**Transient Acoustic Wave Propagation in Porous Media**

Zine El Abiddine Fellah, Mohamed Fellah and Claude Depollier

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/55048

## **1. Introduction**

126 Modeling and Measurement Methods for Acoustic Waves and for Acoustic Microdevices

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A porous material is a two-phase medium consisting of a solid part (skeleton) and a fluid part (pore space). During the propagation of a sound wave in such a medium, interactions between these two phases of different nature take place, giving various physical properties that are unusual in classical media. The large contact area between solid and fluid, which is the main characteristic of porous media induces new phenomena of diffusion and transport in the fluid, in relation to micro-geometry of the pore space. Many applications are concerned with understanding the behavior of acoustic waves in such media. In geophysics, we are interested in the propagation of acoustic waves in porous rocks, for information on soil composition and their fluid content. Oil companies have greatly contributed to the study of acoustic properties of natural porous media. In medicine, the characterization of porous media such as trabecular bone, is useful for diagnosing osteoporosis, bone disease that is manifested by the deterioration of bone microarchitecture. Acoustic characterization of materials is often achieved by measuring the attenuation coefficient and phase velocity in the frequency domain [1] or by solving the direct and inverse problems directly in the time domain [2–4]. The attractive feature of a time domain based approach is that the analysis is naturally limited by the finite duration of ultrasonic pressures and is consequently the most appropriate approach for the transient signal. The objective of this chapter is to show the most recent theoretical and experimental methods developed by the authors for the acoustic characterization of porous materials. The direct and inverse scattering problems are solved in time domain using experimental reflected and transmitted signals. The physical parameters of the porous medium are recovered by solving the inverse problem at the asymptotic domain corresponding to the high frequency range (ultrasound), and at the viscous domain (low frequency range). Figures 1 and 2 give an example of porous materials commonly used in the characterization. The figure 1 shows a sample of air-saturated plastic foam used for sound absorption, and figure 2 shows a sample of cancellous bone used in the diagnosis of the disease of osteoporosis.

©2013 Fellah et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0),which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ©2013 Fellah et al., licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

2 Will-be-set-by-IN-TECH 128 Modeling and Measurement Methods for Acoustic Waves and for Acoustic Microdevices Transient Acoustic Wave Propagation in Porous media 3

finite set of parameters low and high frequencies may be sufficient to describe the dynamic susceptibilities. For a given frequency, one can define a boundary layer thickness, respectively viscous and thermal *δ* = (2*η*/*ωρ f*)1/2, *δ* = (2*κ*/*ρ f Cpω*)1/2, where *η* is the viscosity, *ρ f* the fluid density, *κ* the coefficient of thermal conduction, *Cp* the specific heat at constant pressure and *ω* the angular frequency. The quantities *δ* and *δ* are respectively the penetration depths of isothermal viscous and rotational movement. The boundary layer is small if the penetration depth is small compared to a "radius" characteristic pore. It is then at high frequencies. The opposite situation corresponds to low frequencies. In the high frequencies, inertial effects are described by the tortuosity *α*∞, the visco-thermal losses by viscous and thermal lengths Λ et

, respectively. At low frequencies, inertial effects are represented by the viscous tortuosity

There are several geometrical parameters that can be associated with a porous structure. Some depend only on the shape of the porous solid and do not vary when it expands or changes the scale of the solid. This is the case of the porosity and tortuosity. Others depend on the dimensions, in the case of viscous and thermal permeabilities that vary in shape as an equal

• **Porosity** *φ*: The porosity of a porous material saturated with a fluid is defined by the ratio of the volume of fluid saturating the porous medium to the total volume of the sample. This quantity is dimensionless and can vary from 0 (non-porous solid material) to 1 (free

• **Tortuosity** *α*∞: The tortuosity is a geometric parameter describing the appearance of tortuous pores, and inertial coupling between the fluid and the structure of the material in high-frequency regime. Tortuosity takes its lowest value (*α*∞ = 1) in the case of porous materials having a straight pores, and large values (*α*∞ = 1.5, 2) for the most resistive

• **Inertial factor** *α*0 **(low frequency tortuosity)**: The inertial factor *α*0 corresponds to the low frequency approximation of the dynamic tortuosity given by Norris [10]; *α*0 = *<***v**(**r**)**2***>*

• **The specific flow resistivity** *σ*: When a porous material placed in a tube sealed to the walls, is traversed by a fluid, it is a pressure difference between the two free faces. This means that a proportional relationship exists between flow rate *Qv* and pressure drop Δ*P* given by the Δ*P* = *RQv*, the coefficient R is the resistance to fluid flow and can therefore be defined as the ratio Δ*P*/*Qv*. For a homogeneous porous material, the resistance is proportional to

where *<* **v**(**r**) *>* is the average velocity of the viscous fluid for direct current flow within an elementary volume, small compared to the relevant wavelength, but large compared to the individual grains/pores of the solid. Note that the tortuosity *α*∞ (high frequency parameter) has a similar definition than *α*0, the difference is that for *α*∞, **v**(**r**) corresponds

0: This low frequency parameter is the thermal counterpart of *α*0. Its

*<θ*0*>*2 , where *θ*0 is the scaled response excess temperature field

0. Porosity *φ* is a

Transient Acoustic Wave Propagation in Porous Media 129

*<***v**(**r**)*>***2** ,

*α*0, and visco-thermal exchanges by the visco-thermal permeabilities *k*0 and *k*

surface, and viscous and thermal characteristic lengths that vary as lengths.

parameter playing an important role both in high in low frequency.

**2.2. Parameters describing the porous materials**

to the velocity of the perfect incompressible fluid.

0 = *<θ*2 0*>*

Λ

fluid).

material.

• **Thermal tortuosity** *α*

definition is given by: *α*

in the low frequency range.

**Figure 1.** Air saturated Plastic foam

**Figure 2.** Human cancellous bone sample

## **2. Porous materials in rigid structure: Equivalent fluid model**

## **2.1. Introduction**

The air saturated porous materials are often used as a sound absorber in the fight against noise pollution. The polyurethane foam, felt or glass wools are three examples of common materials used in aerospace and automotive industries and in building construction. Generally, when a sound wave propagates through a porous material saturated with air, the structure remains stationary and non-deformable with respect to the acoustic excitation, this is due to the heaviness and stiffness of the skeleton of the structure relative in air. This is called porous rigid structure and, in this case, we use the equivalent fluid model where viscous and thermal effects are disconnected [5]. The fluid-structure interactions responsible for the sound attenuation (particularly important in porous media), are described by two dynamic susceptibilities, the dynamic tortuosity and compressibility. The dynamic tortuosity [6] decribes the viscous and inertial effects, the dynamic compressibility [7–9] represents the thermal effects. A prediction of the acoustic behavior of porous material requires the determination of both dynamic susceptibilities. The latter can depend only on the physical characteristics of the fluid and the geometry of the fluid domain. Sound propagation in the saturating fluid involves both transport and diffusion characteristics. The phenomenon of transport is the fact that a global movement fluid/solid is induced during the propagation. The phenomenon of diffusion is related to the expansion/compression of the fluid. They result in a change in temperature fluid/solid (the latter remaining at room temperature) driving a heat diffusion, not negligible if the fluid is a gas. Moreover, in the case of sound propagation, the frequency of movement is important. It is clear that the character of the movement is changed to different frequencies, the fluid probe different aspects of micro geometry. In principle one should expect that there are an infinite number of geometric parameters. But on the other hand, a practical viewpoint, the information provided by a finite set of parameters low and high frequencies may be sufficient to describe the dynamic susceptibilities. For a given frequency, one can define a boundary layer thickness, respectively viscous and thermal *δ* = (2*η*/*ωρ f*)1/2, *δ* = (2*κ*/*ρ f Cpω*)1/2, where *η* is the viscosity, *ρ f* the fluid density, *κ* the coefficient of thermal conduction, *Cp* the specific heat at constant pressure and *ω* the angular frequency. The quantities *δ* and *δ* are respectively the penetration depths of isothermal viscous and rotational movement. The boundary layer is small if the penetration depth is small compared to a "radius" characteristic pore. It is then at high frequencies. The opposite situation corresponds to low frequencies. In the high frequencies, inertial effects are described by the tortuosity *α*∞, the visco-thermal losses by viscous and thermal lengths Λ et Λ , respectively. At low frequencies, inertial effects are represented by the viscous tortuosity *α*0, and visco-thermal exchanges by the visco-thermal permeabilities *k*0 and *k* 0. Porosity *φ* is a parameter playing an important role both in high in low frequency.

## **2.2. Parameters describing the porous materials**

2 Will-be-set-by-IN-TECH

**Figure 1.** Air saturated Plastic foam

**Figure 2.** Human cancellous bone sample

**2.1. Introduction**

**2. Porous materials in rigid structure: Equivalent fluid model**

The air saturated porous materials are often used as a sound absorber in the fight against noise pollution. The polyurethane foam, felt or glass wools are three examples of common materials used in aerospace and automotive industries and in building construction. Generally, when a sound wave propagates through a porous material saturated with air, the structure remains stationary and non-deformable with respect to the acoustic excitation, this is due to the heaviness and stiffness of the skeleton of the structure relative in air. This is called porous rigid structure and, in this case, we use the equivalent fluid model where viscous and thermal effects are disconnected [5]. The fluid-structure interactions responsible for the sound attenuation (particularly important in porous media), are described by two dynamic susceptibilities, the dynamic tortuosity and compressibility. The dynamic tortuosity [6] decribes the viscous and inertial effects, the dynamic compressibility [7–9] represents the thermal effects. A prediction of the acoustic behavior of porous material requires the determination of both dynamic susceptibilities. The latter can depend only on the physical characteristics of the fluid and the geometry of the fluid domain. Sound propagation in the saturating fluid involves both transport and diffusion characteristics. The phenomenon of transport is the fact that a global movement fluid/solid is induced during the propagation. The phenomenon of diffusion is related to the expansion/compression of the fluid. They result in a change in temperature fluid/solid (the latter remaining at room temperature) driving a heat diffusion, not negligible if the fluid is a gas. Moreover, in the case of sound propagation, the frequency of movement is important. It is clear that the character of the movement is changed to different frequencies, the fluid probe different aspects of micro geometry. In principle one should expect that there are an infinite number of geometric parameters. But on the other hand, a practical viewpoint, the information provided by a

There are several geometrical parameters that can be associated with a porous structure. Some depend only on the shape of the porous solid and do not vary when it expands or changes the scale of the solid. This is the case of the porosity and tortuosity. Others depend on the dimensions, in the case of viscous and thermal permeabilities that vary in shape as an equal surface, and viscous and thermal characteristic lengths that vary as lengths.

#### 4 Will-be-set-by-IN-TECH 130 Modeling and Measurement Methods for Acoustic Waves and for Acoustic Microdevices Transient Acoustic Wave Propagation in Porous media 5

the length *d* of the sample and inversely to his section *S*. Then can be defined, the resistivity as the specific resistance to the passage of a fluid. Denoted *σ*, the resistivity is defined by *σ* = (*S*/*d*)*R*.

where *j*

in the frequency domain is given by:

 1 − 1 *jx*

*α*(*ω*) = *α*∞

to the tortuosity *α*∞.

*β*(*ω*) = *γ* − (*γ* − 1)/

the parameter *k*�

permeability.

where *k*�

(5):

2 = −1, *η* is the dynamic viscosity of the fluid, *ω* the angular frequency, *ρ f* the

*ρ f k*0 *jω* , *ω* −→ 0, (2)

*σ* (with the dimension of a surface), *σ*

Transient Acoustic Wave Propagation in Porous Media 131

and *M* = 8*k*0*α*∞

0/*φ*, the quantities Λ� and *k*�

, when *ω* −→ 0. (5)

0*Pr ηφ*

and *M*� = 8*k*�

0 *φ*Λ�2 .

, *ω* −→ ∞, (4)

*φ*Λ2 . (3)

0/*φ* for

density of the fluid. In the low frequency limit, the expression of the dynamic tortuosity

is the specific resistance passage of the fluid. The meaning of the quantities *α*∞, Λ, *k*0 have been given previously. Johnson et al [6] imply that these three geometric parameters provide sufficient information about the microgeometry. The proposed general expression for *α*(*ω*) is

A model similar to the previous one has been set for the frequency dependence of the thermal exchanges between the fluid and the structure, because of the heat capacity of the solid part is

thermal effects playing respectively the role of Λ and *k*0 for viscous effects. In a high-frequency

where *Pr* is the number of Prandt and *γ* the adiabatic constant. As before, this behavior implies that the thermal boundary layer thickness *δ*� becomes small compared to any characteristic radius of curvature of the interface. One can see that there is no equivalent of the tortuosity *α*∞ for thermal effects, at very high frequency *β*(*ω*) tends to 1 as *α*(*ω*) tends

0*Pr*

analogous to the parameter *k*0, but adapted to the thermal problem. The proposed general expression for *β*(*ω*) is then that of the simplest analytic function satisfying the limits ((4) and

0, which has the same size (area) that the Darcy permeability of *k*0, is a parameter

where *x*� = *ωρ f k*�

0 introduced by Lafarge called thermal permeability by analogy to the viscous

*ηφ jω*

1/2 1

*jω*

1/2

where *x* = *ωα*∞*ρ f*

*σφ*

*α*(*ω*) = − *ηφ*

where, *k*0 is the viscous permeability given by: *k*0 = *η*

*2.3.2. The equivalent model for the dynamic compressibility*

*β*(*ω*) = 1 − 2(*γ* − 1)

In a low frequency limit, Lafarge [9] showed that:

 1 − 1 *j x*�

significant. The result is a two-parameter model, Λ� and *k*�

limit, Allard and Champoux [7, 8] showed the following behavior of *β*(*ω*):

 *η Prρ f*

Λ�

*β*(*ω*) = *γ* + (*γ* − 1)*ρ f k*�

1 − *M*� 2 *j x*�

then that of the simplest analytic function verifying these limits:

1 − *M* 2 *j x*

## **2.3. Models of dynamic susceptibilities**

There are several models describing the inertial viscous and thermal exchanges between the fluid and the structure. We present here the most elaborate.

### *2.3.1. The model of Johnson et al. for the dynamic tortuosity*

Johnson et al [6] have proposed a simple model describing the function *α*(*ω*), which is the dynamic tortuosity, when the porous structure is saturated with a Newtonian viscous fluid. This model can be applied to the case of our porous structures saturated with air. Two model parameters, the tortuosity *α*∞ and the viscous characteristic length Λ are involved in a high-frequency limit:

$$a(\omega) = a\_{\infty} \left( 1 - \frac{2}{\Lambda} \left( \frac{\eta}{j\omega \rho\_f} \right)^{1/2} \right), \qquad \omega \longrightarrow \infty. \tag{1}$$

where *j* 2 = −1, *η* is the dynamic viscosity of the fluid, *ω* the angular frequency, *ρ f* the density of the fluid. In the low frequency limit, the expression of the dynamic tortuosity in the frequency domain is given by:

$$\mathfrak{a}(\omega) = -\frac{\eta \phi}{\rho\_f k\_0 j \omega'} , \quad \omega \longrightarrow 0,\tag{2}$$

where, *k*0 is the viscous permeability given by: *k*0 = *η σ* (with the dimension of a surface), *σ* is the specific resistance passage of the fluid. The meaning of the quantities *α*∞, Λ, *k*0 have been given previously. Johnson et al [6] imply that these three geometric parameters provide sufficient information about the microgeometry. The proposed general expression for *α*(*ω*) is then that of the simplest analytic function verifying these limits:

$$a(\omega) = a\_{\infty} \left( 1 - \frac{1}{j\mathbf{x}} \sqrt{1 - \frac{M}{2}j\mathbf{x}} \right) \quad \text{where} \quad \mathbf{x} = \frac{\omega a\_{\infty} \rho\_f}{\sigma \phi} \quad \text{and} \quad M = \frac{8k\_0 a\_{\infty}}{\phi \Lambda^2}. \tag{3}$$

#### *2.3.2. The equivalent model for the dynamic compressibility*

4 Will-be-set-by-IN-TECH

• **Viscous permeability** *k*0: Static viscous permeability is related to the flow resistivity by the relation *k*0 = *η*/*σ*, where *η* is the dynamic viscosity of the fluid. Viscous permeability is a geometrical parameter, homogeneous to a surface (it is expressed in *m*2), independent of the nature of the saturating fluid and depending only on the internal geometry of the porous material. This parameter represents the effective pore section for the flow of fluid, it is connected to the low frequency behavior of the fluid exchange between viscous and

• **Viscous characteristic length** Λ: The work of Johnson et al [6] in 1987 on the characterization of viscous effects at high frequencies has introduced the viscous

> � *S u*2 *dS* �

incompressible perfect fluid, *S* the area of the interface between solid and fluid phases and *V* the volume of fluid. The length Λ is a geometrical parameter. This definition applies to a smooth interface fluid/solid, and for a low boundary layer thickness to the radius of curvature characteristic of the interface. When the pore surface has singularities (peaks), this definition of the characteristic length is no longer valid. The parameter Λ is an indicator of the size of the narrow neck of the pore, i.e. the privileged place of viscous

analogy with Johnson *et al* [6], a geometric parameter called thermal characteristic length

contact surface between the solid and fluid. The factor 2 is introduced so that in the case of cylindrical pores, Λ� is the pore radius. The length Λ� is an indicator of the size of large

similarities between these two effects. Starting from the equation of heat diffusion, Lafarge [9] introduced the equivalent of Darcy's law for the temperature excess, by introducing a

There are several models describing the inertial viscous and thermal exchanges between the

Johnson et al [6] have proposed a simple model describing the function *α*(*ω*), which is the dynamic tortuosity, when the porous structure is saturated with a Newtonian viscous fluid. This model can be applied to the case of our porous structures saturated with air. Two model parameters, the tortuosity *α*∞ and the viscous characteristic length Λ are involved in a

�1/2⎞

� *η jωρ f*

*V dV* . In other words Λ� is twice the ratio of fluid volume on the total

0: The study of viscous and thermal effects shows that there are

*V u*2 *dV* where *u* is the speed of a microscopic

: In 1991, Allard and Champoux [8], introduced by

0, which is the inverse of the constant trapping.

⎠ , *ω* −→ ∞. (1)

Λ =

*σ* = (*S*/*d*)*R*.

exchanges.

given by 2

saturating the porous structure.

• **Thermal characteristic length** Λ�

� *S dS* �

pores, privileged place of heat exchange.

new parameter: the thermal permeability *k*�

fluid and the structure. We present here the most elaborate.

*2.3.1. The model of Johnson et al. for the dynamic tortuosity*

*α*(*ω*) = *α*∞

⎛ ⎝1 − 2 Λ

**2.3. Models of dynamic susceptibilities**

Λ� =

• **Thermal permeability** *k*�

high-frequency limit:

characteristic length Λ given by 2

the length *d* of the sample and inversely to his section *S*. Then can be defined, the resistivity as the specific resistance to the passage of a fluid. Denoted *σ*, the resistivity is defined by

> A model similar to the previous one has been set for the frequency dependence of the thermal exchanges between the fluid and the structure, because of the heat capacity of the solid part is significant. The result is a two-parameter model, Λ� and *k*� 0/*φ*, the quantities Λ� and *k*� 0/*φ* for thermal effects playing respectively the role of Λ and *k*0 for viscous effects. In a high-frequency limit, Allard and Champoux [7, 8] showed the following behavior of *β*(*ω*):

$$\beta(\omega) = 1 - \frac{2(\gamma - 1)}{\Lambda'} \left(\frac{\eta}{P\_t \rho\_f}\right)^{1/2} \left(\frac{1}{j\omega}\right)^{1/2}, \qquad \omega \longrightarrow \infty,\tag{4}$$

where *Pr* is the number of Prandt and *γ* the adiabatic constant. As before, this behavior implies that the thermal boundary layer thickness *δ*� becomes small compared to any characteristic radius of curvature of the interface. One can see that there is no equivalent of the tortuosity *α*∞ for thermal effects, at very high frequency *β*(*ω*) tends to 1 as *α*(*ω*) tends to the tortuosity *α*∞.

In a low frequency limit, Lafarge [9] showed that:

$$\mathfrak{G}(\omega) = \gamma + \frac{(\gamma - 1)\rho\_f k\_0' P\_r}{\eta \phi j \omega}, \qquad \text{when} \qquad \omega \longrightarrow 0. \tag{5}$$

where *k*� 0, which has the same size (area) that the Darcy permeability of *k*0, is a parameter analogous to the parameter *k*0, but adapted to the thermal problem. The proposed general expression for *β*(*ω*) is then that of the simplest analytic function satisfying the limits ((4) and (5):

$$\beta(\omega) = \gamma - (\gamma - 1) / \left[ 1 - \frac{1}{j \cdot \mathbf{x'}} \sqrt{1 - \frac{M'}{2} j \mathbf{x'}} \right] \quad \text{where} \quad \mathbf{x'} = \frac{\omega \rho\_f k\_0' P\_r}{\eta \phi} \quad \text{and} \quad M' = \frac{8 k\_0'}{\phi \Lambda'^2}.$$

the parameter *k*� 0 introduced by Lafarge called thermal permeability by analogy to the viscous permeability.

#### 6 Will-be-set-by-IN-TECH 132 Modeling and Measurement Methods for Acoustic Waves and for Acoustic Microdevices Transient Acoustic Wave Propagation in Porous media 7

#### *2.3.3. Extensions of models by pride and Lafarge*

The development of Johnson et al in high frequency for the tortuosity dynamics stops after the term √ 1 *jω* , Pride *et al* [11] proposed a second correction term which is 1 *jω* . The asymptotic expansion of the dynamic tortuosity now is written as follows:

$$a(\omega) = a\_{\infty} \left( 1 - \frac{2}{\Lambda} \left( \frac{\eta}{j\omega \rho\_f} \right)^{1/2} - \frac{\sigma \phi (1 - \wp)}{j\omega \rho\_f \alpha\_{\infty}} \right), \qquad \omega \longrightarrow \infty. \tag{6}$$

The Pride parameter *℘* [11] is a geometrical parameter, dimensionless, connected to a correction term *α*0 playing the role of inertia of fluid at low frequencies:

$$\wp = \frac{M}{4\left(\frac{a\_0}{a\_\infty} - 1\right)}, \qquad \text{where} \qquad M = \frac{8k\_0 a\_\infty}{\phi \Lambda^2}.$$

The expansion of low frequency dynamic tortuosity is given by:

$$\mathfrak{a}(\omega) = -\frac{\eta\phi}{\rho\_f k\_0 j\omega} + \mathfrak{a}\_{0\prime} \qquad \omega \longrightarrow 0 \tag{7}$$

One can notice that the models of Pride and Lafarge are reduced to those of Johnson and Allard when *℘* = 1 and *℘*� = 1. We will see later the influence of these parameters on the

The use of transient signals in the acoustic characterization of porous media is widely used experimentally. These signals have a wide frequency content making some complicated frequency approaches. The frequency methods are very effective for monochromatic signals, however, for transient signals, the temporal approach [2, 12–15] is best suited for several

• Time domain analysis is naturally bounded (limited) by the finite duration of impulses. • Time-modeling is often easier to develop because it is closer to the reality of experiments. • In many situations, the introduction of the "time" parameter make easier the analysis of

• For certain applications, it is fast because it avoids back and forth operations between time

• It provides an elegant solution of the direct problem necessary for the resolution of the

In return for these benefits, it is necessary to use a new mathematical formalism. In the frequency domain approach, the relevant quantities for the characterization of porous media are functions of frequency which can be interpreted as susceptibilities. While frequency methods lead to the frequency responses of the medium, in the temporal approach, one is

In the asymptotic regime corresponding to high frequencies, the fluid-structure interactions are described by expressions (1) et (4) according to the Johnson-Allard model. Writing the equations in the time domain is equivalent to taking the inverse Fourier transform of (1) and (4). The temporal equivalent of *jω* is *∂*/*∂t*, while the temporal equivalent of *jω* is a fraction0al derivative of order 1/2. The definition of fractional derivative of order *ν* is given

> *t* 0

where Γ is the Eulerian function of the second kind [17]. From the definition (9), the expressions of response factors *α*(*ω*) and *β*(*ω*) are then given in the time domain by [2]:

−→ *α*∞*δ*(*t*) + 1

−→ *δ*(*t*) + *Kaχth*(*t*),

*ρ f*

*χv*(*t*),

(*t* − *u*)−*ν*−1*x*(*u*)*du*, (9)

Transient Acoustic Wave Propagation in Porous Media 133

**2.4. Temporal modeling of propagation: asymptotic regime**

• It is well suited for comparison between theory and experiment.

**2.5. Concept of fractional derivative and propagation equation**

*Dν*[*x*(*t*)] = 1

*α*(*ω*) *t*

*β*(*ω*) *t*

Γ(−*ν*)

propagation.

reasons:

by [16]:

experimental results.

inverse problem.

and frequency domains by FFT.

interested in these impulse responses.

The new expression of the analytic function taking into account developments of (6) and (7) is given by:

$$\mathfrak{a}(\omega) = \mathfrak{a}\_{\infty} \left( 1 - \frac{1}{j\chi} \left( 1 - \wp + \wp \sqrt{1 - \frac{M}{2\wp^2} j\,\mathrm{x}} \right) \right) \dots$$

The value of *℘* depends on the geometry of the pores, for example in the case of cylindrical pores of circular section, it is shown that *℘* = 3/4. For rectangular sections, one obtains values close to 3/4.

For thermal effects, Lafarge [9] suggested a similar expression for the dynamic thermal compressibility, also introducing a dimensionless parameter *℘*� and gives a correction to the Allard and Champoux model at high frequencies. The expression of the dynamic compressibility is then given by:

$$\beta(\omega) = 1 - (\gamma - 1) \left[ \frac{2}{\Lambda'} \left( \frac{\eta}{Pr\rho\_f} \right)^{1/2} \left( \frac{1}{j\omega} \right)^{1/2} + \left( \frac{\eta}{\rho\_f Pr j\omega} \right) \cdot \left( \frac{(1 - \wp')\phi}{k\_0'} - \frac{4}{\Lambda'^2} \right) \right], \omega \longrightarrow \infty. \tag{8}$$

The parameter *℘*� is connected to *α*� 0, thermal equivalent of *α*0, which describes the thermal inertia of fluid at low frequency:

$$
\wp' = \frac{M'}{4(\alpha'\_0 - 1)}.
$$

The general expression for the dynamic compressibility is then given by:

$$\beta(\omega) = \gamma - (\gamma - 1) / \left[ 1 - \frac{1}{j \cdot \mathbf{x'}} \left( 1 - \wp' + \wp' \sqrt{1 - \frac{M'}{2 \wp'^2} j \mathbf{x'}} \right) \right].$$

One can notice that the models of Pride and Lafarge are reduced to those of Johnson and Allard when *℘* = 1 and *℘*� = 1. We will see later the influence of these parameters on the propagation.

### **2.4. Temporal modeling of propagation: asymptotic regime**

6 Will-be-set-by-IN-TECH

The development of Johnson et al in high frequency for the tortuosity dynamics stops after

The Pride parameter *℘* [11] is a geometrical parameter, dimensionless, connected to a

The new expression of the analytic function taking into account developments of (6) and (7) is

1 − *℘* + *℘*

The value of *℘* depends on the geometry of the pores, for example in the case of cylindrical pores of circular section, it is shown that *℘* = 3/4. For rectangular sections, one obtains values

For thermal effects, Lafarge [9] suggested a similar expression for the dynamic thermal compressibility, also introducing a dimensionless parameter *℘*� and gives a correction to the Allard and Champoux model at high frequencies. The expression of the dynamic

> *η ρ f Prjω*

� .

� (1 − *℘*�

*k*� 0

0, thermal equivalent of *α*0, which describes the thermal

1 − *M*� 2*℘*�2 *j x*�

)*φ*

− 4 Λ�2

> �� .

�⎤

⎦ , *ω* −→ ∞.

(8)

�

1 − *M* 2*℘*2 *j x*

− *σφ*(1 − *℘*) *jωρ f α*∞

� , where *M* = 8*k*0*α*∞

⎞

*φ*Λ2 .

+ *α*0, *ω* −→ 0 (7)

�� . *jω* . The asymptotic

⎠ , *ω* −→ ∞. (6)

*jω* , Pride *et al* [11] proposed a second correction term which is 1

�1/2

*2.3.3. Extensions of models by pride and Lafarge*

*α*(*ω*) = *α*∞

expansion of the dynamic tortuosity now is written as follows:

� *η jωρ f*

correction term *α*0 playing the role of inertia of fluid at low frequencies:

*α*(*ω*) = − *ηφ*

� 1 − 1 *jx*

�1/2 � 1

*jω*

The general expression for the dynamic compressibility is then given by:

� 1 − 1 *j x*�

�1/2 + �

*℘*� = *M*� 4(*α*� 0 − 1) .

�

1 − *℘*� + *℘*�

�

*ρ f k*0 *jω*

�

⎛ ⎝1 − 2 Λ

*℘* = *M* 4 � *α*0 *α*∞ − 1

*α*(*ω*) = *α*∞

The expansion of low frequency dynamic tortuosity is given by:

the term √

given by:

close to 3/4.

*β*(*ω*)=1−(*γ*−1)

compressibility is then given by:

⎡ ⎣ 2 Λ�

The parameter *℘*� is connected to *α*�

inertia of fluid at low frequency:

� *η Prρ f*

*β*(*ω*) = *γ* − (*γ* − 1)/

1

The use of transient signals in the acoustic characterization of porous media is widely used experimentally. These signals have a wide frequency content making some complicated frequency approaches. The frequency methods are very effective for monochromatic signals, however, for transient signals, the temporal approach [2, 12–15] is best suited for several reasons:

In return for these benefits, it is necessary to use a new mathematical formalism. In the frequency domain approach, the relevant quantities for the characterization of porous media are functions of frequency which can be interpreted as susceptibilities. While frequency methods lead to the frequency responses of the medium, in the temporal approach, one is interested in these impulse responses.

#### **2.5. Concept of fractional derivative and propagation equation**

In the asymptotic regime corresponding to high frequencies, the fluid-structure interactions are described by expressions (1) et (4) according to the Johnson-Allard model. Writing the equations in the time domain is equivalent to taking the inverse Fourier transform of (1) and (4). The temporal equivalent of *jω* is *∂*/*∂t*, while the temporal equivalent of *jω* is a fraction0al derivative of order 1/2. The definition of fractional derivative of order *ν* is given by [16]:

$$[D^{\nu}]\mathbf{x}(t)] = \frac{1}{\Gamma(-\nu)} \int\_{0}^{t} (t-u)^{-\nu-1} \mathbf{x}(u) du,\tag{9}$$

where Γ is the Eulerian function of the second kind [17]. From the definition (9), the expressions of response factors *α*(*ω*) and *β*(*ω*) are then given in the time domain by [2]:

> *α*(*ω*) *t* −→ *α*∞*δ*(*t*) + 1 *ρ f χv*(*t*), *β*(*ω*) *t* −→ *δ*(*t*) + *Kaχth*(*t*),

#### 8 Will-be-set-by-IN-TECH 134 Modeling and Measurement Methods for Acoustic Waves and for Acoustic Microdevices Transient Acoustic Wave Propagation in Porous media 9

where *δ*(*t*) is the Dirac distribution and the operators, *Ka* the modulus of the fluid, *χv*(*t*) and *χth*(*t*) are given by:

$$\chi\_{\upsilon}(t) = \frac{2\rho\_f \alpha\_{\infty}}{\Lambda} \sqrt{\frac{\eta}{\pi \rho\_f}} \, t^{-1/2} \Big/$$

$$\chi\_{th}(t) = \frac{2(\gamma - 1)}{K\_d \Lambda'} \sqrt{\frac{\eta}{\pi P\_r \rho\_f}} \, t^{-1/2} \Big/$$

In this model, the time convolution of *t* −1/2 with a function, is interpreted as an operator of fractional derivative. The acoustic fields satisfy the Euler equation and the constitutive equation [2] given by:

$$
\rho\_f \alpha\_{\infty} \,\partial\_t \mathbf{v}(\mathbf{r}, t) + \int\_0^t \chi\_{\mathcal{V}}(t - t') \,\partial\_t \mathbf{v}(\mathbf{r}, t') \,dt' = -\nabla p(\mathbf{r}, t), \tag{10}
$$

respectively, The coefficient *A* gives the wave velocity *c* = 1/

*p*(*x*, *t*) = *t*

0

*G*(*x*, *t* − *t*

To solve the propagation equation and obtain the Green function of the medium, we solve the equation (13) using the method of Laplace transform [18], taking into account the following

> ∞ 0

*p*(*x*, *t*) = L−1 [*P*(*x*, *z*)] .

*z*

where *H*(*t*) is the Heaviside jump function, the Laplace transform of the wave equation (13)

 1 *c*2 +*B*

*P*(*x*, *z*)=−

where *p*(*x*, *t*) is the sound pressure in the porous medium and *p*(0, *t*) the incident signal.

**2.6. Solution of the propagation equation: Green function (Johnson-Allard**

*p*(*x*, *t*)|*t*=0 = 0 and *∂p*

Denote by *P*(*x*, *z*) the Laplace transform of *p*(*x*, *t*) defined by:

L [*δ*(*t*)] = 1, L [*H*(*t*)] = 1

*P*(*x*, *z*) = L [*p*(*x*, *t*)] =

�

*∂t*

exp(−*zt*)*p*(*x*, *t*)*dt*,

and L

*π z*

 1 √*t* = *π z* ,

*zp*(*x*, 0)+ *∂p*(*x*, 0)

*∂t*

−*Cp*(*x*, 0).

(15)

)*p*(0, *t*)*dt*�

,

changes the speed *c*0 =

**model)**

is:

*∂*2*P*(*x*, *z*) *∂x*2 −*z*2

the porous medium [3, 18], defined by:

initial conditions of causality:

and the inverse Laplace transform by:

Using the following relationship:

 1 *c*2 + *C z* +*B π z*

the porous medium. *α*∞ plays the role of the refractive index of the porous medium that

*Ka*/*ρ f* in free space to *c*0 = *c*/

The other coefficients depend essentially on characteristic lengths Λ and Λ� and express the fluid-structure visco-thermal interactions. The constant *B* describes the dispersion of the signal and *C* the attenuation of the amplitude of the wave (without dispersion). The propagation equation (13) describes the evolution of the acoustic wave inside the porous material. The boundary conditions have not been introduced, the porous medium is assumed to be infinite. The coefficients of the equation of propagation do not depend on the porosity, in fact, porosity appears at the interfaces of the porous material via the relationship of continuity of acoustic flow (the flow rate is equal to the porosity multiplied by the speed). This parameter appears naturally at the reflection and transmission of an acoustic wave through a slice of porous material. The solution of the wave equation (13) is given by the Green function *G* of

*ρ f α*∞/*Ka* in the air saturating

Transient Acoustic Wave Propagation in Porous Media 135

√*α*∞ in the porous medium.

$$\frac{1}{K\_a}\partial\_t p(\mathbf{r}, t) + \int\_0^t \chi\_{th}(t - t')\,\partial\_t p(\mathbf{r}, t')\,dt' = -\nabla.\mathbf{v}(\mathbf{r}, t). \tag{11}$$

These constitutive relations in the time domain satisfy the principle of causality. In these equations, *p* is the acoustic pressure, **v** the particle velocity. The term *α*∞*δ*(*t*) reflects the instantaneous response of the porous medium and describes the inertial coupling between fluid and structure. For instantaneous responses, we mean that the time response is smaller than the characteristic time scale of the change in the acoustic field. The susceptibilities *χv* and *χth* are operators of memory that determine the dispersion of the medium.

We assume that the medium varies with the thickness *x* only, and that the incident wave is plane and normal to the surface of the material. Sound pressure is denoted by *p*(*x*, *t*). We assume that the pressure field is zero for moments earlier to 0. The wave equation for acoustic pressure field of a porous medium having a dispersive rigid structure is obtained from the constitutive equations (10, 11), and is of the form:

$$\left[\partial\_{\mathbf{x}}^{2}p(\mathbf{x},t) - \frac{1}{c\_{0}^{2}}\left[a\_{\infty}\partial\_{t}^{2}p(\mathbf{x},t) + \left(a\_{\infty}K\_{a}\chi\_{th} + \frac{\chi\_{v}}{\rho\_{f}} + c\_{0}^{2}\chi\_{th}\*\chi\_{v}\right)\*\partial\_{t}^{2}p(\mathbf{x},t)\right] = 0,\tag{12}$$

where *c*0 = (*Ka*/*ρ f*)1/2 is the sound speed in the free fluid. The following notation is used for convolution integrals of two causal functions *f*(*x*,*t*) and *g*(*x*, *t*);

$$[f \ast g](\mathbf{x}, t) = \int\_0^t f(\mathbf{x}, t - t') g(\mathbf{x}, t') dt'.$$

The propagation equation (12) can be written as:

$$\frac{\partial^2 p(\mathbf{x}, t)}{\partial \mathbf{x}^2} - A \frac{\partial^2 p(\mathbf{x}, t)}{\partial t^2} - B \int\_0^t \frac{\partial^2 p(\mathbf{x}, t) / \partial t'^2}{\sqrt{t - t'}} dt' - C \frac{\partial p(\mathbf{x}, t)}{\partial t} = 0,\tag{13}$$

where the coefficients *A*, *B* and *C* are constant and given by:

$$A = \frac{\rho\_f a\_{\infty}}{K\_a}, \quad B = \frac{2a\_{\infty}}{K\_a} \sqrt{\frac{\rho\_f \eta}{\pi}} \left(\frac{1}{\Lambda} + \frac{\gamma - 1}{\sqrt{Pr \Lambda'}}\right) \quad \mathbb{C} = \frac{4a\_{\infty}(\gamma - 1)\eta}{K\_a \Lambda \Lambda' \sqrt{Pr}}, \quad \mathbb{C} = \frac{2a\_{\infty}(\gamma - 1)\eta}{\sqrt{Pr \Lambda'}}$$

respectively, The coefficient *A* gives the wave velocity *c* = 1/ *ρ f α*∞/*Ka* in the air saturating the porous medium. *α*∞ plays the role of the refractive index of the porous medium that changes the speed *c*0 = *Ka*/*ρ f* in free space to *c*0 = *c*/ √*α*∞ in the porous medium. The other coefficients depend essentially on characteristic lengths Λ and Λ� and express the fluid-structure visco-thermal interactions. The constant *B* describes the dispersion of the signal and *C* the attenuation of the amplitude of the wave (without dispersion). The propagation equation (13) describes the evolution of the acoustic wave inside the porous material. The boundary conditions have not been introduced, the porous medium is assumed to be infinite. The coefficients of the equation of propagation do not depend on the porosity, in fact, porosity appears at the interfaces of the porous material via the relationship of continuity of acoustic flow (the flow rate is equal to the porosity multiplied by the speed). This parameter appears naturally at the reflection and transmission of an acoustic wave through a slice of porous material. The solution of the wave equation (13) is given by the Green function *G* of the porous medium [3, 18], defined by:

$$p(\mathbf{x},t) = \int\_0^t \mathbf{G}(\mathbf{x}, t - t') p(\mathbf{0}, t) dt',$$

where *p*(*x*, *t*) is the sound pressure in the porous medium and *p*(0, *t*) the incident signal.

### **2.6. Solution of the propagation equation: Green function (Johnson-Allard model)**

To solve the propagation equation and obtain the Green function of the medium, we solve the equation (13) using the method of Laplace transform [18], taking into account the following initial conditions of causality:

$$p(\mathbf{x},t)|\_{t=0} = 0 \quad \text{and} \quad \frac{\partial p}{\partial t}|\_{t=0} = 0. \tag{14}$$

Denote by *P*(*x*, *z*) the Laplace transform of *p*(*x*, *t*) defined by:

$$P(\mathbf{x}, z) = \mathcal{L}\left[p(\mathbf{x}, t)\right] = \int\_0^\infty \exp(-zt)p(\mathbf{x}, t)dt,$$

and the inverse Laplace transform by:

$$p(\mathbf{x}, t) = \mathcal{L}^{-1} \left[ P(\mathbf{x}, z) \right].$$

Using the following relationship:

8 Will-be-set-by-IN-TECH

where *δ*(*t*) is the Dirac distribution and the operators, *Ka* the modulus of the fluid, *χv*(*t*) and

 *η πρ f t* −1/2,

> *η πPrρ f t* −1/2,

−1/2 with a function, is interpreted as an operator

) *dt*� = −∇*p*(**r**, *t*), (10)

) *dt*� = −∇.**v**(**r**, *t*). (11)

2*ρ f α*∞ Λ

*Ka*Λ�

*χv*(*t* − *t* � ) *∂t***v**(**r**, *t* �

*χth*(*t* − *t* � ) *∂t p*(**r**, *t* �

and *χth* are operators of memory that determine the dispersion of the medium.

of fractional derivative. The acoustic fields satisfy the Euler equation and the constitutive

These constitutive relations in the time domain satisfy the principle of causality. In these equations, *p* is the acoustic pressure, **v** the particle velocity. The term *α*∞*δ*(*t*) reflects the instantaneous response of the porous medium and describes the inertial coupling between fluid and structure. For instantaneous responses, we mean that the time response is smaller than the characteristic time scale of the change in the acoustic field. The susceptibilities *χv*

We assume that the medium varies with the thickness *x* only, and that the incident wave is plane and normal to the surface of the material. Sound pressure is denoted by *p*(*x*, *t*). We assume that the pressure field is zero for moments earlier to 0. The wave equation for acoustic pressure field of a porous medium having a dispersive rigid structure is obtained from the

*α*∞*Kaχth* + *χv*

where *c*0 = (*Ka*/*ρ f*)1/2 is the sound speed in the free fluid. The following notation is used for

*f*(*x*, *t* − *t* � )*g*(*x*, *t* � )*dt*� .

*∂*2 *p*(*x*, *t*)/*∂t*

Λ + *γ* − 1 √*Pr*Λ�

�2 √*t* − *t*� *dt*� − *C∂p*(*x*, *t*)

*C* = 4*α*∞(*γ* − 1)*η Ka*ΛΛ�

 *t* 0

> *t* 0

> > 1

*ρ f* + *c*2

0*χth* ∗ *χv*

 ∗ *∂*2 *t p*(*x*, *t*)

*∂t* = 0, (13)

√*Pr* ,

= 0, (12)

*χth*(*t*) = 2(*γ* − 1)

 *t* 0

 *t* 0

*χv*(*t*) =

*χth*(*t*) are given by:

equation [2] given by:

*∂*2

*x p*(*x*, *t*) − 1

*c*2 0

*∂*2 *p*(*x*, *t*)

*A* = *ρ f α*∞ *Ka*

 *α*∞*∂*2

In this model, the time convolution of *t*

1 *Ka*

constitutive equations (10, 11), and is of the form:

The propagation equation (12) can be written as:

*∂x*2 − *A ∂*2 *p*(*x*, *t*)

where the coefficients *A*, *B* and *C* are constant and given by:

, *B* = 2*α*∞

*Ka*

*t p*(*x*, *t*) +

convolution integrals of two causal functions *f*(*x*,*t*) and *g*(*x*, *t*);

[ *f* ∗ *g*](*x*, *t*) =

*∂t*2 − *B*

*ρ f η π*

*ρ f α*∞ *∂t***v**(**r**, *t*) +

*∂t p*(**r**, *t*) +

$$
\mathcal{L}\left[\delta(t)\right] = 1, \qquad \qquad \mathcal{L}\left[H(t)\right] = \frac{1}{z} \qquad \text{and} \qquad \mathcal{L}\left[\frac{1}{\sqrt{t}}\right] = \sqrt{\frac{\pi}{z}}.
$$

where *H*(*t*) is the Heaviside jump function, the Laplace transform of the wave equation (13) is:

$$\frac{\partial^2 P(\mathbf{x}, z)}{\partial \mathbf{x}^2} - z^2 \left( \frac{1}{c^2} + \frac{\mathbb{C}}{z} + B\sqrt{\frac{\pi}{z}} \right) P(\mathbf{x}, z) = -\left( \frac{1}{c^2} + B\sqrt{\frac{\pi}{z}} \right) \left( z p(\mathbf{x}, 0) + \frac{\partial p(\mathbf{x}, 0)}{\partial t} \right) - \mathbb{C} p(\mathbf{x}, 0). \tag{15}$$

#### 10 Will-be-set-by-IN-TECH 136 Modeling and Measurement Methods for Acoustic Waves and for Acoustic Microdevices Transient Acoustic Wave Propagation in Porous media 11

Taking into account the initial conditions (14), equation (15) simplifies to

$$\frac{\partial^2 P(\mathbf{x}, z)}{\partial \mathbf{x}^2} - z^2 \left( \frac{1}{c^2} + \frac{\mathbf{C}}{z} + B\sqrt{\frac{\pi}{z}} \right) P(\mathbf{x}, z) = 0\_\mathbf{v}$$

which is a second order differential equation with real coefficients constant. The general solution of equation is:

$$P(x,z) = e^{-\frac{x}{\varepsilon}}\sqrt{f(z)}\,\varphi(z) + e^{\frac{x}{\varepsilon}}\sqrt{f(z)}\,\psi(z),$$

where *ϕ*(*z*) and *ψ*(*z*) are two independent functions of *x* and:

$$f(z) = z^2 \left( 1 + \frac{\mathbb{C}c^2}{z} + \mathrm{B}c^2 \sqrt{\frac{\pi}{z}} \right) = z(z + b' \sqrt{z} + c'). \tag{16}$$

In equation (16), the constants *b*� and *c*� are positive and are given by:

$$b' = \mathcal{B}c^2 \sqrt{\pi} \quad \text{and} \quad c' = \mathcal{C}c^2. \tag{17}$$

When Δ2 *>* 0, the general solution of the propagation equation is given by:

− *b*�2*x*2 16*c*2(*τ*−*x*/*c*)

1 *ξ*3/2 � 1 −1 exp

⎛

� *μ*Δ

− *b*�2*x*2 16*c*2(*t*−*x*/*c*)

�1/2 *∂*−1/2

*∂*−1*x*(*t*) *∂t*−1 =

> 0 *χ*� *v*(*t* − *t* � ) *∂t***v**(**r**, *t* �

0 *χ*� *th*(*t* − *t* � ) *∂t p*(**r**, *t* �

⎜⎝

×

1 (*τ*−*x*/*c*)3/2 exp �

1 �(*τ* − *ξ*)2 − *x*2/*c*2

In this case, the Green function [18] is given by:

1 (*t*−*x*/*c*)3/2 exp �

0, if 0 ≤ *t* ≤ *x*/*c*,

�

⎛

⎜⎝−

0, if 0 ≤ *t* ≤ *x*/*c*,

�

**2.7. Solution of the propagation equation: Green function (Pride-Lafarge model)** Extensions of Pride-Lafarge (6,8) are expressed in the time domain [3] using the definition of

*∂t*−1/2 + *σφ*(1 − *℘*)

*∂t*−1/2 +

*α*˜(*t*) and *β*˜(*t*) are the operators of tortuosity and compressibility. In these equations, the

� *t* 0 *x*(*t* � )*dt*� ,

�1/2 *∂*−1/2

*ρ f α*∞

� *η ρ f Pr*

+Δ � *τ*−*x*/*c* 0 *h*�

> � *μ*Δ

�(*τ* − *ξ*)2 − *x*2/*c*2 + *b*�

+ Δ � *t*−*x*/*c* 0 *h*�

> *∂*−1 *∂t*−1

⎞ ⎠ ,

� �

−1/2 represents the fractional derivative. In this case, the basic

(1 − *℘*�

*k*� 0 )*φ*

− 4 Λ�2

) *dt*� = −∇*p*(**r**, *t*), (24)

) *dt*� = −∇.**v**(**r**, *t*), (25)

� *∂*−1 *∂t*−1 ⎞ ⎠ ,

(*ξ*)*dξ* �

�(*τ* − *ξ*)2 − *x*2/*c*2 + *b*�

16*ξ*

Transient Acoustic Wave Propagation in Porous Media 137

(*τ* − *ξ*)

8*ξ* − 1

(*ξ*)*dξ* � �2

*p*(0, *t*−*τ*)*dτ*, *t > x*/*c*.

(*τ* − *ξ*)

⎞

⎟⎠

, if *t > x*/*c*. (23)

(22)

�2

*μdμ* �1 − *μ*2

⎞

⎟⎠

.

*p*(*x*, *t*)=

where

*h*�

⎧ ⎨ ⎩

(*ξ*) = − 1

*G*(*x*, *t*) =

*x c* � *t x*/*c* � *b*� 4 √*π*

4*π*3/2

⎧ ⎨ ⎩

fractional derivative:

*α*˜(*t*) = *α*∞

operator *∂*−1/*∂t*

*x c* � *b*� 4 √*π*

⎛

*β*˜(*t*) = *δ*(*t*)+(*γ* − 1)

and semi-operator *∂*−1/2/*∂t*

equations are given by:

⎝*δ*(*t*) + 2

Λ

� *η ρ f*

⎛ ⎝ 2 Λ�

� *η Prρ f*

−1 represents the time integral:

*ρ f α*∞ *∂t***v**(**r**, *t*) + � *t*

*∂t p*(**r**, *t*) + � *t*

1 *Ka*

Retaining the finite solution at infinity, which corresponds to the physical solution of our problem, we have:

*P*(*x*, *z*) = *e* − *x c* √*f*(*z*) *ϕ*(*z*).

The solution of equation (13) is the inverse Laplace transform of *P*(*x*, *z*). We obtain:

$$p(\mathbf{x},t) = \mathcal{L}^{-1}\left(e^{-\frac{\mathbf{z}}{\varepsilon}\sqrt{f(z)}}\rho(z)\right) = \mathcal{L}^{-1}\left(e^{-\frac{\mathbf{z}}{\varepsilon}\sqrt{f(z)}}\right) \* \mathcal{L}^{-1}\left(\rho(z)\right).$$

The core problem is the calculation of the inverse Laplace transform of the term L−1 *e*− *x c* √*f*(*z*) . This has been well studied in reference [18]. By letting:

$$
\Delta^2 = b'^2 - 4c' \,\text{.}\tag{18}
$$

it is easy to check that Δ2 is always positive in the Johnson-Allard model. The expression of *f*(*z*) (Eq. 16) can be written as

$$f(z) = \left(z + \frac{b'}{2}\sqrt{z}\right)^2 - \left(\frac{\Delta\sqrt{z}}{2}\right)^2. \tag{19}$$

When Δ = 0, the solution of the propagation equation(13) is given [18] by:

$$p(\mathbf{x},t) = \begin{cases} 0, & \text{if } \quad 0 \le t \le \mathbf{x}/c, \\ \frac{1}{4\sqrt{\pi}} \frac{b^t \mathbf{x}}{c} \int\_{\mathbf{x}/c}^t \frac{1}{(\tau - \mathbf{x}/c)^{3/2}} \exp\left(-\frac{b^{t2}\mathbf{x}^2}{16c^2(\tau - \mathbf{x}/c)}\right) p(0, t-\tau) d\tau, & \text{if } \quad t > \mathbf{x}/c, \end{cases} \tag{20}$$

where *p*(0, *t*) = L−1 (*ϕ*(*z*)). In this case the Green function is given [18, 19] by:

$$G(\mathbf{x},t) = \begin{cases} 0, & \text{if } \quad 0 \le t \le \mathbf{x}/c, \\ \frac{1}{4\sqrt{\pi}} \frac{b^\circ \mathbf{x}}{c} \frac{1}{(t-\mathbf{x}/c)^{3/2}} \exp\left(-\frac{b^2 \mathbf{x}^2}{16c^2(t-\mathbf{x}/c)}\right), & \text{if } \qquad t > \mathbf{x}/c, \end{cases} \tag{21}$$

When Δ2 *>* 0, the general solution of the propagation equation is given by:

$$p(\mathbf{x},t) = \begin{cases} 0, & \text{if} \quad 0 \le t \le \mathbf{x}/c, \\ \frac{\mathbf{x}}{c} \int\_{\mathbf{x}/c}^{t} \left( \frac{b'}{4\sqrt{\pi}} \frac{1}{(\tau - \mathbf{x}/c)^{3/2}} \exp\left(-\frac{b'^2 \mathbf{x}^2}{16c^2(\tau - \mathbf{x}/c)}\right) + \Delta \int\_{0}^{\tau - \mathbf{x}/c} h'(\xi) d\xi \right) p(0, t - \tau) d\tau, t > \mathbf{x}/c. \end{cases} (22)$$

where

10 Will-be-set-by-IN-TECH

which is a second order differential equation with real coefficients constant. The general

√*f*(*z*)*ϕ*(*z*) + *e*

*π z*

Retaining the finite solution at infinity, which corresponds to the physical solution of our

− *x c* √*f*(*z*)

The core problem is the calculation of the inverse Laplace transform of the term

= L−1 *e* − *x c* √*f*(*z*)

*x c* √*f*(*z*)

= *z*(*z* + *b*�

*ϕ*(*z*).

*P*(*x*, *z*) = 0,

*ψ*(*z*),

*b*� = *Bc*2√*π* and *c*� = *Cc*2. (17)

2

√*z* + *c*�

∗ L−1 (*ϕ*(*z*)).

, (18)

. (19)

*p*(0, *t* − *τ*)*dτ*, if *t > x*/*c*, (20)

, if *t > x*/*c*, (21)

). (16)

Taking into account the initial conditions (14), equation (15) simplifies to

*P*(*x*, *z*) = *e*− *x*

In equation (16), the constants *b*� and *c*� are positive and are given by:

where *ϕ*(*z*) and *ψ*(*z*) are two independent functions of *x* and:

 1 + *Cc*2 *z*

*f*(*z*) = *z*2

*p*(*x*, *t*) = L−1

*f*(*z*) (Eq. 16) can be written as

1 4 √*π b*� *x c t x*/*c* 1

*G*(*x*,*t*) =

 *e* − *x c*

*f*(*z*) =

(*τ*−*x*/*c*)3/2 exp

0, if 0 ≤ *t* ≤ *x*/*c*,

0, if 0 ≤ *t* ≤ *x*/*c*,

1 4 √*π b*� *x c* 1

 *z* + *b*� 2 √*z* 2 − Δ √*z* 2

When Δ = 0, the solution of the propagation equation(13) is given [18] by:

where *p*(0, *t*) = L−1 (*ϕ*(*z*)). In this case the Green function is given [18, 19] by:

(*t*−*x*/*c*)3/2 exp

 1 *c*2 + *C z* + *B π z*

*c*

*P*(*x*, *z*) = *e*

√*f*(*z*)*ϕ*(*z*)

The solution of equation (13) is the inverse Laplace transform of *P*(*x*, *z*). We obtain:

. This has been well studied in reference [18]. By letting:

Δ2 = *b*�2 − 4*c*�

it is easy to check that Δ2 is always positive in the Johnson-Allard model. The expression of

− *b*�2*x*2 16*c*2(*τ*−*x*/*c*)

− *b*�2*x*2 16*c*2(*t*−*x*/*c*)

+ *Bc*2

*∂*2*P*(*x*, *z*) *∂x*2 − *z*2

solution of equation is:

problem, we have:

L−1 *e*− *x c* √*f*(*z*)

*p*(*x*, *t*) =

$$h'(\xi) = -\frac{1}{4\pi^{3/2}} \frac{1}{\sqrt{(\tau-\xi)^2 - \mathbf{x}^2/c^2}} \frac{1}{\xi^{3/2}} \int\_{-1}^1 \exp\left(-\frac{\left(\mu\Delta\sqrt{(\tau-\xi)^2 - \mathbf{x}^2/c^2} + b'(\tau-\xi)\right)^2}{16\xi}\right) d\xi$$

$$\times \left(\frac{\left(\mu\Delta\sqrt{(\tau-\xi)^2 - \mathbf{x}^2/c^2} + b'(\tau-\xi)\right)^2}{8\xi} - 1\right) \frac{\mu d\mu}{\sqrt{1-\mu^2}}.$$

In this case, the Green function [18] is given by:

$$G(\mathbf{x},t) = \begin{cases} 0, & \text{if } \quad 0 \le t \le \mathbf{x}/\mathbf{c} \\ \frac{\mathbf{x}}{\mathbf{c}} \left( \frac{\mathbf{b}^{\prime}}{4\sqrt{\pi}} \frac{1}{(t-\mathbf{x}/\mathbf{c})^{3/2}} \exp\left(-\frac{\mathbf{b}^{2}\mathbf{x}^{2}}{16\mathbf{c}^{2}(t-\mathbf{x}/\mathbf{c})}\right) + \Delta \int\_{0}^{t-\mathbf{x}/\mathbf{c}} h^{\prime}(\boldsymbol{\xi}) d\boldsymbol{\xi} \right), & \text{if } \quad t > \mathbf{x}/\mathbf{c}. \end{cases} \tag{23}$$

#### **2.7. Solution of the propagation equation: Green function (Pride-Lafarge model)**

Extensions of Pride-Lafarge (6,8) are expressed in the time domain [3] using the definition of fractional derivative:

$$\begin{split} \tilde{\alpha}(t) &= a\_{\infty} \left( \delta(t) + \frac{2}{\Lambda} \left( \frac{\eta}{\rho\_f} \right)^{1/2} \frac{\partial^{-1/2}}{\partial t^{-1/2}} + \frac{\sigma \phi (1-\wp)}{\rho\_f \mathfrak{a}\_{\infty}} \frac{\partial^{-1}}{\partial t^{-1}} \right), \\ \tilde{\beta}(t) &= \delta(t) + (\gamma - 1) \left( \frac{2}{\Lambda'} \left( \frac{\eta}{P\_r \rho\_f} \right)^{1/2} \frac{\partial^{-1/2}}{\partial t^{-1/2}} + \left( \frac{\eta}{\rho\_f P\_r} \right) \left[ \frac{(1-\wp')\mathfrak{\phi}}{k\_0'} - \frac{4}{\Lambda'^2} \right] \frac{\partial^{-1}}{\partial t^{-1}} \right), \end{split}$$

*α*˜(*t*) and *β*˜(*t*) are the operators of tortuosity and compressibility. In these equations, the operator *∂*−1/*∂t* −1 represents the time integral:

$$\frac{\partial^{-1}\mathfrak{x}(t)}{\partial t^{-1}} = \int\_0^t \mathfrak{x}(t')dt',$$

and semi-operator *∂*−1/2/*∂t* −1/2 represents the fractional derivative. In this case, the basic equations are given by:

$$
\rho\_f \mathbf{a}\_{\infty} \,\partial\_t \mathbf{v}(\mathbf{r}, t) + \int\_0^t \chi\_v'(t - t') \,\partial\_t \mathbf{v}(\mathbf{r}, t') \,dt' = -\nabla p(\mathbf{r}, t), \tag{24}
$$

$$\frac{1}{K\_a} \partial\_t p(\mathbf{r}, t) + \int\_0^t \chi\_{th}'(t - t') \,\partial\_t p(\mathbf{r}, t') \, dt' = -\nabla \cdot \mathbf{v}(\mathbf{r}, t), \tag{25}$$

#### 12 Will-be-set-by-IN-TECH 138 Modeling and Measurement Methods for Acoustic Waves and for Acoustic Microdevices Transient Acoustic Wave Propagation in Porous media 13

where the operators *χ*� *v*(*t*) and *χ*� *th*(*t*) are given by:

$$\begin{split} \chi'\_{\upsilon}(t) &= \frac{2\rho\_f \mathfrak{a}\_{\infty}}{\Lambda} \sqrt{\frac{\eta}{\pi \rho\_f}} t^{-1/2} + \sigma \phi (1 - \wp) \frac{\mathfrak{d}^{-1}}{\partial t^{-1}}, \\\chi'\_{th}(t) &= \frac{2(\gamma - 1)}{\mathcal{K}\_a \Lambda'} \sqrt{\frac{\eta}{\pi \mathcal{P}\_r \rho\_f}} t^{-1/2} + \left( \frac{\eta(\gamma - 1)}{\mathcal{K}\_a \rho\_f \mathcal{P}\_r} \right) \left[ \frac{(1 - \wp') \mathfrak{d}}{k'\_0} - \frac{4}{\Lambda'^2} \right] \frac{\partial^{-1}}{\partial t^{-1}}. \end{split}$$

In these expressions, the convolutions express the dispersive nature of the porous material. They take into account the memory effect, where the medium response to excitation of the wave is not instantaneous but takes some time.

The propagation equation can be easily obtained from (24, 25). The propagation equation obtained has exactly the same form as the equation (13), the only difference appears at the coefficient *C* = 4*α*∞(*γ*−1)*η Ka*ΛΛ� √*Pr* + *σφ*(1−*℘*) *Ka* + *α*∞(*γ*−1)*η Kak*� 0*Pr* (1−*℘*� )*φ k*� 0 − 4 Λ�2 , However, we will see that its solution is different. Since the coefficient *C* exchange rate, the coefficients *c*� and Δ given by equation ((17, 18) also change. With extensions of Pride-Lafarge, the coefficient Δ2 can be negative [3], it gives:

$$
\Delta^{'2} = -\Delta^2 \Longrightarrow \Delta^{'2} = 4c' - b'^2. \tag{26}
$$

where

A = Δ� *μ*

*F*1(*t*) + *t*−*x*/*c*

**2.8. The reflection and transmission operators**

*pr*

*pt*

The expressions of *R*˜ and *T*˜ are given by (see Ref. [4])

 ∑ *n*≥0

2 ∑ *n*≥0

> √*α*∞ − *φ* √*α*∞ + *φ*

√*α*∞ (*φ* + √*α*∞)2 *G*

the interfaces of the material:

*R*˜(*t*) = −*φ* + √*α*∞

*T*˜(*t*) = 4*φ*

(

are simplified (Ref. [4] ) as follows:

*φ* + √*α*∞

√*α*∞

*R*˜(*t*) =

*T*˜(*t*) = 4*φ*

√*α*∞ + *φ*)

The Green function is given by [3]:

(*t* − *ξ*)

0 *H*(*ξ*, *t*)*dξ* + *j*

the material. These are operators [4] represented in integral form:

0

0

*R*˜(*τ*)*pi*

*T*˜(*τ*)*pi*

*φ* − √*α*∞ *φ* + √*α*∞

 *φ* − √*α*∞ *φ* + √*α*∞

(*x*, *t*) = *t*

(*x*, *t*) = *t*

2 − *x*2/*c*2 and B = *b*�

*G*(*t*, *x*) = 0, if 0 ≤ *t* ≤ *x*/*c*,

*t*−*x*/*c*

For a slice of porous material occupying the domain 0 ≤ *x* ≤ *L*, the incident and scattered fields are connected by scattering operators (i.e. operators of reflection and transmission) of

In equations (30) and (31) functions *R*˜ and *T*˜ represent the kernel of reflection and transmission operators, respectively. These are independently-owned operators of the incident field and depend only on the material properties. To express the operators of reflection and transmission, we assume [4] that the field of acoustic pressure and flow are continuous at

*p*(0+, *t*) = *p*(0−, *t*), *p*(*L*−, *t*) = *p*(*L*+, *t*)

2*n G t* + *L c*0

where *G* is the Green function of the medium. These expressions take into account the multiple *n* reflections within the porous material. Given the high attenuation of acoustic waves in air-saturated porous media, multiple reflections are negligible. For reflections at the interfaces *x* = 0 and *x* = *L*, the expressions of the operators of reflection and transmission

√*α*∞(

(

√*α*∞ − *φ*)

√*α*∞ + *φ*)3 *G*

*δ*(*t*) − 4*φ*

 *t* + *L c* , *L c*

2*n G t*, 2*n L c* − *G*

*t* − *τ* +

*t* − *τ* − *L*

*x c*0

*c* − (*x* − *L*) *c*0

*v*(0−, *t*) = *φv*(0+, *t*), *v*(*L*+, *t*) = *φv*(*L*−, *t*) (32)

,(2*n* + 1)

*L c* .

 *t*, 2*L c*

(*t* − *ξ*).

Transient Acoustic Wave Propagation in Porous Media 139

0 Ξ(*ξ*, *t*)*dξ* if *t > x*/*c*. (29)

*dτ*, (30)

*t*,(2*n* + 2)

. (34)

*L c* ,

, (33)

*dτ*. (31)

In this case, *f*(*z*) given by equation (16) can be written in the form:

$$f(z) = \left(z + \frac{b'}{2}\sqrt{z}\right)^2 + \left(\frac{\Delta'\sqrt{z}}{2}\right)^2. \tag{27}$$

Using expressions (26 and 27) and analytical calculations given in [3], we obtain the solution of the wave equation when Δ2 is negative:

$$p(\mathbf{x},t) = \begin{cases} 0, & \text{if } 0 \le t \le \mathbf{x}/c\\ \int\_{\mathbf{x}/c}^{t} \left[ F\_{\mathbf{f}}(\mathbf{r}) + \int\_{0}^{\mathbf{r}-\mathbf{x}/c} H(\boldsymbol{\xi},\mathbf{r}) d\boldsymbol{\xi} + j \int\_{0}^{\mathbf{r}-\mathbf{x}/c} \Xi(\boldsymbol{\xi},\mathbf{r}) d\boldsymbol{\xi} \right] p(0,t-\mathbf{r}) d\mathbf{r} & \text{if } \quad t > \mathbf{x}/c \end{cases} (28)$$

with

$$F\_1(\tau) = \frac{1}{4\sqrt{\pi}} b^{\prime} \frac{\chi}{c} \frac{1}{(\tau - \mathfrak{x}/c)^{3/2}} \exp\left(-\frac{b^{\prime 2} \mathfrak{x}^2}{16c^2(\tau - \mathfrak{x}/c)}\right), \quad \dot{f}^2 = -1.1$$

The functions *H*(*ξ*, *τ*) and Ξ(*ξ*, *τ*) are given by the following equations where *τ* is replaced by *t*:

$$\begin{split} H(\xi,t) &= \frac{\Delta'}{4\pi\sqrt{\pi}} \frac{x}{c} \frac{1}{\sqrt{(t-\xi)^2 - x^2/c^2}} \frac{1}{\xi^{3/2}} \times \\ &\int\_{-1}^{1} \left[\frac{\mathcal{A}\mathcal{B}}{4\xi} \cos\left(\frac{\mathcal{A}\mathcal{B}}{8\xi}\right) - \left(\frac{\mathcal{B}^2 - \mathcal{A}^2}{8\xi} - 1\right) \sin\left(\frac{\mathcal{A}\mathcal{B}}{8\xi}\right)\right] \exp\left(-\frac{\mathcal{B}^2 - \mathcal{A}^2}{16\xi}\right) \frac{\mu d\mu}{\sqrt{1-\mu^2}} \end{split}$$

and

$$\begin{split} \Xi(\xi,t) &= -\frac{\Delta'}{4\pi\sqrt{\pi}} \frac{\mathbf{x}}{c} \frac{1}{\sqrt{(t-\xi)^2 - \mathbf{x}^2/c^2}} \frac{1}{\xi^{3/2}} \\ &\int\_{-1}^{1} \left[ \left(\frac{\mathcal{B}^2 - \mathcal{A}^2}{8\xi} - 1\right) \cos\left(\frac{\mathcal{A}\mathcal{B}}{8\xi}\right) + \frac{\mathcal{A}\mathcal{B}}{4\xi} \sin\left(\frac{\mathcal{A}\mathcal{B}}{8\xi}\right) \right] \exp\left(-\frac{\mathcal{B}^2 - \mathcal{A}^2}{16\xi}\right) \frac{\mu d\mu}{\sqrt{1-\mu^2}}, \end{split}$$

where

12 Will-be-set-by-IN-TECH

−1/2 + *σφ*(1 − *℘*) *∂*−1

In these expressions, the convolutions express the dispersive nature of the porous material. They take into account the memory effect, where the medium response to excitation of the

The propagation equation can be easily obtained from (24, 25). The propagation equation obtained has exactly the same form as the equation (13), the only difference appears at the

its solution is different. Since the coefficient *C* exchange rate, the coefficients *c*� and Δ given by equation ((17, 18) also change. With extensions of Pride-Lafarge, the coefficient Δ2 can be

Using expressions (26 and 27) and analytical calculations given in [3], we obtain the solution

0, if 0 ≤ *t* ≤ *x*/*c*, *t*

*τ*−*x*/*c*

The functions *H*(*ξ*, *τ*) and Ξ(*ξ*, *τ*) are given by the following equations where *τ* is replaced by

1 *ξ*3/2 ×

> sin

1 *ξ*3/2

0 Ξ(*ξ*, *τ*)*dξ*

− *b*�2*x*2 16*c*2(*τ* − *x*/*c*)

> A B 8*ξ*

A B 8*ξ*

 exp

 exp

*Ka* + *α*∞(*γ*−1)*η Kak*� 0*Pr*

2 = −Δ2 =⇒ Δ�

*∂t*−1 ,

− 4 Λ�2

2

 , *j*

(1 − *℘*�

*k*� 0 )*φ*

2 = 4*c*� − *b*�2. (26)

− 4 Λ�2  *∂*−1 *∂t*−1 ,

, However, we will see that

. (27)

*p*(0, *t*−*τ*)*dτ* if *t > x*/*c*.

2 = −1.

 *μdμ* 1 − *μ*2

 *μdμ* 1 − *μ*2

,

− B2 − A2 16*ξ*

− B2 − A2 16*ξ*

(28)

*η*(*γ* − 1) *Kaρ f Pr*

> (1−*℘*� )*φ k*� 0

*th*(*t*) are given by:

where the operators *χ*�

*χ*� *v*(*t*) =

*χ*�

coefficient *C* =

*p*(*x*, *t*)=

with

*t*:

and

negative [3], it gives:

*v*(*t*) and *χ*�

 *η πρ f t*

> *η πPrρ f t* −1/2 +

+ *σφ*(1−*℘*)

Δ�

*f*(*z*) =

In this case, *f*(*z*) given by equation (16) can be written in the form:

 *z* + *b*� 2 √*z* 2 + Δ� √*z* 2

0 *H*(*ξ*, *τ*)*dξ*+*j*

1 (*τ* − *x*/*c*)3/2 exp

1 (*t* − *ξ*)2 − *x*2/*c*2

B2 − A2

1 (*t* − *ξ*)2 − *x*2/*c*2

> + A B 4*ξ* sin

A B 8*ξ*

8*ξ* − 1

2*ρ f α*∞ Λ

wave is not instantaneous but takes some time.

 4*α*∞(*γ*−1)*η Ka*ΛΛ� √*Pr*

of the wave equation when Δ2 is negative:

*F*1(*τ*)+ *τ*−*x*/*c*

*x*/*c*

*H*(*ξ*, *t*) = Δ�

Ξ(*ξ*, *t*) = − Δ�

B2 − A2

 1 −1

 1 −1  A B 4*ξ* cos

*F*1(*τ*) = 1 4 √*π b*� *x c*

> 4*π* √*π x c*

> > 4*π* √*π x c*

8*ξ* − 1

A B 8*ξ*

 −

 cos

*th*(*t*) = 2(*γ* − 1) *Ka*Λ�

$$\mathcal{A} = \Delta' \mu \sqrt{(t - \xi)^2 - \mathfrak{x}^2/c^2} \qquad \text{and} \qquad \mathcal{B} = b'(t - \xi).$$

The Green function is given by [3]:

$$G(t, \mathbf{x}) = \begin{cases} 0, & \text{if } \quad 0 \le t \le \mathbf{x}/c \\ F\_1(t) + \int\_0^{t-\mathbf{x}/c} H(\xi, t) d\xi + j \int\_0^{t-\mathbf{x}/c} \Xi(\xi, t) d\xi & \text{if } \qquad t > \mathbf{x}/c. \end{cases} \tag{29}$$

#### **2.8. The reflection and transmission operators**

For a slice of porous material occupying the domain 0 ≤ *x* ≤ *L*, the incident and scattered fields are connected by scattering operators (i.e. operators of reflection and transmission) of the material. These are operators [4] represented in integral form:

$$p^r(\mathbf{x}, t) = \int\_0^t \tilde{\mathcal{R}}(\mathbf{r}) p^i \left( t - \mathbf{r} + \frac{\mathbf{x}}{c\_0} \right) d\mathbf{r},\tag{30}$$

$$p^t(\mathbf{x}, t) = \int\_0^t \Upsilon(\tau) p^i \left( t - \tau - \frac{L}{c} - \frac{(\mathbf{x} - L)}{c\_0} \right) d\tau. \tag{31}$$

In equations (30) and (31) functions *R*˜ and *T*˜ represent the kernel of reflection and transmission operators, respectively. These are independently-owned operators of the incident field and depend only on the material properties. To express the operators of reflection and transmission, we assume [4] that the field of acoustic pressure and flow are continuous at the interfaces of the material:

$$\begin{aligned} p(0^+,t) &= p(0^-,t), & p(L^-,t) &= p(L^+,t) \\ v(0^-,t) &= \phi v(0^+,t), & v(L^+,t) &= \phi v(L^-,t) \end{aligned} \tag{32}$$

The expressions of *R*˜ and *T*˜ are given by (see Ref. [4])

$$
\begin{split}
\tilde{\mathcal{R}}(t) &= \left(\frac{-\phi + \sqrt{\kappa\_{\infty}}}{\phi + \sqrt{\kappa\_{\infty}}}\right) \sum\_{n\geq 0} \left(\frac{\phi - \sqrt{\kappa\_{\infty}}}{\phi + \sqrt{\kappa\_{\infty}}}\right)^{2n} \left[G\left(t, 2n\frac{L}{c}\right) - G\left(t, (2n+2)\frac{L}{c}\right)\right], \\
\tilde{\mathcal{T}}(t) &= \frac{4\phi\sqrt{\kappa\_{\infty}}}{\left(\sqrt{\kappa\_{\infty}} + \phi\right)^{2}} \sum\_{n\geq 0} \left(\frac{\phi - \sqrt{\kappa\_{\infty}}}{\phi + \sqrt{\kappa\_{\infty}}}\right)^{2n} G\left(t + \frac{L}{c\_{0}}, (2n+1)\frac{L}{c}\right).
\end{split}
$$

where *G* is the Green function of the medium. These expressions take into account the multiple *n* reflections within the porous material. Given the high attenuation of acoustic waves in air-saturated porous media, multiple reflections are negligible. For reflections at the interfaces *x* = 0 and *x* = *L*, the expressions of the operators of reflection and transmission are simplified (Ref. [4] ) as follows:

$$\tilde{\mathcal{R}}(t) = \frac{\sqrt{\alpha\_{\infty}} - \phi}{\sqrt{\alpha\_{\infty}} + \phi} \delta(t) - \frac{4\phi\sqrt{\alpha\_{\infty}}(\sqrt{\alpha\_{\infty}} - \phi)}{(\sqrt{\alpha\_{\infty}} + \phi)^3} G\left(t, \frac{2L}{c}\right),\tag{33}$$

$$\hat{T}(t) = \frac{4\phi\sqrt{\kappa\_{\infty}}}{(\phi + \sqrt{\kappa\_{\infty}})^2} G\left(t + \frac{L}{c}, \frac{L}{c}\right). \tag{34}$$

#### 14 Will-be-set-by-IN-TECH 140 Modeling and Measurement Methods for Acoustic Waves and for Acoustic Microdevices Transient Acoustic Wave Propagation in Porous media 15

The first term on the second member of the equation (33):

(( √*α*∞ − *φ*)/( √*α*∞ + *φ*)) *δ*(*t*) is equivalent to the instantaneous reflection response of the porous material. This term corresponds to the wave reflected by the first interface x = 0 . It depends only on the porosity and tortuosity of the material. The reflected wave to the first interface has the advantage not to be dispersive, but simply attenuated. This shows that it is possible to measure the porosity and tortuosity of the porous material by measuring just the first reflected wave.

The second term of equation (33): −4*φ* √*α*∞( √*α*∞−*φ*) ( √*α*∞+*φ*)3 *G t*, 2*L c* corresponds to the reflection by the second interface *x* = *L*. This term depends on the Green function of the medium that describes the propagation and scattering of the acoustic wave having made a round trip in the slab of porous material. Green's function depends on the tortuosity, and viscous and thermal characteristic lengths Λ and Λ� material, but does not depend on the porosity. Experimentally, this second contribution to the debate can not be measured for low-resistive materials, because the acoustic signal is very attenuated.

1 2 3 4

Time (s)

0 1 2 3 4

Frequency (Hz)

1 2 3 4

Time (s)

sample M2 less resistive, having the following parameters: thickness 0.8 cm, tortuosity *α*∞ = 1.05, viscous characteristic length Λ = 300 *μ* m, thermal characteristic length Λ 900 = *μ* m,

**Figure 5.** Comparison between simulated transmitted signal corresponding to the real part of the solution of Pride-Lafarge (Eq. 28) for *℘* = *℘* = 0.7 (solid line) and simulated signal corresponding to the

solution of Johnson-Allard (Eq. 22) for *℘* = *℘* = 1 (dotted line) for the sample M1.

−0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8

−15

−0.02 −0.015 −0.01 −0.005 0 0.005 0.01 0.015 0.02

Amplitude (a.u.)

**Figure 4.** Spectrum of the incident signal

−10

Amplitude (dB)

−5

0

**Figure 3.** Incident signal

Amplitude (a.u.)

x 10 −5

Transient Acoustic Wave Propagation in Porous Media 141

x 10 5

x 10−5

Let us study the sensitivity [20] of porosity on the transmission operator *T*˜(*t*). Taking the derivative of *T*˜(*t*) versus the porosity *φ*, we obtain:

$$\frac{\partial \tilde{T}}{\partial \phi} = \frac{4\sqrt{\alpha\_{\infty}}(\sqrt{\alpha\_{\infty}} - \phi)}{\left(\sqrt{\alpha\_{\infty}} + \phi\right)^3} G\left(t + \frac{L}{c}, \frac{L}{c}\right),$$

when *φ* −→ √*α*∞, the derivative *∂T*˜ *∂φ* −→ 0. This shows that when the porous medium tends to a free fluid, or for weakly resistive porous material with low values of porosity and tortuosity (near 1), the sensitivity of the porosity of the transmitted wave tends to zero. More generally, for a wide range of air saturated porous materials, the term *∂T*˜ *∂φ* is very small. Finally, we conclude that the operator of transmission depends on all parameters, but the low sensitivity of the porosity, makes impossible the determination of this parameter from transmitted data.

#### **2.9. Numerical simulations**

Let us consider a sample of air saturated porous M1 having the following physical properties: thickness 0.8 cm, tortuosity *α*∞ = 1.5, viscous characteristic length Λ = 25 *μ* m, thermal characteristic length Λ = 75*μ* m, specific resistance to fluid flow *σ* = 200,000 Nm −4 s, porosity *φ* = 0.82 and thermal permeability *k*� 0 = 2.77 10−10 m 2. A simulated transmitted signal was calculated from the equation (31). The input signal is given in figure 3 and its spectrum in Figure 4. Figure 5 shows a comparison between two signals, the first (solid line) corresponds to the real part of the solution (28) of Pride-Lafarge model when *℘* = *℘*� = 0.7 and the second (dotted) corresponds to the solution (22) model of Johnson-Allard (*℘* = *℘*� = 1). We choose the same values for *℘* and *℘*� to simplify the study. This specific value of 0.7 is only valid for porous materials with circular pores. In the general case, the values of *℘* and *℘*� may be different from 0.7. Note that it is possible to have positive values of Δ (Eq. 18) for other values of *℘* and *℘*� . From Figure 5, we notice a significant change in the wave amplitude. By increasing *℘* and *℘*� from 0.7 to 1, the wave amplitude increases by 65% of its original value. This result can be predicted by the fact that when the values of *℘* and *℘*� increases, the coefficient *C* decreases, and hence the wave amplitude decreases due to interactions; inertial, viscous and thermal between fluid and structure. This phenomenon is much more important for resistive porous materials. Figure 6 shows the same comparison as Figure 5 for another

**Figure 3.** Incident signal

14 Will-be-set-by-IN-TECH

porous material. This term corresponds to the wave reflected by the first interface x = 0 . It depends only on the porosity and tortuosity of the material. The reflected wave to the first interface has the advantage not to be dispersive, but simply attenuated. This shows that it is possible to measure the porosity and tortuosity of the porous material by measuring just the

√*α*∞−*φ*)

√*α*∞+*φ*)3 *G*

the second interface *x* = *L*. This term depends on the Green function of the medium that describes the propagation and scattering of the acoustic wave having made a round trip in the slab of porous material. Green's function depends on the tortuosity, and viscous and thermal characteristic lengths Λ and Λ� material, but does not depend on the porosity. Experimentally, this second contribution to the debate can not be measured for low-resistive materials, because

Let us study the sensitivity [20] of porosity on the transmission operator *T*˜(*t*). Taking the

√*α*∞ − *φ*)

a free fluid, or for weakly resistive porous material with low values of porosity and tortuosity (near 1), the sensitivity of the porosity of the transmitted wave tends to zero. More generally,

conclude that the operator of transmission depends on all parameters, but the low sensitivity of the porosity, makes impossible the determination of this parameter from transmitted data.

Let us consider a sample of air saturated porous M1 having the following physical properties: thickness 0.8 cm, tortuosity *α*∞ = 1.5, viscous characteristic length Λ = 25 *μ* m, thermal characteristic length Λ = 75*μ* m, specific resistance to fluid flow *σ* = 200,000 Nm −4 s, porosity

calculated from the equation (31). The input signal is given in figure 3 and its spectrum in Figure 4. Figure 5 shows a comparison between two signals, the first (solid line) corresponds to the real part of the solution (28) of Pride-Lafarge model when *℘* = *℘*� = 0.7 and the second (dotted) corresponds to the solution (22) model of Johnson-Allard (*℘* = *℘*� = 1). We choose the same values for *℘* and *℘*� to simplify the study. This specific value of 0.7 is only valid for porous materials with circular pores. In the general case, the values of *℘* and *℘*� may be different from 0.7. Note that it is possible to have positive values of Δ (Eq. 18) for other

By increasing *℘* and *℘*� from 0.7 to 1, the wave amplitude increases by 65% of its original value. This result can be predicted by the fact that when the values of *℘* and *℘*� increases, the coefficient *C* decreases, and hence the wave amplitude decreases due to interactions; inertial, viscous and thermal between fluid and structure. This phenomenon is much more important for resistive porous materials. Figure 6 shows the same comparison as Figure 5 for another

. From Figure 5, we notice a significant change in the wave amplitude.

3 *G*

 *t* + *L c* , *L c* ,

*∂φ* −→ 0. This shows that when the porous medium tends to

0 = 2.77 10−10 m 2. A simulated transmitted signal was

√*α*∞ + *φ*)

√*α*∞(

(

√*α*∞(

(

for a wide range of air saturated porous materials, the term *∂T*˜

√*α*∞ + *φ*)) *δ*(*t*) is equivalent to the instantaneous reflection response of the

 *t*, 2*L c*

corresponds to the reflection by

*∂φ* is very small. Finally, we

The first term on the second member of the equation (33):

((

√*α*∞ − *φ*)/(

first reflected wave.

The second term of equation (33): −4*φ*

the acoustic signal is very attenuated.

when *φ* −→ √*α*∞, the derivative *∂T*˜

**2.9. Numerical simulations**

values of *℘* and *℘*�

*φ* = 0.82 and thermal permeability *k*�

derivative of *T*˜(*t*) versus the porosity *φ*, we obtain:

*∂T*˜ *∂φ* = 4

**Figure 4.** Spectrum of the incident signal

**Figure 5.** Comparison between simulated transmitted signal corresponding to the real part of the solution of Pride-Lafarge (Eq. 28) for *℘* = *℘* = 0.7 (solid line) and simulated signal corresponding to the solution of Johnson-Allard (Eq. 22) for *℘* = *℘* = 1 (dotted line) for the sample M1.

sample M2 less resistive, having the following parameters: thickness 0.8 cm, tortuosity *α*∞ = 1.05, viscous characteristic length Λ = 300 *μ* m, thermal characteristic length Λ 900 = *μ* m,

**Figure 6.** Comparison between simulated transmitted signal corresponding to the real part of the solution of Pride-Lafarge (Eq. 28) for *℘* = *℘*� = 0.7 (solid line), and the transmitted signal corresponding to the simulated Allard-Johnson solution (Eq. 22) for *℘* = *℘*� = 1 (dotted line) for the sample M2.

1 2 3 4 5

Time (s)

**Figure 8.** Simulated transmitted signal corresponding to the imaginary part of the solution (28) model of

tan *θ* = *G*2/*G*1 � 1. This leads to the conclusion that all components of the signal have the

The expression (34) shows that the transmitted wave depends on the porosity *φ*, and parameters describing the high frequency acoustic propagation in the material (tortuosity *α*∞

porosity (in the previous paragraph) showed that the effect of propagation described by the Green's function altogether mask the effect of interfaces: recall that the Green function *G* is independent of porosity and the effect of the interface appears through the term 4*φ*

in the expression of the transmission operator (34). The low sensitivity of the porosity in transmission makes its determination impossible. Solving the inverse problem, using the transmitted waves allowed the determination of the tortuosity and viscous and thermal characteristic lengths. Note that the ratio of two lengths was set at 3 [20–23]. Currently the only method for the separate determination of the two lengths is based on the saturation of the porous by two different fluids [24]. The two characteristic lengths are acting the same way on the dispersion and attenuation of the transmitted signal. So it is mathematically impossible to trace two unknowns giving the same physical effect on the waveform. We limit ourselves in our work to set an arbitrary ratio between the two lengths. The tortuosity acts on the speed of the wavefront (delay of the transmitted signal with respect to the incident signal) and the attenuation of the amplitude of the waveform. While the characteristic lengths affect the dispersion and attenuation of the signal. A basic inverse problem associated with a slab of porous material may be stated as follows: from the measurements of the transmitted signals outside the slab, find the values of the parameters of the medium. As shown in the previous section, the solution of the direct problem is the system of two operators expressed as

of the slab is based on a fitting procedure: find the values of the parameters *α*∞, Λ and Λ� such that the transmitted and reflected signal describes the scattering problem in the best possible way (e.g., in the least-squares sense). Experiments were performed in air using a pairs of broadband Ultran transducers NCG200-D13 with a central frequency of 195 kHz and

. The inversion algorithm for finding the values of the parameters

same phase (very small *θ*, which is therefore a physical factor not essential.

−4 −3 −2 −1 0 1 2 3 4 5 x 10−10

and viscous and thermal characteristic lengths Λ and Λ�

Amplitude(a.u.)

Pride-Lafarge.

**2.10. Transmitted wave**

functions on *α*∞, Λ and Λ�

x 10−5

Transient Acoustic Wave Propagation in Porous Media 143

). The study of the sensitivity of the

√*α*∞ (*φ*+√*α*∞)2

**Figure 7.** Spectrum of the transmitted signal simulated for *℘* = *℘*� = 0.7 Eq. (28) (solid line) and simulated transmitted signal spectrum for *℘* = *℘*� = 1 (Eq. 22) (poitillés).

specific resistance to the passage of fluid *σ* =20000*Nm*−4 s, porosity *φ* = 0.96 and thermal permeability *k*� 0 = 2.77 10−9 m 2. In Figure 6, the influence of the parameters *℘* and *℘*� on the attenuation is smaller than that in Figure 5. We can conclude that the parameters *℘* and *℘*� play an important role in the acoustic attenuation, especially for resistive media.

It is possible to see from Figures 5 and 6, when the values of *℘* and *℘*� change from 0.7 to 1, the waveform changes only at the amplitude of the wave, but it is not dispersed. Figure 7 shows the spectra of two simulated signals given in Figure 5. From the spectra of two simulated signals, we can see they have the same bandwidth, which means that there is no dispersion. This last result shows that *℘* and *℘*� play an important role in attenuation the acoustic wave but not on its dispersion. Figure 8 shows the imaginary part of the solution (28) for *℘* = *℘*� = 0.7 (sample M1). The amplitude of the imaginary part of the solution is very small compared to the real part (figure 5), this is why only the real part of the solution corresponding to the physical solution is taken into account when a comparison with the experiment is performed. It is possible to write the Green function given by equation (29) as *G* = *G*1 + *iG*2 =| *G* |e*iθ*,

**Figure 8.** Simulated transmitted signal corresponding to the imaginary part of the solution (28) model of Pride-Lafarge.

tan *θ* = *G*2/*G*1 � 1. This leads to the conclusion that all components of the signal have the same phase (very small *θ*, which is therefore a physical factor not essential.

#### **2.10. Transmitted wave**

16 Will-be-set-by-IN-TECH

1 2 3 4

Time (s)

0 0.5 1 1.5 2 2.5

specific resistance to the passage of fluid *σ* =20000*Nm*−4 s, porosity *φ* = 0.96 and thermal

the attenuation is smaller than that in Figure 5. We can conclude that the parameters *℘* and *℘*�

It is possible to see from Figures 5 and 6, when the values of *℘* and *℘*� change from 0.7 to 1, the waveform changes only at the amplitude of the wave, but it is not dispersed. Figure 7 shows the spectra of two simulated signals given in Figure 5. From the spectra of two simulated signals, we can see they have the same bandwidth, which means that there is no dispersion. This last result shows that *℘* and *℘*� play an important role in attenuation the acoustic wave but not on its dispersion. Figure 8 shows the imaginary part of the solution (28) for *℘* = *℘*� = 0.7 (sample M1). The amplitude of the imaginary part of the solution is very small compared to the real part (figure 5), this is why only the real part of the solution corresponding to the physical solution is taken into account when a comparison with the experiment is performed. It is possible to write the Green function given by equation (29) as *G* = *G*1 + *iG*2 =| *G* |e*iθ*,

**Figure 7.** Spectrum of the transmitted signal simulated for *℘* = *℘*� = 0.7 Eq. (28) (solid line) and

play an important role in the acoustic attenuation, especially for resistive media.

Frequency (Hz)

0 = 2.77 10−9 m 2. In Figure 6, the influence of the parameters *℘* and *℘*� on

**Figure 6.** Comparison between simulated transmitted signal corresponding to the real part of the solution of Pride-Lafarge (Eq. 28) for *℘* = *℘*� = 0.7 (solid line), and the transmitted signal corresponding to the simulated Allard-Johnson solution (Eq. 22) for *℘* = *℘*� = 1 (dotted line) for the sample M2.

−0.4

−20

simulated transmitted signal spectrum for *℘* = *℘*� = 1 (Eq. 22) (poitillés).

−15

−10

Amplitude (dB)

permeability *k*�

−5

0

−0.2

0

0.2

Amplitude (a.u.)

0.4

0.6

x 10−5

x 105

The expression (34) shows that the transmitted wave depends on the porosity *φ*, and parameters describing the high frequency acoustic propagation in the material (tortuosity *α*∞ and viscous and thermal characteristic lengths Λ and Λ� ). The study of the sensitivity of the porosity (in the previous paragraph) showed that the effect of propagation described by the Green's function altogether mask the effect of interfaces: recall that the Green function *G* is independent of porosity and the effect of the interface appears through the term 4*φ* √*α*∞ (*φ*+√*α*∞)2 in the expression of the transmission operator (34). The low sensitivity of the porosity in transmission makes its determination impossible. Solving the inverse problem, using the transmitted waves allowed the determination of the tortuosity and viscous and thermal characteristic lengths. Note that the ratio of two lengths was set at 3 [20–23]. Currently the only method for the separate determination of the two lengths is based on the saturation of the porous by two different fluids [24]. The two characteristic lengths are acting the same way on the dispersion and attenuation of the transmitted signal. So it is mathematically impossible to trace two unknowns giving the same physical effect on the waveform. We limit ourselves in our work to set an arbitrary ratio between the two lengths. The tortuosity acts on the speed of the wavefront (delay of the transmitted signal with respect to the incident signal) and the attenuation of the amplitude of the waveform. While the characteristic lengths affect the dispersion and attenuation of the signal. A basic inverse problem associated with a slab of porous material may be stated as follows: from the measurements of the transmitted signals outside the slab, find the values of the parameters of the medium. As shown in the previous section, the solution of the direct problem is the system of two operators expressed as functions on *α*∞, Λ and Λ� . The inversion algorithm for finding the values of the parameters of the slab is based on a fitting procedure: find the values of the parameters *α*∞, Λ and Λ� such that the transmitted and reflected signal describes the scattering problem in the best possible way (e.g., in the least-squares sense). Experiments were performed in air using a pairs of broadband Ultran transducers NCG200-D13 with a central frequency of 195 kHz and a bandwith of 6 dB extending from 140 kHz to 250 kHz. Pulses of 400 V are provided by a 5058PR Panametrics pulser/receiver. The received signals are filtered above 1 MHz to avoid high frequency noise. Electronic interference is eliminated by 1000 acquisition averages. The experimental setup is shown in Fig. 9. Consider a sample of plastic foam, of thicknesses

6.6 6.8 7 7.2 7.4

Time (s)

**Figure 11.** Comparison between experimental transmitted signal (solid line) and simulated transmitted

[25]. This method is based on a temporal model of direct and inverse scattering problems affecting the propagation of transient ultrasonic waves in a homogeneous isotropic slab of porous material with a rigid frame. Generally, porosity and tortuosity can be evaluated simultaneously by the wave reflected at the first interface at oblique incidence [26–29] but this is not possible at normal incidence [4, 30]. Viscous and thermal characteristic lengths can be estimated only by transmitted waves. Porosity cannot be determined from transmitted waves because of its weak sensitivity in this mode [4]. The advantage of the proposed method is that all the parameters can be determined at normal incidence (the ratio between viscous and thermal lengths is fixed as in classical methods [21–24] based on transmission measurement). The main principle of this method is the experimental detection of reflected contributions from the first and the second interface of the medium. The properties of these two contributions are used to estimate the four acoustical parameters needed for ultrasonic propagation in porous material with a rigid frame, by solving the inverse problem. Studying the sensitivity of each reflected wave parameter demonstrates the importance of each contribution) first and second interface! for the inversion. Numerical and experimental validation for weak resistive

This section concerns the ultrasonic characterization of air-saturated porous materials by solving the inverse problem using experimental data. It is generally easy to solve the inverse problem via transmitted waves, obtaining optimized values of tortuosity, viscous and thermal characteristic lengths, but this is not the case for the porosity because of its weak sensitivity in the transmitted mode. The reflection mode is an alternative to the transmission mode, in that it gives a good estimation of porosity and tortuosity by processing the data relative to measurements of the wave reflected by the first interface. The viscous and thermal characteristic lengths cannot be determined via the first interface reflection. The wave reflected by the second interface can be experimentally detected only for the weakly resistive porous materials. In this case, the characteristic lengths can be estimated. But for common air-saturated porous materials, the second reflection is very damped and its experimental detection is difficult. We solve the inverse problem [31] numerically by the least-squares method, using both reflected and transmitted experimental data. We determine

−0.04 −0.03 −0.02 −0.01 0 0.01 0.02 0.03 0.04

air-saturated industrial plastic foams is given to validate this method.

**2.12. Reflected and transmitted waves**

Amplitude (V)

signal (dashed line).

x 10 −4

Transient Acoustic Wave Propagation in Porous Media 145

**Figure 9.** Experimental set-up of the ultrasonic measurements in transmitted mode.

0.7 ± 0.01*cm*. The measured signal generated by the transducer (measured without the sample) is given in Fig. 10 as a dashed line. The measured transmitted signal (obtained with the sample inserted) is given in the same figure (Fig. 11) as a solid line. By solving the inverse problem using the experimental transmitted data, we find the following optimized values: *α*∞ = 1.26 and Λ = 66*μm*. In Figs. 11, we compare the experimental transmitted signal and simulated transmitted signal for the optimized values of tortuosity and viscous characteristic length. The difference between experiment and theory is slight, which leads us to conclude that the physical parameters are well-identified.

**Figure 10.** Experimental incident signal (dashed line) and transmitted signal (solid line) using the pair of transducers Ultran NCG200-D13 (140 kHz - 250 kHz).

## **2.11. Determination of transport parameters in air-saturated porous materials via reflected ultrasonic waves**

A method for measuring transport parameters in porous materials simultaneously, using measurements of waves reflected at the first and second interface, has been proposed

**Figure 11.** Comparison between experimental transmitted signal (solid line) and simulated transmitted signal (dashed line).

[25]. This method is based on a temporal model of direct and inverse scattering problems affecting the propagation of transient ultrasonic waves in a homogeneous isotropic slab of porous material with a rigid frame. Generally, porosity and tortuosity can be evaluated simultaneously by the wave reflected at the first interface at oblique incidence [26–29] but this is not possible at normal incidence [4, 30]. Viscous and thermal characteristic lengths can be estimated only by transmitted waves. Porosity cannot be determined from transmitted waves because of its weak sensitivity in this mode [4]. The advantage of the proposed method is that all the parameters can be determined at normal incidence (the ratio between viscous and thermal lengths is fixed as in classical methods [21–24] based on transmission measurement). The main principle of this method is the experimental detection of reflected contributions from the first and the second interface of the medium. The properties of these two contributions are used to estimate the four acoustical parameters needed for ultrasonic propagation in porous material with a rigid frame, by solving the inverse problem. Studying the sensitivity of each reflected wave parameter demonstrates the importance of each contribution) first and second interface! for the inversion. Numerical and experimental validation for weak resistive air-saturated industrial plastic foams is given to validate this method.

#### **2.12. Reflected and transmitted waves**

18 Will-be-set-by-IN-TECH

a bandwith of 6 dB extending from 140 kHz to 250 kHz. Pulses of 400 V are provided by a 5058PR Panametrics pulser/receiver. The received signals are filtered above 1 MHz to avoid high frequency noise. Electronic interference is eliminated by 1000 acquisition averages. The experimental setup is shown in Fig. 9. Consider a sample of plastic foam, of thicknesses

> Triggering Sample

**Figure 9.** Experimental set-up of the ultrasonic measurements in transmitted mode.

0.7 ± 0.01*cm*. The measured signal generated by the transducer (measured without the sample) is given in Fig. 10 as a dashed line. The measured transmitted signal (obtained with the sample inserted) is given in the same figure (Fig. 11) as a solid line. By solving the inverse problem using the experimental transmitted data, we find the following optimized values: *α*∞ = 1.26 and Λ = 66*μm*. In Figs. 11, we compare the experimental transmitted signal and simulated transmitted signal for the optimized values of tortuosity and viscous characteristic length. The difference between experiment and theory is slight, which leads us to conclude

6.7 6.8 6.9 7 7.1 7.2

Time (s)

**Figure 10.** Experimental incident signal (dashed line) and transmitted signal (solid line) using the pair

**2.11. Determination of transport parameters in air-saturated porous materials via**

A method for measuring transport parameters in porous materials simultaneously, using measurements of waves reflected at the first and second interface, has been proposed

x 10 −4

Transducers

Computer

Digital oscilloscope

High frequency filtering Pre-amplifier

pulse generator

that the physical parameters are well-identified.

−0.15 −0.1 −0.05 0 0.05 0.1 0.15

Amplitude (V)

of transducers Ultran NCG200-D13 (140 kHz - 250 kHz).

**reflected ultrasonic waves**

This section concerns the ultrasonic characterization of air-saturated porous materials by solving the inverse problem using experimental data. It is generally easy to solve the inverse problem via transmitted waves, obtaining optimized values of tortuosity, viscous and thermal characteristic lengths, but this is not the case for the porosity because of its weak sensitivity in the transmitted mode. The reflection mode is an alternative to the transmission mode, in that it gives a good estimation of porosity and tortuosity by processing the data relative to measurements of the wave reflected by the first interface. The viscous and thermal characteristic lengths cannot be determined via the first interface reflection. The wave reflected by the second interface can be experimentally detected only for the weakly resistive porous materials. In this case, the characteristic lengths can be estimated. But for common air-saturated porous materials, the second reflection is very damped and its experimental detection is difficult. We solve the inverse problem [31] numerically by the least-squares method, using both reflected and transmitted experimental data. We determine simultaneously all the physical parameters intervening in the propagation. The minimization between experiment and theory is made in the time domain. The inverse problem is well posed, and its solution is unique. As with the classic ultrasonic approach for characterizing porous material saturated with one gas, the characteristic lengths are estimated by assuming a given ratio between them. Tests are performed using industrial plastic foams. Experimental and numerical results, and prospects are discussed [31].

solution of the inverse problem. The propagation equation is given (along the *x*−axis) by:

*∂t* + *B ∂*3 *p*(*x*, *t*)

*ρ f k*0 ,

> � − 2 *π* √*π*

−

*γα*∞*ρ*2 *f* Λ2 � *α*0 *α*∞ − 1 �3

> � ∞ *x c* √*b*�

⎤ ⎦

(*t*) fields are related in time domain by the transmission

*du*, *�* =

*∂t* = 0, (37)

, *A* = *ηφ*

0 2 *Pr* 2*ρ f* 2

*ηφk*0*Ka*

− *x*2Δ 16*c*2 *b*� *t*

)ℵ(*ζ*,*s*)*ds*,

) �

*ζ*2 − *x*2 *c*2 *b*�

*D*(*τ*)*G*(*t* − *τ* + *L*/*c*, *L*/*c*)*dτ*, − ∗ (36)

√*s*(*s* + 1/2*b*�

*∂t*3 = 0, (35)

Transient Acoustic Wave Propagation in Porous Media 147

4*ηKa*

�

*yd y* �1 − *y*2 ,

*F*(*ζ*,*t*)

*ζ*2 − *x*2 *c*2 *b*� *dζ* ⎤ ⎦ ,

�

. (38)

*η* �

*φ*3*γσ*3*ρ f* ,

, *B* = 1

*ρ f k*0*γπ η* .

*∂t*2 − *A ∂p*(*x*, *t*)

0

*α*�

The solution of this equation gives the Green function [34] of the porous material:

�

0(*γ* − 1)*k*�

*∂*2 *p*(*x*, *t*) *∂x*2 − 1

1 *c*2 = *ρ f Ka* �

*B* = −

*G*(*x*, *t*) = 1

*F*(*ζ*,*t*) = � ∞

ℵ(*ζ*,*s*) = � 1

*T*˜(*t*) = � *t*

*D*(*t*) = 2

0

*�*√*π*

function of the porous material [33, 35, 36]:

convolution operation. The operator *D*(*t*) is given by

exp � *t B*2 �

The incident *pi*

equation

where *d* = *σφγ*

*D*(*t*) = − 8

*B*2 � 1 + *t B*2 �

√*π x c* √ *b*� ⎡ ⎣ √Δ 4 *b*�

0

−1 cos ⎡ ⎣ √Δ 2*b*� *ζ*

*b*� = *Bc*2, *c*� = *Ac*2, Δ = 1 + 4*b*�

*c*2

*α*0(*γ* − 1)*k*�

exp(−*s t*)

(*t*) and transmitted *pt*

1 *t*3/2 � ∞ 0

*∂*2 *p*(*x*, *t*)

*α*0*γ* − (*γ* − 1)*Prk*�

0*Prρ*2 *f Kaηφ* +

*k*0

1 *t*3/2 exp �

√*s*(*s* + 1/2*b*�

*c*� .

scattering operator 31. Its expression is given by the relation given in Ref. [34]:

�

√*d*

1 *t*3/2 exp �

The transmission scattering operator is *T*˜(*t*) = *D*(*t*) ∗ *G*(*L*,*t*), where \* denotes the time

� +

2 √*π*

Erfc � √*t B*

*u* exp �

In the very low frequency range, the propagation equation 35 is reduced to a diffusive

− *u �* − *u*2 4*t* �

*∂p*(*x*, *t*)

*Ka* the diffusion constant. The solution of this diffusive equation gives the Green

4 *B* √*πt*

− *x*2 *d* 4*t*

> � 2 *t B*2 + 1

�

�

√*s* − *y*

� *u*2 2*t* − 1

> *∂*2 *p*(*x*, *t*) *∂x*2 − *d*

*G*(*x*, *t*) = *x*

## **2.13. Ultrasonic characterization of air-saturated double-layered porous media**

This section concerns a time-domain model of transient wave propagation in double-layered porous materials [32]. 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. the inverse problem is solved using experimental reflected signals at normal incidence. The sensitivity of porosity, tortuosity, and viscous characteristic length of each layer is studied showing their effect on the reflected interface waveforms. The inverse problem is solved numerically by the least-squares method. Five parameters are inverted: porosity and tortuosity of the two layers and the viscous characteristic length of the first layer. The minimization of the discrepancy between experimental and theoretical data is made in the time domain. The inverse problem is shown to be well posed and its solution to be unique. Experimental results for waves reflected by the interfaces of the double-layered porous material are given and compared with theoretical predictions [32].

### **2.14. Acoustic characterization of porous materials at low frequency range**

In this frequency range, the viscous forces are important throughout the fluid, the cycle of compression/expansion in the porous medium is slow enough to allow heat exchange between fluid and structure. At the same time, the temperature of the structure is practically unchanged by the passage of the sound wave due to the large value of its specific heat conducting relation to that of the fluid: the structure acts as a thermostat. In this case the isothermal compression is directly applicable. This range corresponds to frequencies where the viscous skin depth *δ* = (2*η*/*ωρ f*)1/2 is larger than the pore radius.

An acoustic method [33–37] using transmitted and reflected waves is proposed for measuring static viscous permeability *k*0, flow resistivity *σ*, static thermal permeability *k* 0 and the inertial factor *α*0 (low frequency tortuosity), of porous materials having a rigid frame at low frequencies. Flow resistivity of porous material is defined as the ratio between the pressure difference across a sample and the velocity of flow of air through that sample per unit cube. The static thermal permeability of porous material is a geometrical parameter equal to the inverse trapping constant of the solid frame. The inertial factor describes the fluid structure interactions in the low frequency range (1-3 kHz). These parameters are determined from the solution of the inverse problem. The propagation equation is given (along the *x*−axis) by:

20 Will-be-set-by-IN-TECH

simultaneously all the physical parameters intervening in the propagation. The minimization between experiment and theory is made in the time domain. The inverse problem is well posed, and its solution is unique. As with the classic ultrasonic approach for characterizing porous material saturated with one gas, the characteristic lengths are estimated by assuming a given ratio between them. Tests are performed using industrial plastic foams. Experimental

**2.13. Ultrasonic characterization of air-saturated double-layered porous media**

porous material are given and compared with theoretical predictions [32].

the viscous skin depth *δ* = (2*η*/*ωρ f*)1/2 is larger than the pore radius.

**2.14. Acoustic characterization of porous materials at low frequency range**

In this frequency range, the viscous forces are important throughout the fluid, the cycle of compression/expansion in the porous medium is slow enough to allow heat exchange between fluid and structure. At the same time, the temperature of the structure is practically unchanged by the passage of the sound wave due to the large value of its specific heat conducting relation to that of the fluid: the structure acts as a thermostat. In this case the isothermal compression is directly applicable. This range corresponds to frequencies where

An acoustic method [33–37] using transmitted and reflected waves is proposed for measuring

inertial factor *α*0 (low frequency tortuosity), of porous materials having a rigid frame at low frequencies. Flow resistivity of porous material is defined as the ratio between the pressure difference across a sample and the velocity of flow of air through that sample per unit cube. The static thermal permeability of porous material is a geometrical parameter equal to the inverse trapping constant of the solid frame. The inertial factor describes the fluid structure interactions in the low frequency range (1-3 kHz). These parameters are determined from the

0 and the

static viscous permeability *k*0, flow resistivity *σ*, static thermal permeability *k*

This section concerns a time-domain model of transient wave propagation in double-layered porous materials [32]. 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. the inverse problem is solved using experimental reflected signals at normal incidence. The sensitivity of porosity, tortuosity, and viscous characteristic length of each layer is studied showing their effect on the reflected interface waveforms. The inverse problem is solved numerically by the least-squares method. Five parameters are inverted: porosity and tortuosity of the two layers and the viscous characteristic length of the first layer. The minimization of the discrepancy between experimental and theoretical data is made in the time domain. The inverse problem is shown to be well posed and its solution to be unique. Experimental results for waves reflected by the interfaces of the double-layered

and numerical results, and prospects are discussed [31].

$$\begin{split} \frac{\partial^2 p(\mathbf{x}, t)}{\partial \mathbf{x}^2} - \frac{1}{c^2} \frac{\partial^2 p(\mathbf{x}, t)}{\partial t^2} - A \frac{\partial p(\mathbf{x}, t)}{\partial t} + B \frac{\partial^3 p(\mathbf{x}, t)}{\partial t^3} &= 0, \\ \frac{1}{c^2} = \frac{\rho\_f}{K\_d} \left( a\_0 \gamma - \frac{(\gamma - 1) P\_r k\_0'}{k\_0} \right), \quad A = \frac{\eta \phi}{\rho\_f k\_0'}, \\ B = -\frac{a\_0 (\gamma - 1) k\_0' P\_r \rho\_f^2}{K\_d \eta \phi} + \frac{a\_0' (\gamma - 1) k\_0'^2 P\_r^2 \rho\_f^2}{\eta \phi k\_0 K\_d} - \frac{\gamma a\_0 \rho\_f^2 \Lambda^2 \left( \frac{a\_0}{a\_\infty} - 1 \right)^3}{4 \eta K\_d} \end{split}$$

The solution of this equation gives the Green function [34] of the porous material:

$$\begin{split} G(x,t) &= \frac{1}{\sqrt{\pi}} \frac{x}{c} \sqrt{b'} \left[ \frac{\sqrt{\Delta}}{4b'} \frac{1}{b^{3/2}} \exp\left(-\frac{x^2 \Delta}{16c^2 b' t}\right) - \frac{2}{\pi \sqrt{\pi}} \int\_{\frac{x}{c}\sqrt{b'}}^{\infty} \frac{F(\zeta, t)}{\sqrt{\zeta^2 - \frac{x^2}{c^2} b'}} d\zeta \right], \\ F(\zeta, t) &= \int\_0^{\infty} \exp(-st) \sqrt{s} (s + 1/2b') \aleph(\zeta, s) ds, \\ \aleph(\zeta, s) &= \int\_{-1}^1 \exp\left[\frac{\sqrt{\Delta}}{2b'} \zeta \sqrt{s} - y\sqrt{s} (s + 1/2b') \sqrt{\zeta^2 - \frac{x^2}{c^2} b'}\right] \frac{y d\, y}{\sqrt{1 - y^2}}, \\ b' &= Bc^2, c' = Ac^2, \Delta = 1 + 4b'c'. \end{split}$$

The incident *pi* (*t*) and transmitted *pt* (*t*) fields are related in time domain by the transmission scattering operator 31. Its expression is given by the relation given in Ref. [34]:

$$T(t) = \int\_0^t D(\tau)G(t-\tau+L/c, L/c)d\tau, -\ast \tag{36}$$

$$D(t) = \frac{2}{\mathcal{O}\sqrt{\pi}}\frac{1}{t^{3/2}}\int\_0^\infty \left(\frac{u^2}{2t}-1\right)u\exp\left(-\frac{u}{\mathcal{O}}-\frac{u^2}{4t}\right)du, \qquad \mathcal{O} = \sqrt{\frac{\rho\_f k\_0 \gamma \pi}{\eta}}.$$

In the very low frequency range, the propagation equation 35 is reduced to a diffusive equation

$$\frac{\partial^2 p(\mathbf{x}, t)}{\partial \mathbf{x}^2} - d \frac{\partial p(\mathbf{x}, t)}{\partial t} = 0,\tag{37}$$

where *d* = *σφγ Ka* the diffusion constant. The solution of this diffusive equation gives the Green function of the porous material [33, 35, 36]:

$$G(\mathbf{x}, t) = \frac{\mathbf{x}\sqrt{d}}{2\sqrt{\pi}} \frac{1}{t^{3/2}} \exp\left(-\frac{\mathbf{x}^2 d}{4t}\right). \tag{38}$$

The transmission scattering operator is *T*˜(*t*) = *D*(*t*) ∗ *G*(*L*,*t*), where \* denotes the time convolution operation. The operator *D*(*t*) is given by

$$D(t) = -\frac{8}{B^2} \left( 1 + \frac{t}{B^2} \right) \exp\left(\frac{t}{B^2}\right) \text{Erfc}\left(\frac{\sqrt{t}}{B}\right) + \frac{4}{B\sqrt{\pi t}} \left( 2\frac{t}{B^2} + 1 \right), \qquad B = \frac{1}{\eta} \sqrt{\phi^3 \eta \sigma^3 \rho\_{f'}}$$

where Erfc is the complementary error function and *ρ f* is the fluid density.

The inverse problem is to find the parameters *k*� and *α*0 [37] which minimize numerically the discrepancy function *U*(*k*� , *α*0) = ∑*i*=*N i*=1 (*pt exp*(*x*, *ti*) − *pt* (*x*, *ti*))2, wherein *pt exp*(*x*, *ti*)*i*=1,2,...*n* is the discrete set of values of the experimental transmitted signal and *pt* (*x*, *ti*)*i*=1,2,...*n* the discrete set of values of the simulated transmitted signal predicted from Eq. (31). The inverse problem is solved numerically by the least-square method. For its iterative solution, we used the simplex search method (Nedler Mead) [38] which does not require numerical or analytic gradients. Experiments are performed in a guide [33–37] (pipe), having a diameter of 5 cm. The experimental set up is given in Fig. 12. For measuring the static thermal permeability and the inertial factor, a pipe of 3 m long, since the frequencies used in the experiment are between 1kHz and 4 kHz. However, for measuring the viscous permeability (and flow resistivity) [33, 35, 36], a pipe of 50 m long must be used (50Hz-1kHz) to avoid the reflections at its end. In this case, it is not important to keep the pipe straight, it can be rolled in order to save space without perturbations on experimental signals (the cut-off frequency of the tube *fc* ∼ 4kHz). A sound source Driver unit "Brand" constituted by loudspeaker Realistic 40-9000 is used. Bursts are provided by synthesized function generator Standford Research Systems model DS345-30MHz. The signals are amplified and filtered using model SR 650-Dual channel filter, Standford Research Systems. The signals (incident and transmitted) are measured using the same microphone (Bruel&Kjaer, 4190) in the same position in the tube. The incident signal is measured without porous sample, however, the transmitted signal is measured with the porous sample. Consider a cylindrical sample of plastic foam M of diameter 5 cm, porosity *φ* = 0.85 and thickness 4.15 cm. The viscous permeability *k*0 of the porous sample is measured by solving the inverse problem in time domain at viscous domain (very low frequency range)[33, 35, 36]. The obtained value of *k*0 is (0.44 ± 0.02) × 10−9 *m*2. The static thermal permeability is measured using classic Kundt tube (continuous frequency) [39] (continuous frequency) obtaining the value of *k*� 0 = (1.54 ± 0.05) × 10−9, with a ratio of 3.5 between *k*� 0 and *k*0. The tortuosity *α*∞ and the viscous characteristic length Λ are measured in the high frequency range [4] obtaining the values: *α*∞ = 1.2 and Λ = 65*μm*. The value of the thermal tortuosity *α*� 0 is approximated by the relation *α*� 0 � *α*0/*α*∞. Fig. 13 shows the experimental incident signal (solid line) generated by the loudspeaker in the frequency bandwidth (2.5 - 3.5) kHz, and the experimental transmitted signal (dashed line). After solving the inverse problem numerically for the thermal permeability *k*� 0 and the inertial factor *α*0 , we find the following optimized values: *k*� 0 = (1.06 ± 0.2) × 10−9*m*2 and *α*0 = 2.15 ± 0.15. We present in Fig. 14 the variation of the minimization function *U* with the thermal permeability *k*� 0 and tortuosity *α*0. The obtained ratio of 2.4 between the inverted *k*� 0 and *k*0 is in adequacy with the ratio given in literature [9, 39], which is found generally to be between 2 and 4 for plastic foams. In Fig. 15, we show a comparison between an experimental transmitted signal and simulated transmitted signal for the optimized values of thermal permeability and tortuosity. The difference between the two curves is slight, which leads us to conclude that the optimized values of the thermal permeability and inertial factor are correct. This study has been carried out, in the frequency bandwidths (1.5 - 2.5) kHz and has also given good results.

**Figure 12.** Experimental setup of acoustic measurements.

Amplitude (a.u.)

−0.3 −0.2 −0.1 0 0.1 0.2 0.3

0.3 0.4 0.5

0.25

U

**Figure 14.** Minima of the inverse problem.

0.3

U

0.088 0.089 0.09 0.091

Transient Acoustic Wave Propagation in Porous Media 149

Time (s)

1.5 2 2.5

α0

k'0 /k0

skeleton and the saturating fluid. The fluid-structure interactions then play an essential role in the propagation. The study of theses effects has been largely developed by Biot [40, 41] for applications in the domain of oil exploration since 1950. This semi-phenomenological theory provides a rigorous description of the propagation of acoustical waves in porous

1.5 2 2.5 3

**Figure 13.** Experimental incident signal (solid line) and experimental transmitted signal (dashed line).

### **3. Modified Biot theory**

When the structure of a porous material is not rigid, the equivalent fluid model, on which the previous sections are based, no longer applies because the waves propagate in both the

**Figure 12.** Experimental setup of acoustic measurements.

22 Will-be-set-by-IN-TECH

The inverse problem is to find the parameters *k*� and *α*0 [37] which minimize numerically

discrete set of values of the simulated transmitted signal predicted from Eq. (31). The inverse problem is solved numerically by the least-square method. For its iterative solution, we used the simplex search method (Nedler Mead) [38] which does not require numerical or analytic gradients. Experiments are performed in a guide [33–37] (pipe), having a diameter of 5 cm. The experimental set up is given in Fig. 12. For measuring the static thermal permeability and the inertial factor, a pipe of 3 m long, since the frequencies used in the experiment are between 1kHz and 4 kHz. However, for measuring the viscous permeability (and flow resistivity) [33, 35, 36], a pipe of 50 m long must be used (50Hz-1kHz) to avoid the reflections at its end. In this case, it is not important to keep the pipe straight, it can be rolled in order to save space without perturbations on experimental signals (the cut-off frequency of the tube *fc* ∼ 4kHz). A sound source Driver unit "Brand" constituted by loudspeaker Realistic 40-9000 is used. Bursts are provided by synthesized function generator Standford Research Systems model DS345-30MHz. The signals are amplified and filtered using model SR 650-Dual channel filter, Standford Research Systems. The signals (incident and transmitted) are measured using the same microphone (Bruel&Kjaer, 4190) in the same position in the tube. The incident signal is measured without porous sample, however, the transmitted signal is measured with the porous sample. Consider a cylindrical sample of plastic foam M of diameter 5 cm, porosity *φ* = 0.85 and thickness 4.15 cm. The viscous permeability *k*0 of the porous sample is measured by solving the inverse problem in time domain at viscous domain (very low frequency range)[33, 35, 36]. The obtained value of *k*0 is (0.44 ± 0.02) × 10−9 *m*2. The static thermal permeability is measured using classic Kundt tube (continuous frequency) [39] (continuous

and *k*0. The tortuosity *α*∞ and the viscous characteristic length Λ are measured in the high frequency range [4] obtaining the values: *α*∞ = 1.2 and Λ = 65*μm*. The value of the thermal

incident signal (solid line) generated by the loudspeaker in the frequency bandwidth (2.5 - 3.5) kHz, and the experimental transmitted signal (dashed line). After solving the inverse