A time domain model of transient acoustic wave propagation in double-layered porous media

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A time domain model of transient acoustic wave propagation in double-layered porous media

Zine El Abiddine Fellah, Armand Wirgin, Mohamed Fellah, Naima Sebaa, Claude Depollier, Walter Lauriks

To cite this version:

Zine El Abiddine Fellah, Armand Wirgin, Mohamed Fellah, Naima Sebaa, Claude Depollier, et al.. A time domain model of transient acoustic wave propagation in double-layered porous media. Jour- nal of the Acoustical Society of America, Acoustical Society of America, 2005, 118, pp.661-670.

�10.1121/1.1953247�. �hal-00088188�

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A time-domain model of transient acoustic wave propagation in double-layered porous media

Z. E. A. Fellah and A. Wirgin

Laboratoire de Mécanique et d’Acoustique, CNRS-UPR 7051, 31 chemin Joseph Aiguier, Marseille, 13009, France

M. Fellah

Laboratoire de Physique Théorique, Institut de Physique, USTHB, BP 32 El Alia, Bab Ezzouar 16111, Algeria

N. Sebaa and C. Depollier

Laboratoire d’Acoustique de l’Université du Maine, UMR-CNRS 6613, Université du Maine, Avenue Olivier Messiaen, 72085 Le Mans Cedex 09, France

W. Lauriks

Laboratorium voor Akoestiek en Thermische Fysica, Katholieke Universiteit Leuven, Celestijnenlaan 200 D, B-3001 Heverlee, Belgium

This paper concerns a time-domain model of transient wave propagation in double-layered porous materials. An analytical derivation of reflection and transmission scattering operators is given in the time domain. These scattering kernels are the medium’s responses to an incident acoustic pulse. The expressions obtained take into account the multiple reflections occurring at the interfaces of the double-layered material. The double-layered porous media consist of two slabs of homogeneous isotropic porous materials with a rigid frame. Each porous slab is described by a temporal equivalent fluid model, in which the acoustic wave propagates only in the fluid saturating the material. In this model, the inertial effects are described by the tortuosity; the viscous and thermal losses of the medium are described by two susceptibility kernels which depend on the viscous and thermal characteristic lengths. Experimental and numerical results are given for waves transmitted and reflected by double-layered porous media formed by air-saturated plastic foam samples.

I. INTRODUCTION

The ultrasonic characterization of porous materials saturated by air^1,2is of great interest for a large class of industrial applications. These materials are frequently used in the au- tomotive and aeronautics industries and in the building trade.

Ultrasonic characterization of materials is often achieved by measuring the attenuation coefficient and phase velocity in the frequency domain,^3,4or by solving the direct and inverse problems directly in the time domain.^5–13In the frequency domain, measurements of the attenuation coefficient may be more robust than measurements of phase velocity. In these situations, the application of the Kramers–Kronig^11–13dispersion relations may allow the de- termination of the phase velocity from the measured attenuation coefficient.

Many applications, such as medical imaging or inverse scattering,¹⁴require a study of the behavior of pulses travel- ing into porous media.^3,4,8–11 When a broadband ultrasound pulse passes through a layer of a medium, the pulse waveform changes as a result of attenuation and dispersion of the medium. The classic method for predicting a change in the waveform of a signal passing through a medium relies on the system’s impulse response. According to the theory of linear systems,¹⁵ the output signal is a convolution of the input

signal and the system impulse response. Many media, including porous materials and soft tissues, have been observed to have an attenuation function that increases with frequency.¹⁶As a result, higher frequency components of the pulse are attenuated more than lower frequency components.

After passing through the layer, the transmitted pulse is not just a scaled-down version of the incident pulse, but has a different shape. Dispersion refers to the phenomenon observed when the phase velocity of a propagating wave changes with frequency.¹⁷Dispersion causes the propagating pulse waveform to change because wave components with different frequencies travel at different speeds. An under- standing of the interaction of ultrasound with a porous medium in both the time and frequency domains, and the ability to determine the change of waveform when propagating ultrasound pulses, should be useful in designing array transducers and in quantitative ultrasound tissue characterization.^18,19

This time-domain model is an alternative to the classical frequency-domain approach.^1,3,4 It is an advantage of the time domain that the results are immediate and direct.^5–13 The attractive feature of a time-domain-based approach is that the analysis is naturally bounded by the finite duration of ultrasonic pressures, and is consequently the most appropri-

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ate approach for the transient signal. However, for wave propagation generated by time-harmonic incident waves and sources 共monochromatic waves兲, the frequency analysis is more appropriate.¹A time-domain approach differs from frequency analysis in that the susceptibility functions describing viscous and thermal effects are convolution operators acting on velocity and pressure, and therefore a different algebraic formalism must be applied to solve the wave equation. The time-domain response of the material is described by an instantaneous response and a “susceptibility” kernel responsible for memory effects.

In the past, many authors have used fractional calculus²⁰ as an empirical method to describe the properties of vis- coelastic materials, e.g., see Caputo²¹ and Bagley and Torvik.²² The observation that asymptotic expressions of stiffness and damping in porous materials are proportional to the fractional powers of frequency²³ suggests that time derivatives of a fractional order might describe the behavior of sound waves in this kind of material, including relaxation and frequency dependence. In this work, fractional calculus is used to describe viscous and thermal interaction between the fluid and the structure in double-layered porous media consisting of two slabs of homogeneous porous materials.

Given the medium’s response to an incident pulse, reflection and transmission scattering operators are calculated for double-layered porous media. Experimental results are compared with theoretical predictions, giving good correlation.

The outline of this paper is as follows. Section II shows a temporal equivalent fluid model, the connection between the fractional derivatives and the wave propagation in rigid porous media in high-frequency range is established, and the basic equations are written in the time domain. Sections III and IV are devoted to formulating the problem and analytical derivation of the reflection and transmission scattering kernels for double-layered porous media consisting of two slabs of homogeneous porous materials. The scattering responses of the media take into account the multiple reflections at the double-layered porous media interfaces. Finally, in Sec. V, experimental validation using ultrasonic measurement in transmission and reflection is discussed for air-saturated industrial plastic foams.

II. TEMPORAL EQUIVALENT FLUID MODEL

The quantities involved in sound propagation in porous materials can be defined locally, on a microscopic scale.

However, this study is generally difficult because of the com- plicated frame geometries. Only the mean values of the quantities involved are of practical interest. Averaging must be performed on a macroscopic scale, using volumes with large enough dimensions for the average to be significant. At the same time, these dimensions must be much smaller than the wavelength. Even on a macroscopic scale, describing sound propagation in porous material can be very compli- cated, since sound also propagates in the frame of the material. If the frame is motionless, the porous material can be replaced on a macroscopic scale by an equivalent fluid.

In porous material acoustics, a distinction can be made between two situations depending on whether the frame is

moving or not. In the first case, the wave dynamics due to coupling between the solid frame and the fluid is clearly described by the Biot theory.^24,25In air-saturated porous media, the structure is generally motionless and the waves propagate only in the fluid. This case is described by the equivalent fluid model which is a particular case in the Biot model, in which fluid–structure interactions are taken into account by the viscous susceptibility kernel,␹v, and the thermal susceptibility kernel, ␹th, as follows:^8,26

␳f␣_⬁⳵tv共r,t兲+冕0 t

␹v共t−t⬘兲⳵tv共r,t⬘兲dt⬘^{= −} ⵜp共r,t兲, 共1兲

1 Ka

⳵tp共r,t兲+冕0 t

␹th共t−t⬘兲⳵tp共r,t⬘兲dt⬘^{= −} ⵜv共r,t兲. 共2兲 Constitutive relations in the time domain result from ar- guments based on invariance under time translation and causality.^11–13In these equations,pis the acoustic pressure,v is the particle velocity, ␳f andKa are the density and com- pressibility modulus of the fluid, respectively. The parameter

␣_⬁reflects the instantaneous response of the porous medium and describes the inertial coupling between fluid and structure. Instantaneous therefore means that the response time is much shorter than the typical time scale for acoustic field variation. The susceptibility kernels␹v and␹th are memory functions which determine the dispersion of the medium.

The medium varies with depthx only, and the incident wave is planar and normally incident. With no lack of gen- erality, the pressure acoustic field can be assumed to have only one component, denoted p共x,t兲. It is assumed that the pressure field in the medium is zero prior to t= 0. The wave equation for the pressure acoustic field of a porous dispersive medium with a rigid frame is obtained from the constitutive equations 共1兲and共2兲, and is in the form

⳵x2p共x,t兲− 1

c₀²冋^␣^⬁^⳵^t²^p共x,t兲⁺冉^␣^⬁^K^a^␹^th⁺^␹␳^vf

+c₀²␹thⴱ␹v冊^ⴱ^⳵^t²^p^共^x,t^兲册^{= 0,} ^共³^兲

wherec₀=共Ka/␳f兲^1/2is the speed of free fluid. The following notation is used for the convolution integral:

关fⴱg兴共x,t兲=冕0 t

f共x,t−t⬘兲g共x,t⬘兲dt⬘^. 共4兲 Viscous and thermal exchanges between a fluid- saturated porous medium and its structure are responsible for acoustic field damping. In the asymptotic regime, corresponding to high-frequency limit,^8,26viscous and thermal interactions are modeled by the memory relaxation operators

␹v共t兲and␹th共t兲 given by²⁶

␹v共t兲=2␳⌳ f␣⬁冑␲␳^␩f^t^−1/2^, ^共5兲

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␹th共t兲=2共␥− 1兲

Ka⌳⬘ 冑_␲P^␩r␳f

t^−1/2, 共6兲

where Pr is the Prandtl number, ␩ ^and␥ are the fluid vis- cosity and adiabatic constant, respectively. The model’s rel- evant physical parameters are tortuosity␣_⬁, and viscous and thermal characteristic lengths, ⌳ and ⌳⬘. In this model the time convolution oft^−1/2 with a function is interpreted as a semiderivative operator according to the definition of the fractional derivative of order␯ ^given²⁷ ^by

D^␯关x共t兲兴= 1

⌫共−␯兲冕0 t

共t−u兲⁻^␯⁻¹x共u兲du, 共7兲 where⌫共x兲 is the gamma function.

III. FORMULATING THE PROBLEM

Consider a double-layered porous medium consisting of two homogeneous slabs with different acoustic parameters.

The geometry of the problem is shown in Fig. 1. The first porous slab occupies the region 0艋x艋ᐉand the second one occupies the region ᐉ艋x艋L. Each porous slab is assumed to be isotropic and to have a rigid frame. A short sound pulse impinges normally on the medium from the left关free fluid—

region共1兲兴. It gives rise to an acoustic pressure fieldpi共x,t兲, i= 2 , 3 and an acoustic velocity field vi共x,t兲, i= 2 , 3 within each layer 关the symboli= 2 , 3 denotes regions 共2兲 and 共3兲, respectively兴which satisfy the propagation equation共3兲.

It is assumed that the pressure field is continuous at the boundary of each layer

p共0⁺,t兲=p共0⁻,t兲, p共ᐉ⁻,t兲=p共ᐉ⁺,t兲,

p关共ᐉ+L兲⁻,t兴=p关共ᐉ+L兲⁺,t兴共8兲共where ± superscript denotes the limit from left and right, respectively兲and the initial conditions

兩p共x,t兲兩t=0= 0 冏^⳵⳵^p^t冏t=0

= 0, 共9兲

which mean that the medium is idle fort= 0.

If the incident sound wave is launched in region x艋0 关region共1兲兴, then the pressure field in the region on the left of the double-layered material is expressed as the sum of the incident and reflected fields

p₁共x,t兲=pⁱ冉^t⁻c^x₀冊⁺^p^r冉^t⁺c^x₀冊^, ^x^⬍^0, ^共10兲

wherep₁共x,t兲is the field in regionx⬍0,pⁱandp^rdenote the incident and reflected fields, respectively. A transmitted field is also produced in the region on the right of the double- layered material. This takes the form

p₄共x,t兲=p^t冉^t⁻^c^ᐉ2

− L

c₃−x−ᐉ−L

c₀ 冊^, ^x^⬎^ᐉ⁺^L. ^共11兲

关p4共x,t兲 is the field in region共4兲: x⬎ᐉ+L and p^t the transmitted field兴. c₂ and c₃ represent the acoustic velocities in regions 共2兲 and共3兲, respectively, defined by the relation ci

=c₀/共␣i兲^1/2,i= 2 , 3, where␣irepresents the tortuosity of each porous layer.

The incident and scattered fields are related by scattering operators共i.e., reflection and transmission operators兲for the material. These are integral operators represented by

p^r共x,t兲=冕0 t

R˜共␶兲pⁱ冉^t⁻^␶⁺^c^x0冊^d^␶

=R˜共t兲ⴱpⁱ共t兲ⴱ␦冉^t⁺c^x₀冊^, ^共12兲

p^t共x,t兲=冕0 t

T˜共␶兲pⁱ冉^t⁻^c^ᐉ2

− L

c₃−x−ᐉ−L c₀ 冊^d^␶

=T˜共t兲ⴱpⁱ共t兲ⴱ␦冉^t⁻c^ᐉ₂− L

c₃−x−ᐉ−L

c₀ 冊^, ^共13兲

where␦共t兲is the Dirac distribution. In Eqs.共12兲and共13兲the functionsR˜ and˜Tare the respective reflection and transmission kernels for incidence from the left. Note that the lower limit of integration in 共12兲 and 共13兲 is set to 0, which is equivalent to assuming that the incident wavefront first impinges on the material at t= 0. The operators R˜ and T˜ are independent of the incident field used in the scattering ex- periment and depend only on the properties of the materials.

In regionx艋0, the field p₁共x,t兲is given by

p₁共x,t兲=冋^␦冉^t⁻^c^x0冊⁺^R^˜^共t兲^ⴱ^␦冉^t⁺^c^x0冊册^ⴱ^pⁱ^共t兲. ^共14兲

To simplify the analysis, we will use the Laplace transform which is appropriate for our problem. We note P˜

i共x,z兲, i

= 1 , 2 , 3 , 4, the Laplace transform of pi共x,t兲,i= 1 , 2 , 3 , 4 defined by

˜P

i共x,z兲=L关p_i共x,t兲兴=冕0

⬁

exp共−zt兲p_i共x,t兲dt. 共15兲 The Laplace transform of the field outside the double- layered medium is given by

˜P

1共x,z兲=冋^exp冉⁻^z^c^x0冊⁺^R^共^z^兲^exp冉^z^c^x0冊册^␸^共^z^兲^, ^x^艋^0, ^共¹⁶^兲

FIG. 1. Problem geometry.

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P˜

4共x,z兲=T共z兲exp冋⁻^冉^c^ᐉ2

+ L

c₃+x−ᐉ−L

c₀ 冊^z册^␸^共z兲,

x艌ᐉ+L, 共17兲

Here, P˜

1共x,z兲 andP˜

4共x,z兲 are the Laplace transform of the field on the left and right of the double-layered porous media, respectively,␸共z兲 denotes the Laplace transform of the incident fieldpⁱ共t兲, and finallyR共z兲andT共z兲are the Laplace transforms of the reflection and transmission kernels, respectively.

The acoustic pressure fields pi共x,t兲, i= 2 , 3 inside each layer of the porous media 关regions 共2兲 and 共3兲兴 satisfy the propagation equation 共3兲, which can be written in the Laplace domain as

⳵²^P^˜i共x,z兲

⳵^x² ⁻ fi共z兲

c_i² P˜

i共x,z兲= 0, i= 2,3, 0艋x艋ᐉ+L. 共18兲 The function fi共z兲is given by the following expression:

fi共z兲=z²c_i²关␳f␣i+␹^˜vi共z兲兴·关1/Ka+␹^˜thi共z兲兴, i= 2,3, 共19兲 where^˜␹vi共z兲 and␹^˜thi共z兲 represent the Laplace transform of

␹vi共t兲 and ␹thi共t兲, respectively, and their expressions in the time domain are given by

␹vi共t兲=2␳f␣i

⌳i 冑␲␳^␩f

t^−1/2, i= 2,3. 共20兲

␹thi共t兲=2共␥− 1兲

Ka⌳i⬘ 冑_␲P^␩r␳f

t^−1/2, i= 2,3. 共21兲

⌳iand⌳i⬘^,ⁱ= 2 , 3 are the viscous and thermal characteristic lengths of each porous layer.

By developing expression共19兲, we obtain the following relation for fi共z兲:

fi共z兲=z²+ 2冑^␩_␳冉^⌳¹ⁱ⁺^冑^␥^P^{− 1}r⌳i⬘冊^z^冑^z⁺␳^4共f⌳^␥i⌳^{− 1兲}i⬘冑P^␩_rz,

i= 2,3. 共22兲

The solution of Eq.共18兲can be expressed by P˜

i共x,z兲=

冋

^Aⁱ^共z兲exp冉⁻^冑^f^cⁱ^共z兲i

x冊

+Bi共z兲exp冉^冑^f^cⁱ^共z兲i

x冊

册

^␸^共z兲, ⁱ^{= 2,3,} ^共23兲

where coefficientsAi共z兲andBi共z兲can be determined by the physical conditions at the boundary of each layer. This is given in the next section.

IV. REFLECTION AND TRANSMISSION SCATTERING OPERATORS

To derive reflection and transmission coefficients, we use the continuity relations of the acoustic pressure fields 关Eqs.共8兲兴given in the Laplace domain by

˜P

1共0⁻,z兲=P˜

2共0⁺,z兲, P˜

2共ᐉ⁻,z兲=˜P

3共ᐉ⁺,z兲, P˜

3关共ᐉ+L兲⁻,z兴=P˜

4共共ᐉ+L兲⁺,z兲. 共24兲 Using the expressions of the pressure fields in each layer 关Eqs.共16兲,共17兲, and共23兲兴and the conditions共24兲, we obtain the followings relations for the coefficientsA_i共z兲 andB_i共z兲, i= 2 , 3, and the coefficientsR共z兲andT共z兲:

A₂共z兲+B₂共z兲=P˜

2共0,z兲= 1 +R共z兲, 共25兲

A₂共z兲exp冉⁻^冑^f^c²2^共^z^兲

ᐉ冊⁺^B²^共z兲exp冉^冑^f^c²2^共^z^兲

ᐉ冊

=A₃共z兲exp冉⁻^冑^f^c³3^共z兲

ᐉ冊⁺^B³^共z兲exp冉^冑^f^c³3^共z兲

ᐉ冊^, ^共26兲

A₃共z兲exp冉⁻^冑^f^c³3^共z兲

共ᐉ+L兲冊⁺^B³^共z兲exp冉^冑^f^c³3^共z兲

共ᐉ+L兲冊

=T共z兲exp冋⁻^冉^c^ᐉ2

+ L

c₃冊^z册^. ^共27兲

The Euler equation in each region is written as

␳f␣i⳵tvi共x,t兲+␹vi共t兲ⴱ⳵tvi共x,t兲= −⳵xpi共x,t兲, i= 1, . . . ,4.

共28兲 In these equationsvi共x,t兲 i= 1 , . . . , 4 is the acoustic velocity field in regions 共1兲,…,共4兲. Note that in regions共1兲 and共4兲, corresponding to the free fluid, porosity and tortuosity values are 1 共␣1=␣4= 1 and ␾1=␾4= 1兲, and the viscous susceptibility kernel vanishes 共␹vi= 0 ,i= 1 , 4兲 outside the double- layered porous media.

The equation for flow continuity between each interface 共x= 0,x=ᐉ, and x=ᐉ+L兲is given by

␾iv_i共x,t兲=␾i+1v_i+1共x,t兲, i= 1,2,3, 共29兲 where␾i,i= 1 , . . . , 4 is the porosity of each layer.

Using the relations共28兲and共29兲, we obtain the following relations between the acoustic pressurep_i共x,t兲and physical properties of each layer:

␾i+1关␳f␣i⳵xp_i+1共x,t兲+␹vi共t兲ⴱ⳵xp_i+1共x,t兲兴

=␾i关␳f␣i+1⳵xpi共x,t兲+␹v共i+1兲共t兲ⴱ⳵xpi共x,t兲兴, i= 1,2,3.

共30兲 Using the Laplace transform of Eq. 共30兲 and the pressure field expressions for each layer 关Eqs. 共16兲, 共17兲, and 共23兲兴, we obtain the following relations at the interface of each layer:

B₂共z兲−A₂共z兲=K₁共R共z兲− 1兲, 共31兲

B₃共z兲exp冉^冑^f^c³3^共^z^兲

ᐉ冊⁻^A³^共z兲exp冉⁻^冑^f^c³3^共^z^兲

ᐉ冊

=K₂

冋

^B²^共z兲exp冉^冑^f^c²2^共z兲

ᐉ冊⁻^A²^共z兲exp冉⁻^冑^f^c²2^共z兲

ᐉ冊

册

^,

共32兲

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B₃共z兲exp冉^冑^f^c³3^共z兲

共ᐉ+L兲冊⁻^A³^共z兲exp冉⁻ ^冑^f^c³3^共z兲

共ᐉ+L兲冊

=K₃T共z兲exp冋⁻^冉^c^ᐉ2

+ L

c₃冊^z册^, ^共33兲

with

K₁=冑_␣₂

␾2

, K₂=␾2冑_␣₃

␾3冑␣2

, K₃=冑_␣₃

␾3

, K₁K₂=K₃. 共34兲 Using the relations共25兲–共27兲and共31兲–共33兲共see Appen- dix A兲, we obtain the following expressions of the reflection and transmission coefficients:

R共z兲=d₁⌫共z兲

⌶共z兲, 共35兲

T共z兲=h₁exp冋冉^c^ᐉ2

+ L

c₃冊^z册^exp^冉^{− 2}^冑^f^c²²^⌶共z兲^共^z^兲^ᐉ⁻^冑^f^c³³^共^z^兲^L^冊^,

共36兲 with

⌫共z兲= 1 +d₂exp冉^{− 2}^冑^f^c²2^共z兲

ᐉ冊⁺^d³^exp冉^{− 2}^冑^f^c³3^共z兲

L冊

−d₄exp冉^{− 2}^冑^f^c²2^共z兲

ᐉ− 2冑f₃共z兲 c₃ L冊^,

⌶共z兲= 1 +h₂exp冉^{− 2}^冑^f^c²2^共z兲

ᐉ冊⁺^h³^exp冉^{− 2}^冑^f^c³3^共z兲

L冊

+h₄exp冉^{− 2}^冑^f^c²2^共z兲

ᐉ− 2冑f₃共z兲 c₃ L冊^.

d₁=K₁− 1

K₁+ 1, d₂=共K1+ 1兲共K2− 1兲共K1− 1兲共K2+ 1兲,

d₃=共K3− 1兲共K2− 1兲

共K3+ 1兲共K2+ 1兲, d₄= −共K3− 1兲共K1+ 1兲共K3+ 1兲共K1− 1兲,

h₁= 4K₁K₂

共1 +K₃兲共1 +K₁+K₂+K₃兲,

h₂=共1 −K₂兲共1 −K₁兲

共1 +K₁兲共1 +K₂兲, h₃=共K₃− 1兲共1 −K₂兲共K₃+ 1兲共1 +K₂兲,

h₄=共1 −K₃兲共K1− 1兲共1 +K₃兲共K1+ 1兲.

To expressn-multiple reflections in porous layers, we shall write the reflection and transmission coefficients as follows:

R共z兲=d₁⌫共z兲兺

n艌0共− 1兲ⁿ共⌶共z兲− 1兲ⁿ, 共37兲

+ L

c₃冊^z册^exp冉^{− 2}^冑^f^c²2^共z兲

ᐉ

−冑f₃共z兲 c₃ 共L兲冊n^兺艌0

共− 1兲ⁿ共⌶共z兲− 1兲ⁿ. 共38兲

Using the identity 共x+y+z兲ⁿ=_n 兺

1+n₂+n₃=n

n!

n₁!n₂!n₃!xⁿ¹yⁿ²zⁿ³, 共39兲 where n! =⌫共n+ 1兲, the reflection and transmission expressions become

R共z兲=d₁⌫共z兲_n兺

艌0

共− 1兲ⁿn!_n 兺

1+n₂+n₃=n

h₂ⁿ¹h₃ⁿ²h₄ⁿ³ n₁!n₂!n₃!

⫻exp冉^{− 2}^冑^f^c²2^共z兲

共n1+n₃兲ᐉ冊

⫻exp冉^{− 2}^冑^f^c³3^共z兲

共n2+n₃兲L冊^, ^共40兲

and

+ L c₃冊^z册

⫻_n兺_艌0^共^{− 1}^兲ⁿ^n!_n 兺

1+n₂+n₃=n

⫻exp冉⁻^冑^f^c²2^共z兲

共2n1+ 2n₃+ 1兲ᐉ冊

⫻exp冉⁻^冑^f^c³3^共z兲

共2n2+ 2n₃+ 1兲L冊^. ^共41兲

By setting z=j␻^{, where} ^j²^{= −1 and} ␻ is the angular frequency, we can easily deduce the expressions of the reflection and transmission coefficients in the frequency domain.

Recall that the inverse Laplace transforms of exp关−共ᐉ/c₂兲冑^f2共z兲兴 and exp关−共L/c₃兲冑^f3共z兲兴 are the Green function²⁸ of the first and second porous slab, respectively.

In the time domain, the transmission scattering operator is expressed as

T˜共t兲=h₁_n兺

艌0

共− 1兲ⁿn!_n 兺

1+n₂+n₃=n

h₂ⁿ¹h₃ⁿ²h₄ⁿ³

n₁!n₂!n₃!F₂冋^t⁺^c^ᐉ2

,共2n1

+ 2n₃+ 1兲ᐉ

c₂册^ⴱ^F³冋^t⁺^c^L3

,共2n2+ 2n₃+ 1兲L c₃册^,

共42兲 whereF_i,i= 2 , 3 is the Green function²⁸of porous layers共2兲 and共3兲, respectively共see Appendix B兲.

The reflection scattering operator is expressed by the relation

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R˜共t兲=d₁_n兺_艌0^共^{− 1}^兲ⁿ^n!_n 兺

1+n₂+n₃=n

⫻冋^F²冋^t,2共n¹⁺ⁿ³^兲^c^ᐉ2册^ⴱ^F³冋^t,2共n²⁺ⁿ³^兲^c^L3册

+d₂F₂冋^t,2共n¹⁺ⁿ³^{+ 1兲}^c^ᐉ2册^ⴱ^F³冋^t,2共n²⁺ⁿ³^兲^c^L3册

+d₃F₂冋^t,2^共ⁿ¹⁺ⁿ³^兲^c^ᐉ2册^ⴱ^F³冋^t,2^共ⁿ²⁺ⁿ³^{+ 1}^兲^c^L3册

+d₄F₂冋^t,2共n¹⁺ⁿ³^{+ 1兲}^c^ᐉ2册

ⴱF₃冋^t,2共n²⁺ⁿ³^{+ 1兲}^c^L3册册^. ^共43兲

If only the first reflections at interfacesx= 0,x=ᐉ andx=L are taken into account, the reflection scattering kernel expression becomes

R˜共t兲=d₁␦共t兲+d₁共d₂−h₂兲F₂冋^t,2^c^ᐉ2册⁺^d¹^共^d⁴⁻^h⁴⁻^d²^h³

−d₃h₂+ 2h₂h₃兲F2冋^t,2^c^ᐉ2册^ⴱ^F³冋^t,2^c^L3册^. ^共44兲

The transmission scattering kernel describing the direct wave transmitted through two layers of porous materials without internal reflection is expressed by

T˜共t兲=h₁F₂冋^t⁺^c^ᐉ2

,ᐉ

c₂册^ⴱ^F³冋^t⁺^c^L3

,L c₃册

=h₁冕0 t

F₂冋^␶⁺^c^ᐉ2

,ᐉ

c₂册^F³冋^t⁻^␶⁺^c^L3

,L

c₃册^d^␶^. ^共45兲

The first term on the right-hand side of Eq. 共44兲, d₁␦共t兲

=关共冑_␣₂−␾2兲/共冑_␣₂+␾2兲兴␦共t兲, is equivalent to the instantaneous reflected response of the first layer 共region 2兲. The part of the wave which is equivalent to this term corresponds to the wave reflected by the first interfacex= 0 of the first porous layer. It depends only on the porosity and tortuosity of the first porous slab. The wave reflected by the first interface has the advantage of not being dispersive, but simply attenuated by factor d₁. This result is in agreement with the conclusions obtained in other works for wave reflected by a slab of porous material.^9,29,30This shows that it is possible to measure the porosity and tortuosity of the first porous layer just by measuring its first reflected wave.

The second term on the right-hand side of Eq. 共44兲, d₁共d₂−h₂兲F₂关t, 2共ᐉ/c₂兲兴, corresponds to the second interface reflection contribution, x=ᐉ. This term depends on the porosity and tortuosity of the two porous layers. The Green’s function F₂ describes the propagation and dispersion of an acoustic wave making one round trip inside the first porous slab. This Green’s function depends on the viscous and thermal characteristics lengths⌳and⌳⬘of the first layer共region 2兲. This result means that it can be possible to get informa- tion of all the acoustical properties共porosity, tortuosity, vis-

cous and thermal characteristics lengths兲 of the first porous layer共region 2兲, and also, the porosity and tortuosity of the second porous layer共region 3兲, but the viscous and thermal characteristics lengths of the second porous layer 共region 3兲 do not intervene.

Finally, the term d₁共d₄−h₄−d₂h₃−d₃h₂ + 2h₂h₃兲F2关t, 2共ᐉ/c₂兲兴ⴱF₃关t, 2共L/c₃兲兴 represents the reflection contribution of the third interface,x=L. The correspond- ing wave makes one round trip inside the two porous layers.

Evidently this wave contribution depends on all acoustical parameters of each porous layer.

The advantage of the obtained time-domain expression of the reflection and transmission scattering operators关Eqs.

共44兲and共45兲兴is to show analytically the effect of the acoustical parameters 共porosity, tortuosity, viscous and thermal characteristic lengths兲of each porous layer on the reflection contributions by the interfaces of the double-layered media.

V. EXPERIMENTAL VALIDATION

In application of this model, several numerical simula- tions for reflected and transmitted acoustic waves by two layered porous materials are compared to experimental data.

Experiments are performed in the air using two broadband Ultran NCT202 transducers with a central frequency of 190 kHz in air and a bandwidth of 6 dB from 150 to 230 kHz. Pulses of 400 V are provided by a 5052PR Panametrics pulser/receiver. The signals received are ampli- fied to 90 dB and filtered above 1 MHz to avoid high-

FIG. 2. Experimental setup of the ultrasonic measurements in transmitted mode.

FIG. 3. Incident signal given out by the transducer in transmitted mode.

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frequency noise 共energy is totally filtered by the sample in this upper-frequency domain兲. Electronic interference is eliminated by 1000 acquisition averages. The experimental setup is shown in Fig. 2. The experimental incident signal generated by the transducer is given in Fig. 3. The amplitude is represented by an arbitrary unit共a.u.兲 and the point number represented in the abscissa is proportional to time. Signal duration is important as its spectrum must verify the condi- tion of high-frequency approximation^8,26 referred to in Sec.

II. The spectrum of the incident signal is given in Fig. 4.

Measurements were made on plastic foam samples M1–

M4. Their acoustic characteristics were determined indepen- dently using classical methods⁸共which were developed for a slab of porous material兲. The acoustical parameters of the plastic foam samples are given in Table I.

Three samples of double-layered porous materials were considered, the first consists of 0.86 cm of M1 and 0.81 cm of M2, the second of 4.13 cm of M1 and 1.99 cm of M3, and finally the third of 2.98 cm of M1 and 1.99 cm of M3. Nu- merical simulation and experimental results共transmitted sig- nal兲 for the three samples of double-layered materials are presented in Figs. 5–7, respectively. The numerical results are obtained from convolution of the transmission operator 关Eq.共45兲兴with the signal generated by the transducer shown in Fig. 3. The reader can see, from Figs. 5–7, the good correlation obtained between the experimental transmitted sig- nal共solid line兲and simulated signal共dashed line兲. This result validates the expression of the transmission scattering opera- tor关Eq.共45兲兴.

Reflected waves were processed by another experimental setup shown in Fig. 8. One transducer was used alterna- tively as a transmitter and receiver to detect the reflected

wave. The experimental incident signal used in reflected mode is given in Fig. 9 and its spectrum in Fig. 10.

A double-layered porous medium consisting of 1.11 cm of M4 and 0.87 cm of M1 was considered. Figure 11 shows a comparison between a simulated reflected signal 共dashed line兲 and an experimental reflected signal 共solid line兲. The simulated signal was obtained through convolution of the reflection scattering kernel given in Eq. 共44兲 with the incident signal given in Fig. 9. Three reflected signals can be seen in Fig. 11. The first corresponds to the reflection of the first porous layer M4 共x= 0兲 by the first interface, this reflected wave corresponds to the first term on the right-hand side of Eq. 共44兲: d₁␦共t兲=关共冑_␣₂−␾2兲/共冑_␣₂+␾2兲兴 ␦共t兲. The second reflected wave given in Fig. 11 corresponds to the reflection between the second M4 interface and the first M1 interface共x=ᐉ兲; this wave corresponds to the second term on the right-hand side of Eq. 共44兲:d₁共d2−h₂兲F2关t, 2共ᐉ/c₂兲兴. Fi- nally, the third signal corresponds to reflection by the second M1 interface 共x=L兲, which is given by the term d₁共d4−h₄

−d₂h₃−d₃h₂+ 2h₂h₃兲F2关t, 2共ᐉ/c₂兲兴ⴱF₃关t, 2共L/c₃兲兴. Gener- ally, it is not possible to see the other reflection contributions experimentally because of the high damping of ultrasonic waves in air-saturated plastic foams. However, the third reflection contribution at x=L is not always seen experimen-

FIG. 4. Spectrum of the incident signal generated by the transducer in transmitted mode.

FIG. 6. Comparison between experimental transmitted signal 共solid line兲 and simulated transmitted signal共dashed line兲for a double-layered medium consisting of 4.13 cm of M1 and 1.99 cm of M3.

TABLE I. Acoustical characteristics of the plastic foams samples.

Material M1 M2 M3 M4

Viscous characteristic length关⌳共␮m兲兴 200 30 330 230 Thermal characteristic length关⌳⬘共␮^m兲兴 600 90 990 690

Tortuosity共␣兲 1.07 1.4 1.02 1.05

Porosity共␾兲 0.97 0.85 0.90 0.98

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tally; for example, Fig. 12 shows a comparison between theoretical predictions 共dashed line兲 and experimental results 共solid line兲 for a double-layered medium consisting of 0.88 cm of M1 and 0.83 cm of M2. Acoustic attenuation in plastic foam sample M2 is higher than in the other samples.

Sample M2 has high tortuosity and low characteristic lengths compared to those of the other plastic foam samples, which indicates high acoustic damping. In Fig. 12, we can only see the two reflected waves corresponding to reflection by the first M1共first layer兲interface 共x= 0兲and reflection between the second M1 interface and the first M2 共second layer兲 in- terface共x=ᐉ兲, respectively. The reflected wave of the second M2 interface共x=L兲 is fully absorbed by the two layers, M1 and M2. We can also see in Fig. 12 that the amplitude of the second reflected wave is greater than that of the first reflected wave. This is due to the high resistivity of sample M2 near sample M1 and the thickness of M1, which also plays an important part in attenuating the wave reflected at the second interface,x=ᐉ.

VI. CONCLUSION

In this paper the analytical expressions of reflection and transmission scattering operators are derived for double-

layered porous media consisting of two homogeneous isotropic materials. Simple relationships are given between these operators and the acoustic parameters of the medium. It is shown that the scattering operators are equal to the sum of the contribution of each interface to the double-layered porous medium. The advantages of the analytical expressions of reflection and transmission scattering operators in the time domain is to show easily the effect of the acoustical parameters on the multiple reflections at the interfaces of the double-layered medium.

Ultrasonic measurements in the transmission and reflection mode were processed using different experimental set- ups. A slight difference was observed between theoretical predictions and experimental data in the two modes共reflec- tion and transmission兲. This leads to the conclusion that the expressions of scattering operators obtained are correct. Fu- ture studies will concentrate on the inverse problem, and methods and inversion algorithms will be developed to opti- mize the acoustic properties of double-layered air-saturated porous media for reflected and transmitted ultrasonic waves.

APPENDIX A: EXPRESSION OF THE REFLECTION AND TRANSMISSION OPERATORS

Using Eqs. 共25兲 and 共31兲, we can write the following equation system given the relations betweenA₂共z兲,B₂共z兲, and R共z兲:

FIG. 8. Experimental setup of the ultrasonic measurements in reflected mode.

FIG. 9. Incident signal given out by the transducer in reflected mode.

FIG. 10. Spectrum of the incident signal generated by the transducer in reflected mode.