2. Model

In the acoustics of porous materials, a distinction can be made between two situations depending on whether the frame is moving or not. In the general case (when the frame is moving), as for cancellous bone and rock and all porous media saturated by a liquid, the dynamics of the waves due to the coupling between the solid frame and the fluid are clearly described by the Biot theory [17, 18]. In air-saturated porous media, as plastic foams or fibrous materials that are used in sound absorption, the structure is generally motionless and the waves propagate only in the fluid. This case is described by the model of equivalent fluid, which is a particular case in the Biot model, in which the interactions between the fluid and the structure are taken into account in two frequency response factors: the dynamic tortuosity of the medium α ωð Þ given by Johnson et al. [4] and the dynamic compressibility of the air included in the porous material β ωð Þ given by Allard and Champoux [5]. In the frequency domain, these factors multiply the density of the fluid and its compressibility, respectively, and represent the deviation from the behavior of the fluid in free space as the frequency increases. In the time domain, they act as operators, and in the high-frequency approximation, their expressions are given by [11]

$$
\tilde{\alpha}(t) = a\_{\ast\ast} \left( \delta(t) + \frac{2}{\Lambda} \left( \frac{\eta}{\pi \rho\_f} \right)^{1/2} t^{-1/2} \right),
\tag{1}
$$

$$\tilde{\beta}(t) = \left(\delta(t) + \frac{2(\gamma - 1)}{\Lambda'} \left(\frac{\eta}{\pi Pr \rho\_f}\right)^{1/2} t^{-1/2}\right). \tag{2}$$

In these equations, δð Þt is the Dirac function, Pr is the Prandtl number, η and r<sup>f</sup> are fluid viscosity and fluid density, respectively, and γ is the adiabatic constant. The most important physical parameters of the model are the medium's tortuosity α<sup>∞</sup> initially introduced by Zwikker and Kosten [2] and viscous and the thermal characteristic lengths Λ and Λ<sup>0</sup> introduced by Johnson et al. [4] and Allard and Champoux [5]. In this model, the time convolution of t �1=<sup>2</sup> with a function is interpreted as a semi-derivative operator according to the definition of the fractional derivative of order ν given in Samko et al. [19]

$$D^{\nu}[\mathbf{x}(t)] = \frac{1}{\Gamma(-\nu)} \int\_{0}^{t} (t-u)^{-\nu-1} \mathbf{x}(u) du,\tag{3}$$

where Γð Þx is the Gamma function.

pressure change and volume change are monitored using a U-shaped fluid-filled manometer. Leonard [6] has given an alternate dynamic method for measuring porosity. Other techniques using water as the pore-filling fluid, rather than air, are common in geophysical studies [7, 8]. Mercury has been used as the pore-filling fluid in other applications [9]. However, for many materials, the introduction of liquids into the material is not appropriate. Recently, a similar device to that of Beranek [3], involving the use of an electronic pressure transducer, was introduced by Champoux et al. [10]. This device can be used to measure very slight changes in

Generally, the most methods used for measuring the porosity cited previously do not use acoustic waves. Here, we present an ultrasonic method for measuring porosity using ultrasonic reflected waves by the porous material. The direct and inverse problem is solved in the time domain using experimental reflected data. The inverse problem is solved directly in time domain using the waveforms. The attractive feature of a time domain-based approach [11–16] is that the analysis is naturally limited by the finite duration of ultrasonic pressures and is

In the acoustics of porous materials, a distinction can be made between two situations depending on whether the frame is moving or not. In the general case (when the frame is moving), as for cancellous bone and rock and all porous media saturated by a liquid, the dynamics of the waves due to the coupling between the solid frame and the fluid are clearly described by the Biot theory [17, 18]. In air-saturated porous media, as plastic foams or fibrous materials that are used in sound absorption, the structure is generally motionless and the waves propagate only in the fluid. This case is described by the model of equivalent fluid, which is a particular case in the Biot model, in which the interactions between the fluid and the structure are taken into account in two frequency response factors: the dynamic tortuosity of the medium

pressure accurately, and the output can be recorded by a computer.

consequently the most appropriate approach for the transient signal.

2. Model

Figure 1. Air saturated plastic foam.

112 Porosity - Process, Technologies and Applications

In this framework, the basic equations of our model can be expressed as follows

$$
\rho\_f \ddot{\alpha}(t) \* \frac{\partial v\_i}{\partial t} = -\nabla\_i p \quad \text{and} \quad \frac{\ddot{\beta}(t)}{K\_a} \* \frac{\partial p}{\partial t} = -\nabla.v,\tag{4}
$$

where <sup>∗</sup> denotes the time convolution operation, p is the acoustic pressure, v is the particle velocity and Ka is the bulk modulus of the air. The first equation is the Euler equation, the second is a constitutive equation obtained from the equation of mass conservation associated with the behavior (or adiabatic) equation.

In the plane (xoz), the constitutive equation (4) can be written as

$$\begin{split} \rho\_{\boldsymbol{f}}a\_{\boldsymbol{\alpha}}\frac{\partial v\_{\boldsymbol{x}}(\mathbf{x},z,t)}{\partial t} + \frac{2\rho\_{\boldsymbol{f}}a\_{\boldsymbol{\alpha}}}{\Lambda} \left(\frac{\eta}{\pi\rho\_{\boldsymbol{f}}}\right)^{1/2} \int\_{0}^{t} \frac{\partial v\_{\boldsymbol{x}}(\mathbf{x},z,t')/\partial t'}{\sqrt{t-t}} dt' &= -\frac{\partial p(\mathbf{x},z,t)}{\partial \mathbf{x}},\\ \rho\_{\boldsymbol{f}}a\_{\boldsymbol{\alpha}}\frac{\partial v\_{\boldsymbol{x}}(\mathbf{x},z,t)}{\partial t} + \frac{2\rho\_{\boldsymbol{f}}a\_{\boldsymbol{\alpha}}}{\Lambda} \left(\frac{\eta}{\pi\rho\_{\boldsymbol{f}}}\right)^{1/2} \int\_{0}^{t} \frac{\partial v\_{\boldsymbol{x}}(\mathbf{x},z,t')/\partial t'}{\sqrt{t-t}} dt' &= -\frac{\partial p(\mathbf{x},z,t)}{\partial \mathbf{z}},\\ \frac{1}{K\_{\mathbf{d}}}\frac{\partial p(\mathbf{x},z,t)}{\partial t} + \frac{2(\gamma-1)}{K\_{\mathbf{d}}\Lambda'} \left(\frac{\eta}{\pi\rho\_{\boldsymbol{f}}P\_{\boldsymbol{r}}}\right)^{1/2} \int\_{0}^{t} \frac{\partial p(\mathbf{x},z,t')/\partial t'}{\sqrt{t-t}} dt' &= -\frac{\partial v\_{\mathbf{x}}(\mathbf{x},z,t)}{\partial \mathbf{x}} - \frac{\partial v\_{\mathbf{z}}(\mathbf{x},z,t)}{\partial z},\end{split} \tag{5}$$

where vx and vz are the components of particle velocity along axes x and z. In these equations, the integrals express the fractional derivatives, defined mathematically by convolutions and describe the dispersive nature of the porous material. They take into account the memory effects due to the fluid-structure interactions and to the viscous and thermal losses in the medium.

The problem geometry is shown in Figure 2. A homogeneous porous material occupies the region 0 ≤ x ≤ L. This medium is assumed to be isotropic and to have a rigid frame. A short sound pulse impinges at oblique incidence on the medium from the left, giving rise to an acoustic pressure field p xð Þ ; z; t and an acoustic velocity field vð Þ x; z; t within the material, which satisfies the system of Eq. (5) that can be written as:

$$
\rho\_f \alpha\_\ast \frac{\partial v\_x(\mathbf{x}, z, t)}{\partial t} + \frac{2\rho\_f \alpha\_\ast}{\Lambda} \left(\frac{\eta}{\rho\_f}\right)^{1/2} D^{1/2} [v\_x(\mathbf{x}, z, t)] = -\frac{\partial p(\mathbf{x}, z, t)}{\partial \mathbf{x}},\tag{6}
$$

where c<sup>0</sup> is the velocity in the porous material (0 ≤ x ≤ L), and the refraction angle θ<sup>0</sup> is given by

<sup>¼</sup> sin <sup>θ</sup><sup>0</sup>

and <sup>∂</sup><sup>p</sup>

2r<sup>f</sup> α<sup>∞</sup> Λ

By using Eqs. (6), (8) and (14), the equation systems (6), (7) and (8) can thus be simplified to

D<sup>1</sup>=<sup>2</sup>

D<sup>1</sup>=<sup>2</sup>

2r<sup>f</sup> α<sup>∞</sup> Λ

From Eqs. (15) and (16), we derive the fractional propagation wave equation in the time domain

The solution of the wave Eq. (17) with suitable initial and boundary conditions is by using the

0 if 0 ≤ t ≤ k

<sup>Ξ</sup>ð Þþ <sup>t</sup> <sup>Δ</sup>Ð<sup>t</sup>�<sup>k</sup>

η rf !<sup>1</sup>=<sup>2</sup>

η r<sup>f</sup> Pr !<sup>1</sup>=<sup>2</sup>

r<sup>f</sup> α<sup>∞</sup> þ

η rf

!<sup>1</sup>=<sup>2</sup>

<sup>∂</sup><sup>z</sup> ¼ � sin <sup>θ</sup> c0

D�1=<sup>2</sup>

½ �¼� vxð Þ x; z; t

½ �¼ p xð Þ ; z; t

D�1=<sup>2</sup>

<sup>Λ</sup> <sup>þ</sup> <sup>γ</sup> � <sup>1</sup> ffiffiffiffiffi Pr <sup>p</sup> <sup>Λ</sup><sup>0</sup> � �D<sup>3</sup>=<sup>2</sup>

<sup>0</sup> h tð Þ ; ξ dξ if t ≥ k

1 A �1

η rf

ffiffiffiffiffiffiffi r<sup>f</sup> η π r 1

!<sup>1</sup>=<sup>2</sup>

∂p ∂t

1 A

�1 ∗ ∂p ∂t

∂p xð Þ ; z; t

<sup>∗</sup> <sup>∂</sup>p xð Þ ; <sup>z</sup>; <sup>t</sup> <sup>∂</sup><sup>t</sup> :

∂ ∂t

<sup>c</sup><sup>0</sup> (11)

Ultrasound Measuring of Porosity in Porous Materials http://dx.doi.org/10.5772/intechopen.72696 115

, (12)

: (13)

: (14)

<sup>∂</sup><sup>x</sup> , (15)

½ �¼ p xð Þ ; z; t 0, (17)

(16)

(18)

sin θ c0

To simplify the system of Eq. (5), we can then use the following property

∂vz

<sup>∂</sup><sup>z</sup> ¼ � sin <sup>2</sup><sup>θ</sup> c2 0

∂vxð Þ x; z; t ∂t þ

þ

From Eqs. (7) and (13), we obtain the relation

∂vz

r<sup>f</sup> α<sup>∞</sup>

∂p xð Þ ; z; t ∂t þ

� <sup>∂</sup>vxð Þ <sup>x</sup>; <sup>z</sup>; <sup>t</sup> ∂x

> <sup>α</sup><sup>∞</sup> � sin <sup>2</sup> <sup>θ</sup> � � <sup>∂</sup><sup>2</sup>

1 Ka

> c2 0

along the x-axis

p xð Þ ; z; t <sup>∂</sup>x<sup>2</sup> � <sup>1</sup>

∂2

<sup>∂</sup><sup>z</sup> ¼ � sin <sup>θ</sup> c0

∂

∂vz ∂t

r<sup>f</sup> α<sup>∞</sup> þ

0 @

2r<sup>f</sup> α<sup>∞</sup> Λ

2ð Þ γ � 1 KaΛ<sup>0</sup>

sin <sup>2</sup>θ c2 0

0 @

p xð Þ ; z; t ∂t

Laplace transform. F is the medium's Green function [12, 20] given by

(

F tð Þ¼ ; k

<sup>2</sup> � <sup>2</sup>α<sup>∞</sup> Ka

<sup>∂</sup><sup>z</sup> ¼ � sin <sup>θ</sup> c0

the Descartes-Snell law [27, 28]:

which implies

$$
\rho\_f \alpha\_\ast \frac{\partial v\_z(\mathbf{x}, z, t)}{\partial t} + \frac{2\rho\_f \alpha\_\ast}{\Lambda} \left(\frac{\eta}{\rho\_f}\right)^{1/2} D^{1/2} [v\_z(\mathbf{x}, z, t)] = -\frac{\partial p(\mathbf{x}, z, t)}{\partial z},\tag{7}
$$

$$\frac{1}{K\_{\mathfrak{a}}} \frac{\partial p(\mathbf{x}, z, t)}{\partial t} + \frac{2(\mathbf{y} - \mathbf{1})}{K\_{\mathfrak{a}} \Lambda'} \left(\frac{\eta}{\rho\_f P\_r}\right)^{1/2} D^{1/2} [p(\mathbf{x}, z, t)] = -\frac{\partial v\_{\mathbf{x}}(\mathbf{x}, z, t)}{\partial \mathbf{x}} - \frac{\partial v\_z(\mathbf{x}, z, t)}{\partial z}. \tag{8}$$

In the region x ≤ 0, the incident pressure wave is given by

$$p^i(\mathbf{x}, z, t) = p^i \left( t - \frac{\mathbf{x} \cos \theta}{c\_0} - \frac{z \sin \theta}{c\_0} \right), \tag{9}$$

where c<sup>0</sup> is the velocity of the free fluid (x ≤ 0); c<sup>0</sup> ¼ ffiffiffiffiffiffiffiffiffiffiffiffi Ka=r<sup>f</sup> q . In the region 0 ≤ x ≤ L, the pressure wave is given by

$$p(\mathbf{x}, z, t) = p\left(t - \frac{\mathbf{x}\cos\theta'}{c'} - \frac{z\sin\theta'}{c'}\right),\tag{10}$$

Figure 2. Problem geometry.

where c<sup>0</sup> is the velocity in the porous material (0 ≤ x ≤ L), and the refraction angle θ<sup>0</sup> is given by the Descartes-Snell law [27, 28]:

$$\frac{\sin \theta}{c\_0} = \frac{\sin \theta'}{c'} \tag{11}$$

To simplify the system of Eq. (5), we can then use the following property

$$\frac{\partial}{\partial z} = -\frac{\sin \theta}{c\_0} \frac{\partial}{\partial t}'\tag{12}$$

which implies

where vx and vz are the components of particle velocity along axes x and z. In these equations, the integrals express the fractional derivatives, defined mathematically by convolutions and describe the dispersive nature of the porous material. They take into account the memory effects due to the fluid-structure interactions and to the viscous and thermal losses in the medium.

The problem geometry is shown in Figure 2. A homogeneous porous material occupies the region 0 ≤ x ≤ L. This medium is assumed to be isotropic and to have a rigid frame. A short sound pulse impinges at oblique incidence on the medium from the left, giving rise to an acoustic pressure field p xð Þ ; z; t and an acoustic velocity field vð Þ x; z; t within the material,

which satisfies the system of Eq. (5) that can be written as:

2r<sup>f</sup> α<sup>∞</sup> Λ

2r<sup>f</sup> α<sup>∞</sup> Λ

η r<sup>f</sup> Pr !<sup>1</sup>=<sup>2</sup>

η rf

η rf !<sup>1</sup>=<sup>2</sup>

ð Þ¼ <sup>x</sup>; <sup>z</sup>; <sup>t</sup> <sup>p</sup><sup>i</sup> <sup>t</sup> � <sup>x</sup> cos <sup>θ</sup>

p xð Þ¼ ; <sup>z</sup>; <sup>t</sup> p t � <sup>x</sup> cos <sup>θ</sup><sup>0</sup>

D<sup>1</sup>=<sup>2</sup>

!<sup>1</sup>=<sup>2</sup>

D<sup>1</sup>=<sup>2</sup>

D<sup>1</sup>=<sup>2</sup>

c0

� �

½ �¼� p xð Þ ; z; t

½ �¼� vxð Þ x; z; t

½ �¼� vzð Þ x; z; t

� <sup>z</sup> sin <sup>θ</sup> c0

.

ffiffiffiffiffiffiffiffiffiffiffiffi Ka=r<sup>f</sup> q

<sup>c</sup><sup>0</sup> � <sup>z</sup> sin <sup>θ</sup><sup>0</sup> c0

� �

∂p xð Þ ; z; t

∂p xð Þ ; z; t

<sup>∂</sup><sup>x</sup> � <sup>∂</sup>vzð Þ <sup>x</sup>; <sup>z</sup>; <sup>t</sup>

, (9)

, (10)

∂vxð Þ x; z; t

<sup>∂</sup><sup>x</sup> , (6)

<sup>∂</sup><sup>z</sup> , (7)

<sup>∂</sup><sup>z</sup> : (8)

∂vxð Þ x; z; t ∂t þ

∂vzð Þ x; z; t ∂t þ

> 2ð Þ γ � 1 KaΛ<sup>0</sup>

In the region x ≤ 0, the incident pressure wave is given by

pi

where c<sup>0</sup> is the velocity of the free fluid (x ≤ 0); c<sup>0</sup> ¼

In the region 0 ≤ x ≤ L, the pressure wave is given by

r<sup>f</sup> α<sup>∞</sup>

114 Porosity - Process, Technologies and Applications

r<sup>f</sup> α<sup>∞</sup>

∂p xð Þ ; z; t ∂t þ

1 Ka

Figure 2. Problem geometry.

$$\frac{\partial v\_z}{\partial z} = -\frac{\sin \theta}{c\_0} \frac{\partial v\_z}{\partial t} \quad \text{and} \quad \frac{\partial p}{\partial z} = -\frac{\sin \theta}{c\_0} \frac{\partial p}{\partial t} . \tag{13}$$

From Eqs. (7) and (13), we obtain the relation

$$\frac{\partial v\_z}{\partial z} = -\frac{\sin^2 \theta}{c\_0^2} \left(\rho\_f \alpha\_o + \frac{2\rho\_f \alpha\_o}{\Lambda} \left(\frac{\eta}{\rho\_f}\right)^{1/2} D^{-1/2}\right)^{-1} \ast \frac{\partial p}{\partial t}.\tag{14}$$

By using Eqs. (6), (8) and (14), the equation systems (6), (7) and (8) can thus be simplified to

$$
\rho\_f \alpha\_\ast \frac{\partial v\_x(\mathbf{x}, z, t)}{\partial t} + \frac{2\rho\_f \alpha\_\ast}{\Lambda} \left(\frac{\eta}{\rho\_f}\right)^{1/2} D^{1/2} [v\_x(\mathbf{x}, z, t)] = -\frac{\partial p(\mathbf{x}, z, t)}{\partial \mathbf{x}},\tag{15}
$$

$$\begin{split} \frac{1}{K\_{a}} \frac{\partial p(\mathbf{x}, z, t)}{\partial t} + \frac{\mathcal{D}(\mathbf{y} - 1)}{K\_{a} \Lambda'} \left( \frac{\eta}{\rho\_{f} P\_{r}} \right)^{1/2} D^{1/2} [p(\mathbf{x}, z, t)] &= \\ -\frac{\partial v\_{\mathbf{x}}(\mathbf{x}, z, t)}{\partial \mathbf{x}} + \frac{\sin^{2} \theta}{c\_{0}^{2}} \left( \rho\_{f} \alpha\_{\text{ov}} + \frac{2 \rho\_{f} \alpha\_{\text{ov}}}{\Lambda} \left( \frac{\eta}{\rho\_{f}} \right)^{1/2} D^{-1/2} \right)^{-1} \ast \frac{\partial p(\mathbf{x}, z, t)}{\partial t} . \end{split} \tag{16}$$

From Eqs. (15) and (16), we derive the fractional propagation wave equation in the time domain along the x-axis

$$\frac{\partial^2 p(\mathbf{x}, z, t)}{\partial \mathbf{x}^2} - \frac{1}{c\_0^2} (a\_\mathbf{w} - \sin^2 \theta) \frac{\partial^2 p(\mathbf{x}, z, t)}{\partial t^2} - \frac{2a\_\mathbf{w}}{\mathcal{K}\_d} \sqrt{\frac{\rho\_f \eta}{\pi}} \left( \frac{1}{\Lambda} + \frac{\gamma - 1}{\sqrt{\mathbf{Pr}} \Lambda'} \right) \mathcal{D}^{3/2} [p(\mathbf{x}, z, t)] = \mathbf{0}, \tag{17}$$

The solution of the wave Eq. (17) with suitable initial and boundary conditions is by using the Laplace transform. F is the medium's Green function [12, 20] given by

$$F(t,k) = \begin{cases} 0 & \text{if } \quad 0 \le t \le k \\\\ \Xi(t) + \Delta \int\_0^{t-k} h(t,\xi) d\xi & \text{if } \quad t \ge k \end{cases} \tag{18}$$

$$\Xi(t) = \frac{b'}{4\sqrt{\pi}} \frac{k}{(t-k)^{3/2}} \exp\left(-\frac{b'^2 k^2}{16(t-k)}\right),\tag{19}$$

the first reflections at interfaces x ¼ 0 and x ¼ L, the expression of the reflection operator will

� �δð Þ<sup>t</sup> and <sup>ℜ</sup>ð Þ¼� <sup>t</sup>; <sup>θ</sup> <sup>4</sup>Eð Þ <sup>1</sup> � <sup>E</sup>

where r tð Þ ; θ is the instantaneous response of the porous material corresponding to the reflection contribution at the first interface (x ¼ 0). ℜð Þ t; θ is equivalent to reflection at the interface x ¼ L, which is the bulk contribution to reflection. The part of the wave corresponding to r tð Þ ; θ

Figure 3 shows the variation of the reflection coefficient at the first interface r with the incident angle θ, for a fixed porosity value ϕ ¼ 0:9, and for different values of tortuosity α<sup>∞</sup> ¼ 1:99 (solid line), α<sup>∞</sup> ¼ 1:75 (star), α<sup>∞</sup> ¼ 1:5 (dashed line), α<sup>∞</sup> ¼ 1:24 (dash-dot line) and α<sup>∞</sup> ¼ 1 (circle).

Figure 4 shows the variation of r with the incident angle, for a fixed tortuosity of α<sup>∞</sup> ¼ 1:1, and for different values of porosity ϕ ¼ 0:99 (solid line), ϕ ¼ 0:75 (star), ϕ ¼ 0:50 (dashed line),

Figure 3. Variation of the reflection coefficient at the first interface r with the incident angle θ, for a fixed porosity value ϕ ¼ 0:9, and for different values of tortuosity α<sup>∞</sup> ¼ 1:99 (solid line), α<sup>∞</sup> ¼ 1:75 (star), α<sup>∞</sup> ¼ 1:5 (dashed line), α<sup>∞</sup> ¼ 1:24

is not subjected to dispersion but simply multiplied by the factor 1ð Þ � E =ð Þ 1 þ E .

The reflection coefficient at the first interface vanishes for a critical angle θ<sup>c</sup>

r tð Þ¼ ; θ 0 ) sin θ<sup>c</sup> ¼

R t <sup>~</sup> ð Þ¼ ; <sup>θ</sup> r tð Þþ ; <sup>θ</sup> <sup>ℜ</sup>ð Þ <sup>t</sup>; <sup>θ</sup> , (25)

ð Þ <sup>1</sup> <sup>þ</sup> <sup>E</sup> <sup>3</sup> F t;

:

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>α</sup><sup>∞</sup> <sup>α</sup><sup>∞</sup> � <sup>ϕ</sup><sup>2</sup> � � α2 <sup>∞</sup> � <sup>ϕ</sup><sup>2</sup>

s

2L c

Ultrasound Measuring of Porosity in Porous Materials http://dx.doi.org/10.5772/intechopen.72696

� �, (26)

117

be given by

r tð Þ¼ ; <sup>θ</sup> <sup>1</sup> � <sup>E</sup>

ϕ ¼ 0:25 (dash-dot line) and ϕ ¼ 0:01 (circle).

(dash-dot line) and α<sup>∞</sup> ¼ 1 (circle).

1 þ E

with

where hð Þ τ; ξ has the following form:

$$h(\xi,\tau) = -\frac{1}{4\pi^{3/2}} \frac{1}{\sqrt{\left(\pi-\xi\right)^2 - k^2}} \frac{1}{\xi^{3/2}} \int\_{-1}^{1} \exp\left(-\frac{\chi\left(\mu,\tau,\xi\right)}{2}\right) \left(\chi\left(\mu,\tau,\xi\right) - 1\right) \frac{\mu d\mu}{\sqrt{1-\mu^2}} \tag{20}$$

$$\text{and } \chi(\mu, \tau, \xi) = \left(\Delta\mu\sqrt{\left(\tau-\xi\right)^2 - k^2} + b'(\tau-\xi)\right)^2 / 8\xi,\\ b' = Bc\_0^2\sqrt{\pi} \text{ and } \Delta = b'^2.$$

If the incident sound wave is launched in region x ≤ 0, then the expression of the pressure field in the region on the left of the material is the sum of the incident and reflected fields

$$p\_1(\mathbf{x}, t, \theta) = p^i \left( t - \frac{\mathbf{x} \cos \theta}{c\_0} \right) + p^r \left( t + \frac{\mathbf{x} \cos \theta}{c\_0} \right), \qquad \mathbf{x} < 0. \tag{21}$$

Here, <sup>p</sup>1ð Þ <sup>x</sup>; <sup>t</sup>; <sup>θ</sup> is the field in the region <sup>x</sup> <sup>&</sup>lt; 0, pi and <sup>p</sup><sup>r</sup> denote the incident and reflected fields, respectively.

The incident and reflected fields are related by the scattering operator (i.e., the reflection operator) for the material. This is an integral operator represented by

$$p^r(\mathbf{x}, t, \theta) = \int\_0^t \tilde{R}(\tau, \theta) p^i \left(t - \tau + \frac{\mathbf{x} \cos \theta}{c\_0} \right) d\tau,\tag{22}$$

In Eq. (22), the function R~ is the reflection kernel for incidence from the left. Note that the lower limit of integration in Eq. (22) is set to 0, which is equivalent to assuming that the incident wavefront impinges on the material at t ¼ 0:

Expression of the reflection-scattering operator taking into account the n-multiple reflections in the material is given by

$$\tilde{R}(t,\theta) = \left(\frac{1-E}{1+E}\right) \sum\_{n\geq 0} \left(\frac{1-E}{1+E}\right)^{2n} \left[F\left(t, 2n\frac{L}{c}\right) - F\left(t, \left(2n+2\right)\frac{L}{c}\right)\right].\tag{23}$$

with

$$E = \frac{\phi \sqrt{1 - \frac{\sin^2 \theta}{a\_{\ast}}}}{\sqrt{a\_{\ast}} \cos \theta},\tag{24}$$

Generally [21], in air-saturated porous materials, acoustic damping is very important, and the multiple reflections are thus negligible inside the material. So, by taking into account only

with

the first reflections at interfaces x ¼ 0 and x ¼ L, the expression of the reflection operator will be given by

$$
\ddot{R}(t,\theta) = r(t,\theta) + \mathcal{R}(t,\theta), \tag{25}
$$

with

with

ΞðÞ¼ t

1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ <sup>τ</sup> � <sup>ξ</sup> <sup>2</sup> � <sup>k</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ <sup>τ</sup> � <sup>ξ</sup> <sup>2</sup> � <sup>k</sup>

<sup>p</sup>1ð Þ¼ <sup>x</sup>; <sup>t</sup>; <sup>θ</sup> pi <sup>t</sup> � <sup>x</sup> cos <sup>θ</sup>

operator) for the material. This is an integral operator represented by

ðt 0

ð Þ¼ x; t; θ

pr

wavefront impinges on the material at t ¼ 0:

<sup>~</sup> ð Þ¼ ; <sup>θ</sup> <sup>1</sup> � <sup>E</sup>

1 þ E � �X

n ≥ 0

1 � E 1 þ E � �<sup>2</sup><sup>n</sup>

> E ¼ ϕ

the material is given by

with

R t

where hð Þ τ; ξ has the following form:

116 Porosity - Process, Technologies and Applications

1 4π<sup>3</sup>=<sup>2</sup>

q

q

hð Þ¼� ξ; τ

and χ μ; <sup>τ</sup>; <sup>ξ</sup> � � <sup>¼</sup> <sup>Δ</sup><sup>μ</sup>

respectively.

b0 4 ffiffiffi <sup>π</sup> <sup>p</sup>

2

2

� �<sup>2</sup>

þ b<sup>0</sup>

ð Þ τ � ξ

in the region on the left of the material is the sum of the incident and reflected fields

c0 � �

If the incident sound wave is launched in region x ≤ 0, then the expression of the pressure field

Here, <sup>p</sup>1ð Þ <sup>x</sup>; <sup>t</sup>; <sup>θ</sup> is the field in the region <sup>x</sup> <sup>&</sup>lt; 0, pi and <sup>p</sup><sup>r</sup> denote the incident and reflected fields,

The incident and reflected fields are related by the scattering operator (i.e., the reflection

<sup>R</sup><sup>~</sup> ð Þ <sup>τ</sup>; <sup>θ</sup> pi <sup>t</sup> � <sup>τ</sup> <sup>þ</sup>

In Eq. (22), the function R~ is the reflection kernel for incidence from the left. Note that the lower limit of integration in Eq. (22) is set to 0, which is equivalent to assuming that the incident

Expression of the reflection-scattering operator taking into account the n-multiple reflections in

F t; 2n

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � sin <sup>2</sup><sup>θ</sup> α<sup>∞</sup>

q

ffiffiffiffiffiffi α<sup>∞</sup>

Generally [21], in air-saturated porous materials, acoustic damping is very important, and the multiple reflections are thus negligible inside the material. So, by taking into account only

L c � �

� F t;ð Þ <sup>2</sup><sup>n</sup> <sup>þ</sup> <sup>2</sup> <sup>L</sup>

<sup>p</sup> cos <sup>θ</sup> , (24)

� � � �

c

, (23)

<sup>þ</sup> pr <sup>t</sup> <sup>þ</sup>

1 ξ<sup>3</sup>=<sup>2</sup> ð1 �1

k

ð Þ <sup>t</sup> � <sup>k</sup> <sup>3</sup>=<sup>2</sup> exp � <sup>b</sup>0<sup>2</sup>

exp � χ μ; <sup>τ</sup>; <sup>ξ</sup> � � 2 � �

<sup>=</sup>8ξ, <sup>b</sup><sup>0</sup> <sup>¼</sup> Bc<sup>2</sup>

x cos θ c0 � �

> x cos θ c0

� �

0 ffiffiffi

k 2 16ð Þ t � k

, (19)

<sup>1</sup> � <sup>μ</sup><sup>2</sup> <sup>p</sup> (20)

χ μ; <sup>τ</sup>; <sup>ξ</sup> � � � <sup>1</sup> � � <sup>μ</sup>d<sup>μ</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffi

.

, x < 0: (21)

dτ, (22)

<sup>π</sup> <sup>p</sup> and <sup>Δ</sup> <sup>¼</sup> <sup>b</sup>0<sup>2</sup>

!

$$r(t, \theta) = \left(\frac{1 - E}{1 + E}\right) \delta(t) \quad \text{and} \quad \Re(t, \theta) = -\frac{4E(1 - E)}{\left(1 + E\right)^3} F\left(t, \frac{2L}{c}\right),\tag{26}$$

where r tð Þ ; θ is the instantaneous response of the porous material corresponding to the reflection contribution at the first interface (x ¼ 0). ℜð Þ t; θ is equivalent to reflection at the interface x ¼ L, which is the bulk contribution to reflection. The part of the wave corresponding to r tð Þ ; θ is not subjected to dispersion but simply multiplied by the factor 1ð Þ � E =ð Þ 1 þ E .

The reflection coefficient at the first interface vanishes for a critical angle θ<sup>c</sup>

$$r(t, \theta) = 0 \Rightarrow \sin \theta\_c = \sqrt{\frac{\alpha\_\infty \left(\alpha\_\infty - \phi^2\right)}{\alpha\_\infty^2 - \phi^2}}.$$

Figure 3 shows the variation of the reflection coefficient at the first interface r with the incident angle θ, for a fixed porosity value ϕ ¼ 0:9, and for different values of tortuosity α<sup>∞</sup> ¼ 1:99 (solid line), α<sup>∞</sup> ¼ 1:75 (star), α<sup>∞</sup> ¼ 1:5 (dashed line), α<sup>∞</sup> ¼ 1:24 (dash-dot line) and α<sup>∞</sup> ¼ 1 (circle).

Figure 4 shows the variation of r with the incident angle, for a fixed tortuosity of α<sup>∞</sup> ¼ 1:1, and for different values of porosity ϕ ¼ 0:99 (solid line), ϕ ¼ 0:75 (star), ϕ ¼ 0:50 (dashed line), ϕ ¼ 0:25 (dash-dot line) and ϕ ¼ 0:01 (circle).

Figure 3. Variation of the reflection coefficient at the first interface r with the incident angle θ, for a fixed porosity value ϕ ¼ 0:9, and for different values of tortuosity α<sup>∞</sup> ¼ 1:99 (solid line), α<sup>∞</sup> ¼ 1:75 (star), α<sup>∞</sup> ¼ 1:5 (dashed line), α<sup>∞</sup> ¼ 1:24 (dash-dot line) and α<sup>∞</sup> ¼ 1 (circle).

where <sup>p</sup><sup>r</sup> <sup>x</sup>; <sup>θ</sup><sup>i</sup> ð Þ ; ti represents the discrete set of values of the experimental reflected signal for different incident angles <sup>θ</sup>i, <sup>r</sup> <sup>θ</sup><sup>i</sup> ð Þ ; ti is the reflection coefficient at the first interface, and <sup>p</sup><sup>i</sup> <sup>x</sup>; <sup>θ</sup><sup>i</sup> ð Þ ; ti is the experimental incident signal. The term <sup>r</sup> <sup>θ</sup><sup>i</sup> ð Þ ; ti <sup>∗</sup>pi <sup>x</sup>; <sup>θ</sup><sup>i</sup> ð Þ ; ti represents the predicted

Ultrasound Measuring of Porosity in Porous Materials http://dx.doi.org/10.5772/intechopen.72696 119

Let us consider three samples of plastic foam M1, M2, and M3. Their tortuosity and porosity were measured using classical methods [22–31] (Table 1). We solved the inverse problem for these samples via waves reflected at the first interface and for different incident angles.

Experiments were performed in air with two broadband Ultran NCT202 transducers with a 190 kHz central frequency in air and a 6 dB bandwidth extending from 150 to 230 kHz. Pulses of 400 V were provided by a 5052PR panametrics pulser/receiver. An optical goniometer was used to position the transducers. The received signals were amplified to 90 dB and filtered above 1 MHz to avoid high-frequency noise. Electronic interference was removed by 1000

Figures 6 and 7 show a variation of the cost function, U, with tortuosity and porosity, for samples M1, M2, and M3, respectively. The reconstructed values of porosity and tortuosity corresponding to the positions of the minima of these cost functions are given in Table 2.

Figure 5. Experimental set-up of the ultrasonic measurements in reflected mode. PG: pulse generator; HFFPA: high-

frequency filtering-pre-amplifier; DO: digital oscilloscope; C: computer; T: transducer; S: sample.

Material M1 M2 M3 Tortuosity 1.1 1.1 1.5 Porosity 0.95 0.99 0.86

reflected signal. The inverse problem is solved numerically by the least-square method.

acquisition averages. The experimental setup is shown in Figure 5.

Table 1. Measured values of porosity and tortuosity by classical methods [22–31].

Figure 4. Variation of the reflection coefficient at the first interface with the incident angle, for a fixed tortuosity value α<sup>∞</sup> ¼ 1:1, and for different values of porosity ϕ ¼ 0:99 (solid line), ϕ ¼ 0:75 (star), ϕ ¼ 0:50 (dashed line), ϕ ¼ 0:25 (dashdot line) and ϕ ¼ 0:01 (circle).

When the incident angle is θ < θc, the reflection coefficient decreases slowly with the incident angle, and when it is θ > θc, the reflection coefficient increases quickly with the angle. It can be seen also from Figures 2 and 3 that the sensitivity of porosity variation is more important than the sensitivity of tortuosity on the reflection coefficient at the first interface.

### 3. Inverse problem

The inverse problem is solved using transmitted waves and an estimation of the tortuosity, viscous and thermal characteristic length is given. However, the porosity cannot be inverted in transmission since its sensitivity is low. In this work, we determine porosity and tortuosity by solving the inverse problem for waves reflected by the first interface and by taking into account experimental data concerning all measured incident angles (Figure 4). The propagation of acoustic waves in a slab of porous material in the high-frequency asymptotic domain is characterized by four parameters: porosity ϕ, tortuosity α∞, viscous characteristic length Λ, and thermal characteristic length Λ<sup>0</sup> , the values of which are crucial to the behavior of sound waves in such materials. It is of some importance to work out new experimental methods and efficient tools for their estimation. The basic inverse problem associated with the slab may be stated as follows: from the measurement of the signals transmitted and/or reflected outside the slab, find the values of the medium's parameters. The inverse problem is to find values of parameters ϕ, α∞, which minimize the function:

$$\mathcal{U}\{\phi, a\_{\ast\ast}\} = \sum\_{\theta\_i} \sum\_{t\_i} \left[ p^r(\mathbf{x}, \theta\_i, t\_i) - r(\theta\_i, t\_i) \ast p^i(\mathbf{x}, \theta\_i, t\_i) \right]^2 \tag{27}$$

where <sup>p</sup><sup>r</sup> <sup>x</sup>; <sup>θ</sup><sup>i</sup> ð Þ ; ti represents the discrete set of values of the experimental reflected signal for different incident angles <sup>θ</sup>i, <sup>r</sup> <sup>θ</sup><sup>i</sup> ð Þ ; ti is the reflection coefficient at the first interface, and <sup>p</sup><sup>i</sup> <sup>x</sup>; <sup>θ</sup><sup>i</sup> ð Þ ; ti is the experimental incident signal. The term <sup>r</sup> <sup>θ</sup><sup>i</sup> ð Þ ; ti <sup>∗</sup>pi <sup>x</sup>; <sup>θ</sup><sup>i</sup> ð Þ ; ti represents the predicted reflected signal. The inverse problem is solved numerically by the least-square method.

Let us consider three samples of plastic foam M1, M2, and M3. Their tortuosity and porosity were measured using classical methods [22–31] (Table 1). We solved the inverse problem for these samples via waves reflected at the first interface and for different incident angles.

Experiments were performed in air with two broadband Ultran NCT202 transducers with a 190 kHz central frequency in air and a 6 dB bandwidth extending from 150 to 230 kHz. Pulses of 400 V were provided by a 5052PR panametrics pulser/receiver. An optical goniometer was used to position the transducers. The received signals were amplified to 90 dB and filtered above 1 MHz to avoid high-frequency noise. Electronic interference was removed by 1000 acquisition averages. The experimental setup is shown in Figure 5.

Figures 6 and 7 show a variation of the cost function, U, with tortuosity and porosity, for samples M1, M2, and M3, respectively. The reconstructed values of porosity and tortuosity corresponding to the positions of the minima of these cost functions are given in Table 2.


Table 1. Measured values of porosity and tortuosity by classical methods [22–31].

When the incident angle is θ < θc, the reflection coefficient decreases slowly with the incident angle, and when it is θ > θc, the reflection coefficient increases quickly with the angle. It can be seen also from Figures 2 and 3 that the sensitivity of porosity variation is more important than

Figure 4. Variation of the reflection coefficient at the first interface with the incident angle, for a fixed tortuosity value α<sup>∞</sup> ¼ 1:1, and for different values of porosity ϕ ¼ 0:99 (solid line), ϕ ¼ 0:75 (star), ϕ ¼ 0:50 (dashed line), ϕ ¼ 0:25 (dash-

The inverse problem is solved using transmitted waves and an estimation of the tortuosity, viscous and thermal characteristic length is given. However, the porosity cannot be inverted in transmission since its sensitivity is low. In this work, we determine porosity and tortuosity by solving the inverse problem for waves reflected by the first interface and by taking into account experimental data concerning all measured incident angles (Figure 4). The propagation of acoustic waves in a slab of porous material in the high-frequency asymptotic domain is characterized by four parameters: porosity ϕ, tortuosity α∞, viscous characteristic length Λ, and

in such materials. It is of some importance to work out new experimental methods and efficient tools for their estimation. The basic inverse problem associated with the slab may be stated as follows: from the measurement of the signals transmitted and/or reflected outside the slab, find the values of the medium's parameters. The inverse problem is to find values of

, the values of which are crucial to the behavior of sound waves

, (27)

pr <sup>x</sup>; <sup>θ</sup><sup>i</sup> ð Þ� ; ti <sup>r</sup> <sup>θ</sup><sup>i</sup> ð Þ ; ti <sup>∗</sup>pi <sup>x</sup>; <sup>θ</sup><sup>i</sup> ð Þ ; ti � �<sup>2</sup>

the sensitivity of tortuosity on the reflection coefficient at the first interface.

3. Inverse problem

dot line) and ϕ ¼ 0:01 (circle).

118 Porosity - Process, Technologies and Applications

thermal characteristic length Λ<sup>0</sup>

parameters ϕ, α∞, which minimize the function:

U ϕ; α<sup>∞</sup> � � <sup>¼</sup> <sup>X</sup>

θi

X ti

Figure 5. Experimental set-up of the ultrasonic measurements in reflected mode. PG: pulse generator; HFFPA: highfrequency filtering-pre-amplifier; DO: digital oscilloscope; C: computer; T: transducer; S: sample.

A comparison is given in Figures 8–10, between simulated reflection coefficient at the first interface using reconstructed values of porosity and tortuosity (solid line) and experimental data of the reflection coefficient at the first interface (circle) for plastic foams M1, M2, and M3, respectively.

Ultrasound Measuring of Porosity in Porous Materials http://dx.doi.org/10.5772/intechopen.72696 121

The correspondence between experiment and theory is good, which leads us to conclude that this method based on the solution of the inverse problem is appropriate for estimating the porosity and tortuosity of porous materials with a rigid frame. This method is very interesting

Figure 9. Comparison between simulated reflection coefficient at the first interface using reconstructed values of the porosity and tortuosity (solid line) and experimental data of the reflection coefficient at the first interface (circle) for the

Figure 10. Comparison between simulated reflection coefficient at the first interface using reconstructed values of the porosity and tortuosity (solid line) and experimental data of the reflection coefficient at the first interface (circle) for the

comparing the classical one using non-acoustic waves.

plastic foam M2.

plastic foam M3.

Figure 6. Variation of the cost function U with tortuosity for the plastic foams M1, M2, and M3.

Figure 7. Variation of the cost function U with porosity for the plastic foams M1, M2, and M3.


Table 2. Reconstructed values of porosity and tortuosity by solving the inverse problem.

Figure 8. Comparison between simulated reflection coefficient at the first interface using reconstructed values of the porosity and tortuosity (solid line) and experimental data of the reflection coefficient at the first interface (circle) for the plastic foam M1.

A comparison is given in Figures 8–10, between simulated reflection coefficient at the first interface using reconstructed values of porosity and tortuosity (solid line) and experimental data of the reflection coefficient at the first interface (circle) for plastic foams M1, M2, and M3, respectively.

The correspondence between experiment and theory is good, which leads us to conclude that this method based on the solution of the inverse problem is appropriate for estimating the porosity and tortuosity of porous materials with a rigid frame. This method is very interesting comparing the classical one using non-acoustic waves.

Figure 6. Variation of the cost function U with tortuosity for the plastic foams M1, M2, and M3.

120 Porosity - Process, Technologies and Applications

Figure 7. Variation of the cost function U with porosity for the plastic foams M1, M2, and M3.

Table 2. Reconstructed values of porosity and tortuosity by solving the inverse problem.

plastic foam M1.

Material M1 M2 M3 Tortuosity 1.12 1.1 1.6 Porosity 0.96 0.99 0.85

Figure 8. Comparison between simulated reflection coefficient at the first interface using reconstructed values of the porosity and tortuosity (solid line) and experimental data of the reflection coefficient at the first interface (circle) for the

Figure 9. Comparison between simulated reflection coefficient at the first interface using reconstructed values of the porosity and tortuosity (solid line) and experimental data of the reflection coefficient at the first interface (circle) for the plastic foam M2.

Figure 10. Comparison between simulated reflection coefficient at the first interface using reconstructed values of the porosity and tortuosity (solid line) and experimental data of the reflection coefficient at the first interface (circle) for the plastic foam M3.
