }\varphi ^{s}+{\mathbf \nabla }\times {\bm \Psi ^{\bf s}},\end{equation}
u=ϕs+×Ψs,
(8)
\begin{equation}{\mathbf U}={\mathbf \nabla }\varphi ^{f}+{\mathbf \nabla }\times {\bm \Psi ^{\bf f}},\end{equation}
U=ϕf+×Ψf,
(9)

where the superscripts s and f, are, respectively, associated with the solid-borne and fluid-borne waves. The solid-borne wave propagates in the two phases with similar amplitude, while the fluid-borne wave propagates mainly in the fluid phase, and is usually strongly damped. The above resolutions are substituted into Biot's field equations of motion. Then employing the vector identities, ∇(∇ · φs) = ∇2φs, ∇ × (φs) = 0 and

$\nabla \times \nabla \times {\bm \Psi }^{s}=\nabla (\nabla \cdot {\bm \Psi}^{s})-\nabla ^2 {\mathbf \Psi}^{s}$
××Ψs=(·Ψs)2Ψs and since φ and
$\bm {\Psi}$
Ψ
must satisfy the wave equations the following relations are obtained,

\begin{equation}P\nabla ^2\varphi ^{s} + Q\nabla ^2\varphi ^{f}+\omega ^{2}(\tilde{\rho }_{11}\varphi ^{s}+\tilde{\rho }_{12}\varphi ^{f})=0\end{equation}
P2ϕs+Q2ϕf+ω2(ρ̃11ϕs+ρ̃12ϕf)=0
(10)
\begin{equation}Q\nabla ^2\varphi ^{s}+R\nabla ^2\varphi ^{f} +\omega ^{2}(\tilde{\rho }_{12}\varphi ^{s}+\tilde{\rho }_{22}\varphi ^{f})=0,\end{equation}
Q2ϕs+R2ϕf+ω2(ρ̃12ϕs+ρ̃22ϕf)=0,
(11)

Eqns. (10) and (11) can be rewritten in matrix form as follows

\begin{equation}-\omega ^2[\rho ][\varphi ]=[M]\nabla ^2[\varphi ],\end{equation}
ω2[ρ][ϕ]=[M]2[ϕ],
(12)

where [φ] = [φs, φf]T, [ρ] and [M] are respectively

\begin{equation}[\rho ]=\left[\begin{array}{cc}\rho _{11} & \rho _{12} \\\rho _{12} & \rho _{22} \end{array}\right],\quad [M]=\left[\begin{array}{cc}P & Q \\Q & R \end{array}\right]\end{equation}
[ρ]=ρ11ρ12ρ12ρ22,[M]=PQQR
(13)

By imposing that the vector potential

$\bm {\Psi}$
Ψ be a solution we obtain two equations that provide the rotation wave equations,

\begin{equation}N\nabla ^2 {\bm \Psi}^{s}+\omega ^{2}(\tilde{\rho }_{11} {\bm \Psi}^{s}+\tilde{\rho }_{12} {\bm \Psi}^{f})=0,\end{equation}
N2Ψs+ω2(ρ̃11Ψs+ρ̃12Ψf)=0,
(14)
\begin{equation}\omega ^{2}(\tilde{\rho }_{12} {\bm \Psi}^{s}+\tilde{\rho }_{22} {\bm \Psi}^{f})=0.\end{equation}
ω2(ρ̃12Ψs+ρ̃22Ψf)=0.
(15)

The Helmholtz type of equations are obtained after solving the corresponding eigenvalue problems for the longitudinal and shear waves,

\begin{eqnarray}\displaystyle \nonumber \nabla ^{2}\varphi _{1,2}^{s}+k_{1,2}^{2}\varphi _{1,2}^{s} &=& 0 \\\nabla ^{2} {\bm \Psi}^{s}+k_{3}^{2} {\bm \Psi}^{s} &=& 0\end{eqnarray}
2ϕ1,2s+k1,22ϕ1,2s=02Ψs+k32Ψs=0
(16)

where ∇2 is the Laplacian and the propagation constants k1 and k2, which characterize two compressional waves, are found from

\begin{equation}\displaystyle k^2_{1,2}=\frac{\omega ^{2}}{2(PR-Q^{2})}\left(P\tilde{\rho }_{22}+R\tilde{\rho }_{11}-2Q\tilde{\rho }_{12}\pm \sqrt{\Delta }\right),\end{equation}
k1,22=ω22(PRQ2)Pρ̃22+Rρ̃112Qρ̃12±Δ,
(17)

with

$\Delta =\left(P\tilde{\rho }_{22}+R\tilde{\rho }_{11}-2Q\tilde{\rho }_{12}\right)^{2}$
Δ=Pρ̃22+Rρ̃112Qρ̃122-
$4\left(PR-Q^{2}\right)\left(\tilde{\rho }_{11}\tilde{\rho }_{22}-\tilde{\rho }_{12}^{2}\right)$
4PRQ2ρ̃11ρ̃22ρ̃122

Each wave propagates in both phases with different velocities and amplitudes. The ratio between displacement of fluid and solid phases, respectively denoted μi is given by

\begin{equation}\displaystyle \mu _{i}=\frac{\varphi _{i}^{f}}{\varphi _{i}^{s}}=\frac{Pk_{i}^{2}-\omega ^{2}\tilde{\rho }_{11}}{\omega ^{2}\tilde{\rho }_{12}-Qk_{i}^{2}},\,\, i=1,2.\end{equation}
μi=ϕifϕis=Pki2ω2ρ̃11ω2ρ̃12Qki2,i=1,2.
(18)

Likewise, a shear wave propagates in the porous media and is characterized by its wave number

\begin{equation}\displaystyle k_{3}^{2}=\frac{\omega ^{2}}{N}\left(\frac{\tilde{\rho }_{11}\tilde{\rho }_{22}-\tilde{\rho }_{12}^{2}}{\tilde{\rho }_{22}}\right),\end{equation}
k32=ω2Nρ̃11ρ̃22ρ̃122ρ̃22,
(19)

and the ratio μ3 of displacement amplitude of the fluid and of the frame is (using Eqn. (15))

\begin{equation}\displaystyle \mu _{3}=\frac{\Psi ^{f}}{\Psi ^{s}}=-\frac{\tilde{\rho }_{12}}{\tilde{\rho }_{22}}.\end{equation}
μ3=ΨfΨs=ρ̃12ρ̃22.
(20)

The fundamental field equations in cylindrical coordinate for the solid and fluid displacements in r and ψ direction in terms of displacement potentials in the poroelastic cylinder are written18 

\begin{eqnarray}\displaystyle \nonumber u_{r} &=& \frac{\partial \varphi ^{s}}{\partial r}+\frac{1}{r}\frac{\partial {\bm \Psi}^{s}}{\partial \psi },\,\, \\u_{\psi } &=& \frac{1}{r}\frac{\partial \varphi ^{s}}{\partial \psi }-\frac{\partial {\bm \Psi}^{s}}{\partial r},\end{eqnarray}
ur=ϕsr+1rΨsψ,uψ=1rϕsψΨsr,
(21)

and

\begin{eqnarray}\displaystyle \nonumber U_{r}&=&\frac{\partial \varphi ^{f}}{\partial r}+\frac{1}{r}\frac{\partial {\bm \Psi}^{f}}{\partial \psi },\,\, \\U_{\psi }&=&\frac{1}{r}\frac{\partial \varphi ^{f}}{\partial \psi }-\frac{\partial {\bm \Psi}^{f}}{\partial r},\end{eqnarray}
Ur=ϕfr+1rΨfψ,Uψ=1rϕfψΨfr,
(22)

where

$\displaystyle {\varphi ^{s}=\varphi }_{1}^{s}+\varphi _{2}^{s}$
ϕs=ϕ1s+ϕ2s⁠,
$\displaystyle {\varphi ^{f}=\varphi }_{1}^{f}+\varphi _{2}^{f}=\mu _{1}\varphi _{1}^{s}+\mu _{2}\varphi _{2}^{s}$
ϕf=ϕ1f+ϕ2f=μ1ϕ1s+μ2ϕ2s
. Also, the general stress-strain relations in the Biot theory are

\begin{eqnarray}\displaystyle \nonumber \sigma _{ij}^{s}&=&\left[(P-2N)e+Q\varepsilon \right]\delta _{ij}+2Ne_{ij}^{s},\\\sigma _{ii}^{f}&=&(-\phi p_{p_f})=Qe+R\varepsilon ,\end{eqnarray}
σijs=(P2N)e+Qɛδij+2Neijs,σiif=(φppf)=Qe+Rɛ,
(23)

where

$p_{p_f}$
ppf is the pore fluid pressure and
$\displaystyle e_{i,j}^{s}=\frac{u_{i,j}+u_{j,i}}{2}$
ei,js=ui,j+uj,i2
. Employing Eqns. (8), (9), and (16), the expressions for the frame and fluid dilatations are given by

\begin{equation}e=\nabla \times {\mathbf u}=\nabla ^{2}{\varphi ^{s}=\nabla ^{2}\varphi }_{1}^{s}+\nabla ^{2}\varphi _{2}^{s}={-k}_{1}^{2}\varphi _{1}^{s}-k_{2}^{2}\varphi _{2}^{s}\end{equation}
e=×u=2ϕs=2ϕ1s+2ϕ2s=k12ϕ1sk22ϕ2s
(24)
\begin{equation}\varepsilon =\nabla \times {\mathbf U}=\nabla ^{2}{\varphi ^{f}={\mu _{1}\nabla }_{1}^{2}\varphi }_{1}^{s}+\mu _{2}\nabla ^{2}\varphi _{2}^{s}={-\mu _{1}k}_{1}^{2}\varphi _{1}^{s}-\mu _{2}k_{2}^{2}\varphi _{2}^{s}.\end{equation}
ɛ=×U=2ϕf=μ112ϕ1s+μ22ϕ2s=μ1k12ϕ1sμ2k22ϕ2s.
(25)

The general expressions for the relevant stress components and the pore fluid pressure, are derived by substituting Eqns. (24) and (25) into Eqn. (23),

\begin{eqnarray}\nonumber \sigma _{rr}^{s} &=& a_{1} k_{1}^{2}\varphi _{1}^{s}+a_{2}k_{2}^{2}\varphi _{2}^{s}+2N(\frac{\partial u_{r}}{\partial r}), \\\nonumber \sigma _{r\psi }^{s} &=& \frac{N}{r}\left(\frac{\partial u_{r}}{\partial \psi }+r\frac{\partial u_{\psi }}{\partial r}-u_{\psi }\right), \\\phi p_{fp} &=& b_{1}k_{1}^{2}\varphi _{1}^{s}+b_{2}k_{2}^{2}\varphi _{2}^{s},\end{eqnarray}
σrrs=a1k12ϕ1s+a2k22ϕ2s+2N(urr),σrψs=Nrurψ+ruψruψ,φpfp=b1k12ϕ1s+b2k22ϕ2s,
(26)

where a1, 2 = 2NPQμ1, 2 and b1, 2 = Q + Rμ1, 2. Finally, the orthogonal field expansions for the fast and slow dilatational and shear waves propagating in the poroelastic cylinder may be expressed in the general form

\begin{eqnarray}\nonumber \varphi _{1}^{s}(r,\psi ,\omega )&=&\sum _{m=-\infty }^{\infty }B_{m}(\omega )J_{m}(k_{1}r)\exp (im\psi ), \quad \vec{r}\in \Omega _1\\\nonumber \varphi _{2}^{s}(r,\psi ,\omega )&=&\sum _{m=-\infty }^{\infty }C_{m}(\omega )J_{m}(k_{2}r)\exp (im\psi ),\quad \vec{r}\in \Omega _1\\\Psi ^{s}(r,\psi ,\omega )&=&\sum _{m=-\infty }^{\infty }D_{m}(\omega )J_{m}(k_{3}r)\exp (im\psi ), \quad \vec{r}\in \Omega _1\end{eqnarray}
ϕ1s(r,ψ,ω)=m=Bm(ω)Jm(k1r)exp(imψ),rΩ1ϕ2s(r,ψ,ω)=m=Cm(ω)Jm(k2r)exp(imψ),rΩ1Ψs(r,ψ,ω)=m=Dm(ω)Jm(k3r)exp(imψ),rΩ1
(27)

where Bm(ω) , Cm(ω) and Dm(ω) are unknown coefficients to be determined using the boundary conditions.

The total acoustic field potential (for

$\vec{r}\in \Omega _0$
rΩ0⁠) is given by

\begin{equation}\Phi (r,\psi ,\omega )=\sum _{m=-\infty }^{\infty }i^{m}J_{m}(k_{0}r)\exp (im\psi )+\sum _{m=-\infty }^{\infty }A_{m}H_{m}(k_{0}r)\exp (im\psi ).\end{equation}
Φ(r,ψ,ω)=m=imJm(k0r)exp(imψ)+m=AmHm(k0r)exp(imψ).
(28)

Identically, the corresponding expression for the fluid particle normal displacement dn is found as

\begin{equation}d_{n}(r,\psi ,\omega )=\frac{1}{-i\omega }\left\lbrace \sum _{m=-\infty }^{\infty }i^{m}\dot{J}_{m}(k_{0}r)\exp (im\psi )+\sum _{m=-\infty }^{\infty }A_{m}\dot{H}_{m}(k_{0}r)\exp (im\psi )\right\rbrace , \quad \vec{r}\in \Omega _0.\end{equation}
dn(r,ψ,ω)=1iωm=imJ̇m(k0r)exp(imψ)+m=AmḢm(k0r)exp(imψ),rΩ0.
(29)

The boundary conditions that have to be satisfied at the surface of the poroelastic cylinder to yield a unique solution for the scattering problem are

  1. the continuity of pressure, i.e.,

    $p_{f_p}=p$
    pfp=p⁠.

  2. the compatibility of total stresses, i.e.,

    $\sigma _{rr}^{s}=-(1-\phi )p$
    σrrs=(1φ)p and
    $\sigma _{r\psi }^{s}=0$
    σrψs=0
    .

  3. the conservation of fluid volume, i.e.,

    $\displaystyle (1-\phi )\dot{u}_{r}+\phi \dot{U}_{r}=\dot{d}_{n}$
    (1φ)u̇r+φU̇r=ḋn⁠.

Incorporation of Eqns. (21)–(29), in the above boundary conditions yields the linear systems of equations

\begin{equation}i\omega \rho _{0}\phi \left(A_{m}H_{m}(k_{0}r_c)\right)-B_{m}b_{1}k_{1}^{2}J_{m}(k_{1}r_c)-C_{m}b_{2}k_{2}^{2}J_{m}(k_{2}r_c)=-i\omega \rho _{0}\phi i^{m}J_{m}(k_{0}r_c)\end{equation}
iωρ0φAmHm(k0rc)Bmb1k12Jm(k1rc)Cmb2k22Jm(k2rc)=iωρ0φimJm(k0rc)
(30)
\begin{eqnarray}\nonumber i\omega \rho _{0}(1-\phi )A_{m}H_{n}(k_{0}r_c)\!+\!B_{m}k_{1}^{2}\left[a_{1}J_{m}(k_{1}r_c)+\!2N\ddot{J}_{m}(k_{1}r_c)\right]\!+\!C_{m}k_{2}^{2}\left[a_{2}J_{m}(k_{2}r_c)+2N\ddot{J}_{m}(k_{2}r_c)\right]\\\nonumber +\!\left(\frac{2iN}{r_c^{2}}\right)\!D_{m}m\left[k_{3}r_c\dot{J}_{m}(k_{3}r_c)-\!J_{m}(k_{3}r_c)\right]\!=\! -i\omega \rho _{0}(1\!-\phi )i^{m}J_{m}(k_{0}r_c)\\\end{eqnarray}
iωρ0(1φ)AmHn(k0rc)+Bmk12a1Jm(k1rc)+2NJ̈m(k1rc)+Cmk22a2Jm(k2rc)+2NJ̈m(k2rc)+2iNrc2Dmmk3rcJ̇m(k3rc)Jm(k3rc)=iωρ0(1φ)imJm(k0rc)
(31)
\begin{eqnarray}\nonumber 2iB_{m}m\left[k_{1}r_c\dot{J}_{m}(k_{1}r_c)-J_{m}(k_{1}r_c)\right]+2iC_{m}m\left[k_{2}r_c\dot{J}_{m}(k_{2}r_c)-J_{m}(k_{2}r_c)\right]\\+D_{m}\left[-m^{2}J_{m}(k_{3}r_c)+k_{3}r_c\dot{J}_{m}(k_{3}r_c)-k_{3}^{2}r_c^{2}\ddot{J}_{m}(k_{3}r_c)\right] & = & 0\end{eqnarray}
2iBmmk1rcJ̇m(k1rc)Jm(k1rc)+2iCmmk2rcJ̇m(k2rc)Jm(k2rc)+Dmm2Jm(k3rc)+k3rcJ̇m(k3rc)k32rc2J̈m(k3rc)=0
(32)
\begin{eqnarray}\nonumber A_{m}k_{0}\dot{H}_{m}(k_{0}r_c)+i\omega \left[1+\phi (\mu _{1}-1)\right]B_{m}k_{1}\dot{J}_{m}(k_{1}r_c)+i\omega \left[1+\phi (\mu _{2}-1)\right]C_{m}k_{2}\dot{J}_{m}(k_{2}r_c)\\\nonumber -(\frac{\omega }{r_c})\left[1+\phi (\mu _{3}-1)\right]D_{m}J_{m}(k_{3}r_c)=-i^{m}k_{0}\dot{J}_{m}(k_{0}r_c)\\\end{eqnarray}
Amk0Ḣm(k0rc)+iω1+φ(μ11)Bmk1J̇m(k1rc)+iω1+φ(μ21)Cmk2J̇m(k2rc)(ωrc)1+φ(μ31)DmJm(k3rc)=imk0J̇m(k0rc)
(33)

The resulting matrix equation can be written as

\begin{equation}\mathbf {X} \mathbf {y}=\mathbf {z},\end{equation}
Xy=z,
(34)

The elements of the matrix equation that enable to determine the unknown coefficient vectors

$\mathbf {y}$
y and consequently to solve the forward-scattering problem, notably for the prediction of the scattered pressure field in the host medium (air) are detailed in Appendix  B. The method is implemented in Maple19 and the matrix equation solved using the LinearSolve function. The scattered field measured in the host medium around the cylinder by the microphone is determined from the value of Am, consequently this is the only coefficient extracted from the matrix equation.

The total pressure amplitude is finally given by,

\begin{equation}p(r,\psi ,\omega )=i\omega \rho _{0}\Phi (r,\psi ,\omega ), \quad \vec{r}\in \Omega _0.\end{equation}
p(r,ψ,ω)=iωρ0Φ(r,ψ,ω),rΩ0.
(35)

The inverse acoustic diffraction problem is formulated as follows. Given: (i) the diffracted wavefield, (ii) the material properties of the saturating fluid and host medium, reconstruct the poroelastic parameters of the diffracting cylinder. The radius of the cylinder is determined by measurement using vernier calipers.

Precisely, the inverse problem herein involves the recovery of the mechanical parameters of the skeleton (density ρb, Young's modulus Eb, and Poisson ratio νb) of the poroelastic cylinder from the measured diffracted field and the MBT wave/poroelastic cylinder interaction model. The other parameters of the model are assumed to be known.

A function, constituting the measure of the discrepancy between the interaction model diffracted field and the experimental diffracted field is computed for each set of trial values of the parameters. We first divide [0, 2π] into Ns sectors so as to discretize ψ as: ψn = (n − 1)δ, n = 1…Ns where

$\displaystyle \delta ={2\pi }/{N_s}$
δ=2π/Ns⁠. Then we compute the discretized form of the cost function,fdc given by

\begin{equation}\displaystyle f_{dc}(\omega ,\psi _n,\rho _b, E_b, \rho _b)=p_{sc}^{theory}(\omega ,\psi _n,\rho _b, E_b, \rho _b)-p_{sc}^{experiment}(\omega ,\psi _n),\end{equation}
fdc(ω,ψn,ρb,Eb,ρb)=psctheory(ω,ψn,ρb,Eb,ρb)pscexperiment(ω,ψn),
(36)

where psc is the scattered pressure field.

The cost function

$\mathfrak {J}$
J is then,

\begin{equation}\displaystyle \mathfrak {J}=\sum _{n=1}^{N_s}\left( f_{dc}(\omega ,\psi _n,\rho _b, E_b, \rho _b)\right)^2,\end{equation}
J=n=1Nsfdc(ω,ψn,ρb,Eb,ρb)2,
(37)

The model parameters are listed in Table (I).

Table I.

Poroelastic material parameters for melamine foam, 1.5 m long, having a diameter of 24.0 cm. The values marked with ? are those to be recovered by solving the inverse problem.

Parametersymbolmelamine
Density of the solid (kg/m3ρs ρb/(1 − ϕ) 
Density of porous skeletal frame (kg/m3ρb 
Young's modulus of the solid (Pa) Es Eb/(1 − ϕ)2 
Young's modulus of the porous skeletal frame (Pa) Eb 
Poisson ratio of the porous skeletal frame νb 
Porosity ϕ 0.99 
Tortuosity α 1.01 
Viscous characteristic length (μm) Λ 100 
Thermal characteristic length (μm) Λ′ 3 × Λ 
Airflow resistivity (Pa.s.m−2σ 12500 
Density of the pore fluid (kg/m3ρ0 1.297 
Fluid bulk modulus (Pa) Kf 1.42 × 105 
Viscosity of pore fluid (Pa s) η 1.78 × 10−5 
Parametersymbolmelamine
Density of the solid (kg/m3ρs ρb/(1 − ϕ) 
Density of porous skeletal frame (kg/m3ρb 
Young's modulus of the solid (Pa) Es Eb/(1 − ϕ)2 
Young's modulus of the porous skeletal frame (Pa) Eb 
Poisson ratio of the porous skeletal frame νb 
Porosity ϕ 0.99 
Tortuosity α 1.01 
Viscous characteristic length (μm) Λ 100 
Thermal characteristic length (μm) Λ′ 3 × Λ 
Airflow resistivity (Pa.s.m−2σ 12500 
Density of the pore fluid (kg/m3ρ0 1.297 
Fluid bulk modulus (Pa) Kf 1.42 × 105 
Viscosity of pore fluid (Pa s) η 1.78 × 10−5 

The geometric parameters of the melamine foam characteristics were derived using ultrasonic wave propagation methods as reported in.7 The results of the inversions are presented by plotting the cost/objective functions for the three mechanical parameters using a 500 Hz probe (Fig. 2(a)–2(c)). The retrieved values (ρb, 8.30 kg/m3, Eb, 187 ±5 kPa, and νb, 0.483), are solutions of the inverse problem. The recovered density was in excellent agreement with that obtained from the calculated volume and weight (7.95 kg/m3) of the cylinder measured on an electronic balance (Denver Instruments SI-4200). The density given by BASF (Ludwigshafen, Germany) for Basotec foam is 8.35 kg/m3. The value of the recovered Eb is in good agreement with that in reference 7 obtained using ultrasonic waves. Sharper minima for νb is obtained at 1500 Hz (Fig. 3). A νb of 0.48 ± 0.01 is recovered. This is also in good agreement with those obtained in reference 7. A sensitivity study of the mechanical parameters on the diffracted field has been undertaken after which it has been remarked that small variations in Poisson ratio provoke significant changes to the diffracted field.

FIG. 2.

Cost functions for the mechanical properties of the frame of an air-saturated poroelastic cylinder for a 500 Hz acoustic probe (a) Density. (b) Young's modulus. (c) Poisson coefficient. The correct minima are shown by the arrows.

FIG. 2.

Cost functions for the mechanical properties of the frame of an air-saturated poroelastic cylinder for a 500 Hz acoustic probe (a) Density. (b) Young's modulus. (c) Poisson coefficient. The correct minima are shown by the arrows.

Close modal
FIG. 3.

Cost functions for the Poisson ratio of the frame of an air-saturated poroelastic cylinder for 1500 Hz. Poisson ratio is 0.48±0.01

FIG. 3.

Cost functions for the Poisson ratio of the frame of an air-saturated poroelastic cylinder for 1500 Hz. Poisson ratio is 0.48±0.01

Close modal

The theoretically computed fields using the recovered values are finally compared against the measured total and diffracted fields. In Fig. 4(a) and 4(b) the measured total field and diffracted fields are compared with the theoretical ones. The diffracted field computed using the equivalent fluid model at this frequency is also shown in Fig. 4(c). The modified Biot method is shown to better reconstruct the back scattered field amplitude than the EFM at this frequency. The other comparisons with experimental data for 1500 Hz is depicted on Fig. (5). The measurement units are in Pascal. There is a good agreement as is with the EFM results in reference 2.

FIG. 4.

Air-saturated poroelastic cylinder: amplitudes for a 500 Hz acoustic probe (solid line - theory and dashed line - experiment) (a) Total acoustic pressure field. (b) Diffracted field (c) Comparison of the same experimental data with the Equivalent fluid theory1 (dashed-dotted line).

FIG. 4.

Air-saturated poroelastic cylinder: amplitudes for a 500 Hz acoustic probe (solid line - theory and dashed line - experiment) (a) Total acoustic pressure field. (b) Diffracted field (c) Comparison of the same experimental data with the Equivalent fluid theory1 (dashed-dotted line).

Close modal
FIG. 5.

Air-saturated poroelastic cylinder: amplitudes (a) Total acoustic pressure field. (b) Diffracted field for a 1500 Hz acoustic probe (solid line - theory and dashed line - experiment).

FIG. 5.

Air-saturated poroelastic cylinder: amplitudes (a) Total acoustic pressure field. (b) Diffracted field for a 1500 Hz acoustic probe (solid line - theory and dashed line - experiment).

Close modal

The measured, total acoustic fields depicted on Figures 4(a) and 5(b), are slightly shifted as compared to the theoretically computed ones. The total scattered pressure field which is a direct measurement, is affected by the degree of anechoicity of the room, that is, stray acoustic fields diffracted by objects that are not necessarily the test target but are present in the room e.g the turn-table. The diffracted field, which is the difference between the incident and scattered fields agree well with the theory i.e is not shifted because the “noise” present in the two latter fields have been subtracted. The difficulty in centering of the suspended target with respect to the table's center of rotation and also on the perpendicularity of the vertical plane of the object and the horizontal plane of the floor on which the turn-table is resting, can also contribute to the shift.

Even though the back-scattered field (from object to source) is very small compared to the forward one (downstream after the object) due to absorption by the poroelastic cylinder, the diffracted fields (Fig. 4(b) and Fig. 5(b)) are still important. Consequently it is necessary for applications necessitating acoustic (or electromagnetic wave) discretion to control the scattered field by other means other than the passive application of absorbing liners or paints.

The transition frequency which separates low-frequency viscosity dominated flow from the high-frequency inertia dominated one is given by

\begin{equation}f_{T_v}=\frac{\phi \eta }{2\pi \alpha _\infty q_0\rho _0}=\frac{\phi \sigma }{2\pi \alpha _\infty \rho _0},\end{equation}
fTv=φη2παq0ρ0=φσ2παρ0,
(38)

where q0 is the viscous permeability. The transition frequency for the melamine foam employed in this study having a flow resistivity (σ) of 12500 N

$\text{m}^{-4}$
m4s,2 is
$f_{T_v}\approx$
fTv
1500 Hz. Characterization has been done in the low frequency range, that is ⩽ 1500 Hz. Refinements in the low frequency bulk modulus predictions20 can be attained by employing the Lafarge model12,21. The only inconvenience is that the Lafarge model necessities knowledge of an extra parameter, the thermal permeability
$q^{\prime }_0$
q0
and the inertial factor.22 This refinement will constitute the topic of future studies.

Most foams are anisotropic to some extent due to the foaming process23 but in this study we have supposed an isotropic model.

This study highlights that employing an incident acoustic wave initially propagating in the air medium is able to carry some information on the mechanical properties of the frame (Young's modulus, Poisson ratio and density) of a diffracting poroelastic cylinder. Usually the solid phase is considered as immobile when solicited from air generated waves and approximation of the rigid frame is often assumed, i.e. the wave inside the porous material propagates only inside the fluid phase. Implicitly, if the waves carry some information on the elastic properties of the solid phase, this means that the waves are also propagating in the solid part and that the approximation of the rigid frame is valid only as a first approximation. This is the first time the modified Biot model is shown to be able to recover the poroelastic material parameters of soft plastic open cell foams from measurements of diffracted acoustic waves in air, in the frequency domain where these materials are employed.

Johnson et al.24 developed an expression for the complex effective density matching the high and low frequency regimes written as follows

\begin{equation}\rho (\omega ) = \rho _{0}\left(\alpha _{\infty }-\frac{v\phi }{i\omega q_{0} }G_{j}(\omega )\right),\end{equation}
ρ(ω)=ρ0αvφiωq0Gj(ω),
(A1)

where ρ0 is the density of the saturating fluid, ϕ is the porosity, α is the tortuosity of the porous material, v = η/ρ0 is kinematic viscosity and q0 = η/σ is the static viscous permeability (η is the shear viscosity of the saturating fluid and σ the static flow resistivity). The viscous permeability q0 is an intrinsic parameter depending only on the geometry of the pores. Gj(ω) is the viscodynamic function that provides the description of both the magnitude and phase of the exact dynamic tortuosity of large networks formed from a distribution of random radii and is given by

\begin{equation}G_{j}(\omega )=\left(1-\frac{i\omega }{v}\left(\frac{2\alpha _{\infty }q_0}{\Lambda \phi }\right)^2\right)^{\frac{1}{2}},\end{equation}
Gj(ω)=1iωv2αq0Λφ212,
(A2)

where Λ is the viscous characteristic length (related to the size of the inter-connection between two pores).

The thermal transfer due to the interaction between the fluid (in our case air) and the structure is modeled in the frequency domain through the complex bulk modulus of the porous material using the Champoux-Allard model3 and is given by

\begin{equation}K_{f}(\omega )=\gamma P_{0}{\left\lbrace \gamma -(\gamma -1)\left(1-\frac{v^{\prime }\phi }{i\omega q^{\prime }_0}{G^{\prime }}_{j}(\omega )\right)^{-1}\right\rbrace }^{-1},\end{equation}
Kf(ω)=γP0γ(γ1)1vφiωq0Gj(ω)11,
(A3)

where P0 is the ambient pressure, γ is the specific heat ratio,

$q^{\prime }_0=\phi \Lambda ^{\prime }/8$
q0=φΛ/8⁠, Λ′ is the thermal characteristic length (related to the size of the cell, the edge breadth of the ligament25), Pr is the Prandt number and v′ = v/Pr2. Gj(ω) is the thermal correction factor which is written as follows

\begin{equation}{G^{\prime }}_{j}(\omega )=\left(1-\frac{i\omega }{v^{\prime }}\left(\frac{\Lambda ^{\prime }}{4}\right)^2\right)^{\frac{1}{2}}.\end{equation}
Gj(ω)=1iωvΛ4212.
(A4)

In this study and also as in reference 7, Λ′ = 3Λ (classically fixed from 2 to 3 for plastic foams26).

The elements of the 4 × 4 matrix

$\mathbf {X}$
X whose elements are Xmn, denoting element of row m and column n are given by

\begin{eqnarray}\nonumber {X}_{11} &=& i\omega \rho _{0}\phi H_{m}(k_{0}r_c), \\\nonumber {X}_{12} &=& -b_{1}k_{1}^{2}J_{m}(k_{1}r_c),\\\nonumber {X}_{13} &=& -b_{2}k_{2}^{2}J_{m}(k_{2}r_c),\\\nonumber {X}_{14} &=& 0,\\\nonumber {X}_{21} &=& i\omega \rho _{0}(1-\phi )H_{m}(k_{0}r_c),\\\nonumber {X}_{22} &=& k_{1}^{2}\left[a_{1}J_{m}(k_{1}r_c)+2N\ddot{J}_{m}(k_{1}r_c)\right],\\\nonumber {X}_{23} &=& k_{2}^{2}\left[a_{2}J_{m}(k_{2}r_c)+2N\ddot{J}_{m}(k_{2}r_c)\right] ,\\\nonumber {X}_{24} &=& \left(\frac{2iN}{r_c^{2}}\right)m\left[k_{3}r_c\dot{J}_{m}(k_{3}r_c)-J_{m}(k_{3}r_c)\right],\\\nonumber {X}_{31} &=& 0 ,\\\nonumber {X}_{32} &=& 2im\left[k_{1}r_c\dot{J}_{m}(k_{1}r_c)-J_{m}(k_{1}r_c)\right] ,\\\nonumber {X}_{33} &=& 2im\left[k_{2}r_c\dot{J}_{m}(k_{2}r_c)-J_{m}(k_{2}r_c)\right],\\\nonumber {X}_{34} &=& \left[-m^{2}J_{m}(k_{3}r_c)+k_{3}r_c\dot{J}_{m}(k_{3}r_c)-k_{3}^{2}r_c^{2}\ddot{J}_{m}(k_{3}r_c)\right],\\\nonumber {X}_{41} &=& k_{0}\dot{H}_{m}(k_{0}r_c),\\\nonumber {X}_{42} &=& i\omega \left[1+\phi (\mu _{1}-1)\right]k_{1}\dot{J}_{m}(k_{1}r_c),\\\nonumber {X}_{43} &=& i\omega \left[1+\phi (\mu _{2}-1)\right]k_{2}\dot{J}_{m}(k_{2}r_c),\\{X}_{44} &=& -(\frac{\omega }{r_c})\left[1+\phi (\mu _{3}-1)\right]mJ_{m}(k_{3}r_c) ,\end{eqnarray}
X11=iωρ0φHm(k0rc),X12=b1k12Jm(k1rc),X13=b2k22Jm(k2rc),X14=0,X21=iωρ0(1φ)Hm(k0rc),X22=k12a1Jm(k1rc)+2NJ̈m(k1rc),X23=k22a2Jm(k2rc)+2NJ̈m(k2rc),X24=2iNrc2mk3rcJ̇m(k3rc)Jm(k3rc),X31=0,X32=2imk1rcJ̇m(k1rc)Jm(k1rc),X33=2imk2rcJ̇m(k2rc)Jm(k2rc),X34=m2Jm(k3rc)+k3rcJ̇m(k3rc)k32rc2J̈m(k3rc),X41=k0Ḣm(k0rc),X42=iω1+φ(μ11)k1J̇m(k1rc),X43=iω1+φ(μ21)k2J̇m(k2rc),X44=(ωrc)1+φ(μ31)mJm(k3rc),
(B1)

The elements of the vector

$\mathbf {y}$
y are

\begin{eqnarray}\nonumber {y}_{1} &=& A_m,\\\nonumber {y}_{2} &=& B_m,\\\nonumber {y}_{3} &=& C_m,\\{y}_{4} &=& D_m,\end{eqnarray}
y1=Am,y2=Bm,y3=Cm,y4=Dm,
(B2)

Finally, the elements of the vector

$\mathbf {z}$
z are given by

\begin{eqnarray}\nonumber \nonumber {z}_{1} &=& -i\omega \rho _{0}\phi i^m J_{m}(k_{0}r_c), \\\nonumber {z}_{2} &=& -i\omega \rho _{0}(1-\phi )i^{m}J_{m}(k_{0}r_c),\\\nonumber {z}_{3} &=& 0,\\{z}_{4} &=& -i^{m}k_{0}\dot{J}_{m}(k_{0}r_c),\end{eqnarray}
z1=iωρ0φimJm(k0rc),z2=iωρ0(1φ)imJm(k0rc),z3=0,z4=imk0J̇m(k0rc),
(B3)
1.
E.
Ogam
,
C.
Depollier
, and
Z. E. A.
Fellah
.
The direct problem of acoustic diffraction of an audible probe radiation by an air-saturated porous cylinder
.
J. Appl. Phys.
,
108
(
113519
):
9
pages,
2010
.
2.
E.
Ogam
,
C.
Depollier
, and
Z. E. A.
Fellah
.
The direct and inverse problems of an air-saturated porous cylinder submitted to acoustic radiation
.
Rev. Sci. Inst.
,
81
(
094902
):
9
pages,
2010
.
3.
J.-F
Allard
and
N.
Atalla
.
Propagation of Sound in Porous Media: Modelling Sound Absorbing Materials
.
John Wiley and Sons
, 2nd edition,
2009
.
4.
J.-P.
Groby
,
E.
Ogam
,
L.
De Ryck
,
N.
Sebaa
, and
W.
Lauriks
.
Analytical method for the ultrasonic characterization of homogeneous rigid porous materials from transmitted and reflected coefficients
.
J. Acoust. Soc. Am.
,
127
(
2
):
764
772
,
2010
.
5.
Z. E. A.
Fellah
,
F. G.
Mitri
,
M.
Fellah
,
E.
Ogam
, and
C.
Depollier
.
Ultrasonic characterization of porous absorbing materials : Inverse problem
.
J. Sound Vib.
,
302
(
4-5
):
746
759
,
2007
.
6.
Ph.
Leclaire
,
L.
Kelders
,
W.
Lauriks
,
J. F.
Allard
, and
C.
Glorieux
.
Ultrasonic wave propagation in reticulated foams saturated by different gases: High frequency limit of the classical models
.
Appl. Phys. Lett.
,
69
(
18
):
2641
2643
,
1996
.
7.
E.
Ogam
,
Z. E. A.
Fellah
,
N.
Sebaa
, and
J.-P
Groby
.
Non-ambiguous recovery of Biot poroelastic parameters of cellular panels using ultrasonic waves
.
J. Sound and Vib.
,
330
(
6
):
1074
1090
,
2011
.
8.
M. A.
Biot
.
Theory of propagation of elastic waves in a fluid-saturated porous solid. I. Low-Frequency Range
.
J. Acoust. Soc. Am.
,
28
(
2
):
168
178
,
1956
.
9.
M. A.
Biot
.
Theory of propagation of elastic waves in a fluid-saturated porous solid. II. Higher Frequency Range
.
J. Acoust. Soc. Am.
,
28
(
2
):
179
191
,
1956
.
10.
Y.
Champoux
and
J.-F.
Allard
.
Dynamic tortuosity and bulk modulus in air-saturated porous media
.
J. Appl. Phys.
,
70
:
1975
1979
,
1991
.
11.
Y.
Champoux
and
Michael R.
Stinson
.
On acoustical models for sound propagation in rigid frame porous materials and the influence of shape factors
.
J. Acoust. Soc. Am.
,
92
(
2
):
1120
1131
,
1992
.
12.
D.
Lafarge
,
P.
Lemarinier
,
J.-F.
Allard
, and
V.
Tarnow
.
Dynamic compressibility of air in porous structures at audible frequencies
.
J. Acoust. Soc. Am.
,
102
(
4
):
1995
2006
,
1997
.
13.
J.-F.
Allard
and
Y.
Champoux
.
New empirical equations for sound propagation in rigid frame fibrous materials
.
J. Acoust. Soc. Am.
,
91
(
6
):
3346
3353
,
1992
.
14.
S. M.
Hasheminejad
and
M. A.
Alibakhshi
.
Diffraction of sound by a poroelastic cylindrical absorber near an impedance plane
.
Int. J. Mech. Sci.
,
49
(
1
):
1
12
,
2007
.
15.
M.
Abramowitz
and
I. A.
Stegun
.
Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables
.
Dover
,
New York
, Ninth Dover printing, tenth GPO printing edition,
1964
.
16.
Jin Au
Kong
.
Electromagnetic Wave Theory
.
EMW publishing
,
Cambridge,. Massachusetts, USA
,
2008
.
17.
P. M.
Morse
and
K. U.
Ingard
.
Theoretical acoustics
.
McGraw-Hill, Inc
,
New York
,
1968
.
18.
J. D.
Achenbach
.
Wave propagation in elastic solids
.
North-Holland
,
New York
,
1973
.
19.
Keith O.
Geddes
,
George
Labahn
, and
Michael
Monagan
.
Maple 12 Advanced Programming Guide
.
Computer Science Department, University of Waterloo
,
Ontario, Canada
,
2010
.
20.
X.
Olny
and
R.
Panneton
.
Acoustical determination of the parameters governing thermal dissipation in porous media
.
J. Acoust. Soc. Am.
,
123
(
2
):
814
824
,
2008
.
21.
D.
Lafarge
.
Matériaux et acoustique 1
, volume
1
, chapter Milieux poreux et poreux stratifiés. Modèles linéaires de propagation, pages
143
188
.
lavoisier
,
Paris
,
2006
.
22.
A. N.
Norris
.
On the viscodynamic operator in Biot's equations of poroelasticity
.
Journal of Wave-Material Interaction
,
1
(
4
):
365
380
,
1986
.
23.
L. J.
Gibson
and
M. F.
Ashby
.
Cellular solids : Structure and properties
.
Cambridge Solid State Science. Cambridge University Press
, second edition,
1997
.
24.
D. L.
Johnson
,
J.
Koplik
, and
R.
Dashen
.
Theory of dynamic permeability and tortuosity in fluid-saturated porous media
.
J. fluid mech.
,
176
(
1
):
379
402
,
1987
.
25.
C.
Perrot
,
F.
Chevillotte
, and
R.
Panneton
.
Bottom-up approach for microstructure optimization of sound absorbing materials
.
J. Acoust. Soc. Am.
,
124
(
2
):
940
948
,
2008
.
26.
A.
Moussatov
,
C.
Ayrault
, and
B.
Castagnède
.
Porous material characterization - ultrasonic method for estimation of tortuosity and characteristic length using a barometric chamber
.
Ultrasonics
,
39
(
3
):
195
202
,
2001
.