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

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I. INTRODUCTION
The ultrasonic characterization of porous materials saturated by air 1,2 is of great interest for a large class of industrial applications.These materials are frequently used in the automotive and aeronautics industries and in the building trade.
][10][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 understanding 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. 18,19his time-domain model is an alternative to the classical frequency-domain approach. 1,3,4[7][8][9][10][11][12][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-ate approach for the transient signal.However, for wave propagation generated by time-harmonic incident waves and sources ͑monochromatic waves͒, 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 calculus 20 as an empirical method to describe the properties of viscoelastic materials, e.g., see Caputo 21 and Bagley and Torvik. 22The observation that asymptotic expressions of stiffness and damping in porous materials are proportional to the fractional powers of frequency 23 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 complicated 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 complicated, 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 2][13] In these equations, p is the acoustic pressure, v is the particle velocity, f and K a are the density and compressibility modulus of the fluid, respectively.The parameter ␣ ϱ 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 depth x only, and the incident wave is planar and normally incident.With no lack of generality, the pressure acoustic field can be assumed to have only one component, denoted p͑x , t͒.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 ͑1͒ and ͑2͒, and is in the form

͑3͒
where c 0 = ͑K a / f ͒ 1/2 is the speed of free fluid.The following notation is used for the convolution integral: Viscous and thermal exchanges between a fluidsaturated porous medium and its structure are responsible for acoustic field damping.In the asymptotic regime, corresponding to high-frequency limit, 8,26 viscous and thermal interactions are modeled by the memory relaxation operators v ͑t͒ and th ͑t͒ given by 26 where P r is the Prandtl number, and ␥ are the fluid viscosity and adiabatic constant, respectively.The model's relevant physical parameters are tortuosity ␣ ϱ , and viscous and thermal characteristic lengths, ⌳ and ⌳Ј.In this model the time convolution of t −1/2 with a function is interpreted as a semiderivative operator according to the definition of the fractional derivative of order given 27 by where ⌫͑x͒ 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 ഛ x ഛ ᐉ and the second one occupies the region ᐉ ഛ x ഛ 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 ͓free fluidregion ͑1͔͒.It gives rise to an acoustic pressure field p i ͑x , t͒, i =2,3 and an acoustic velocity field v i ͑x , t͒, i =2,3 within each layer ͓the symbol i =2,3 denotes regions ͑2͒ and ͑3͒, respectively͔ which satisfy the propagation equation ͑3͒.
It is assumed that the pressure field is continuous at the boundary of each layer ͑where ± superscript denotes the limit from left and right, respectively͒ and the initial conditions which mean that the medium is idle for t =0.
If the incident sound wave is launched in region x ഛ 0 ͓region ͑1͔͒, 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 where p 1 ͑x , t͒ is the field in region x Ͻ 0, p i and p r denote the incident and reflected fields, respectively.A transmitted field is also produced in the region on the right of the doublelayered material.This takes the form ͓p 4 ͑x , t͒ is the field in region ͑4͒: x Ͼ ᐉ + L and p t the transmitted field͔.c 2 and c 3 represent the acoustic velocities in regions ͑2͒ and ͑3͒, respectively, defined by the relation , where ␣ i represents the tortuosity of each porous layer.
The incident and scattered fields are related by scattering operators ͑i.e., reflection and transmission operators͒ for the material.These are integral operators represented by where ␦͑t͒ is the Dirac distribution.In Eqs.͑12͒ and ͑13͒ the functions R ˜and T ˜are the respective reflection and transmission kernels for incidence from the left.Note that the lower limit of integration in ͑12͒ and ͑13͒ 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 experiment and depend only on the properties of the materials.
In region x ഛ 0, the field p 1 ͑x , t͒ is given by To simplify the analysis, we will use the Laplace transform which is appropriate for our problem.We note P ˜i͑x , z͒, i =1,2,3,4, the Laplace transform of p i ͑x , t͒, i =1,2,3,4 defined by The Laplace transform of the field outside the doublelayered medium is given by Here, P ˜1͑x , z͒ and P ˜4͑x , z͒ are the Laplace transform of the field on the left and right of the double-layered porous media, respectively, ͑z͒ denotes the Laplace transform of the incident field p i ͑t͒, and finally R͑z͒ and T͑z͒ are the Laplace transforms of the reflection and transmission kernels, respectively.The acoustic pressure fields p i ͑x , t͒, i =2,3 inside each layer of the porous media ͓regions ͑2͒ and ͑3͔͒ satisfy the propagation equation ͑3͒, which can be written in the Laplace domain as

͑18͒
The function f i ͑z͒ is given by the following expression:

͑19͒
where ˜vi ͑z͒ and ˜thi ͑z͒ represent the Laplace transform of vi ͑t͒ and thi ͑t͒, respectively, and their expressions in the time domain are given by ⌳ i and ⌳ i Ј, i =2,3 are the viscous and thermal characteristic lengths of each porous layer.By developing expression ͑19͒, we obtain the following relation for f i ͑z͒: The solution of Eq. ͑18͒ can be expressed by where coefficients A i ͑z͒ and B i ͑z͒ 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 ͓Eqs.͑8͔͒ given in the Laplace domain by P ˜1͑0 − ,z͒ = P ˜2͑0 + ,z͒, P ˜2͑ᐉ − ,z͒ = P ˜3͑ᐉ + ,z͒, Using the expressions of the pressure fields in each layer ͓Eqs.͑16͒, ͑17͒, and ͑23͔͒ and the conditions ͑24͒, we obtain the followings relations for the coefficients A i ͑z͒ and B i ͑z͒, i =2,3, and the coefficients R͑z͒ and T͑z͒:

͑27͒
The Euler equation in each region is written as
The equation for flow continuity between each interface ͑x =0, x = ᐉ, and x = ᐉ + L͒ is given by where i , i = 1 , . . ., 4 is the porosity of each layer.
Using the relations ͑28͒ and ͑29͒, we obtain the following relations between the acoustic pressure p i ͑x , t͒ and physical properties of each layer:

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

͑34͒
Using the relations ͑25͒-͑27͒ and ͑31͒-͑33͒ ͑see Appendix A͒, we obtain the following expressions of the reflection and transmission coefficients: To express n-multiple reflections in porous layers, we shall write the reflection and transmission coefficients as follows: Using the identity x n 1 y n 2 z n 3 , ͑39͒ where n!=⌫͑n +1͒, the reflection and transmission expressions become By setting z = j, where j 2 = −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 exp͓−͑ᐉ / c 2 ͒ ͱ f 2 ͑z͔͒ and exp͓−͑L / c 3 ͒ ͱ f 3 ͑z͔͒ are the Green function 28 of the first and second porous slab, respectively.In the time domain, the transmission scattering operator is expressed as

͑42͒
where F i , i =2,3 is the Green function 28 of porous layers ͑2͒ and ͑3͒, respectively ͑see Appendix B͒.The reflection scattering operator is expressed by the relation

͑43͒
If only the first reflections at interfaces x =0, x = ᐉ and x = L are taken into account, the reflection scattering kernel expression becomes The transmission scattering kernel describing the direct wave transmitted through two layers of porous materials without internal reflection is expressed by The first term on the right-hand side of Eq. ͑44͒, neous reflected response of the first layer ͑region 2͒.The part of the wave which is equivalent to this term corresponds to the wave reflected by the first interface x =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 d 1 .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. ͑44͒, d 1 ͑d 2 − h 2 ͒F 2 ͓t ,2͑ᐉ / c 2 ͔͒, 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 F 2 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 ⌳Ј of the first layer ͑region 2͒.This result means that it can be possible to get information of all the acoustical properties ͑porosity, tortuosity, vis-cous and thermal characteristics lengths͒ of the first porous layer ͑region 2͒, and also, the porosity and tortuosity of the second porous layer ͑region 3͒, but the viscous and thermal characteristics lengths of the second porous layer ͑region 3͒ do not intervene.
Finally, the term represents the reflection contribution of the third interface, x = L.The corresponding 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 ͓Eqs.͑44͒ and ͑45͔͒ is to show analytically the effect of the acoustical parameters ͑porosity, tortuosity, viscous and thermal characteristic lengths͒ 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 simulations 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 frequency noise ͑energy is totally filtered by the sample in this upper-frequency domain͒.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 ͑a.u.͒ and the point number represented in the abscissa is proportional to time.Signal duration is important as its spectrum must verify the condition 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 independently using classical methods 8 ͑which were developed for a slab of porous material͒.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.Numerical simulation and experimental results ͑transmitted sig-nal͒ 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 ͓Eq.͑45͔͒ 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 signal ͑solid line͒ and simulated signal ͑dashed line͒.This result validates the expression of the transmission scattering operator ͓Eq.͑45͔͒.
Reflected waves were processed by another experimental setup shown in Fig. 8.One transducer was used alternatively 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 ͑dashed line͒ and an experimental reflected signal ͑solid line͒.The simulated signal was obtained through convolution of the reflection scattering kernel given in Eq. ͑44͒ 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 ͑x =0͒ by the first interface, this reflected wave corresponds to the first term on the right-hand side of Eq. ͑44͒:  tally; for example, Fig. 12 shows a comparison between theoretical predictions ͑dashed line͒ and experimental results ͑solid line͒ 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 ͑first layer͒ interface ͑x =0͒ and reflection between the second M1 interface and the first M2 ͑second layer͒ interface ͑x = ᐉ͒, respectively.The reflected wave of the second M2 interface ͑x = L͒ 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 setups.A slight difference was observed between theoretical predictions and experimental data in the two modes ͑reflection and transmission͒.This leads to the conclusion that the expressions of scattering operators obtained are correct.Future studies will concentrate on the inverse problem, and methods and inversion algorithms will be developed to optimize 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.͑25͒ and ͑31͒, we can write the following equation system given the relations between A 2 ͑z͒, B 2 ͑z͒, and R͑z͒:

APPENDIX B: GREEN FUNCTION OF THE MEDIUM
The Green function 28 F i , i =2,3 of the porous layer ͑2͒ and ͑3͒, respectively is given by where i ͑ ,͒, i=2,3 has the following form: and where when k → ϱ, J i and i ͑ , ͒ tends to zero, then the Green function F i ͑t , k͒ also tends to zero. 1

FIG. 3 .
FIG. 2. Experimental setup of the ultrasonic measurements in transmitted mode.
FIG. 4. Spectrum of the incident signal generated by the transducer in transmitted mode.

F
FIG. 11.Comparison between experimental reflected signal ͑solid line͒ and simulated reflected signal ͑dashed line͒ for a double-layered medium consisting of 1.11 cm of M4 and 0.87 cm of M1.

TABLE I .
Acoustical characteristics of the plastic foams samples.Comparison between experimental transmitted signal ͑solid line͒ and simulated transmitted signal ͑dashed line͒ for a double-layered medium consisting of 0.86 cm of M1 and 0.81 cm of M2.