Improving acoustic wave propagation models in highly attenuating porous materials.

This article presents an improved and extended modeling approach for acoustic wave propagation in rigid porous materials, focusing on examples, such as plastic foams used for noise reduction in automotive applications. We demonstrate that the classical model (Johnson-Champoux-Allard) in the asymptotic high-frequency limit, widely employed in the literature, fails to accurately reconstruct the transmitted acoustic signal through high absorbent porous materials characterized by significant wave attenuation. The study focuses on the airborne ultrasonic frequency range (30-200 kHz). To address this limitation, we introduce new non-acoustic parameters Σ and V for viscous effects, and Σ' and V' for thermal effects, with surface and volumetric dimensions, respectively, allowing for the reconstruction of the transmitted signal and accurate modeling of the pronounced acoustic attenuation within the material. These parameters are incorporated into the expansion on skin depths of the dynamic tortuosity α(ω) and thermal tortuosity α' (ω) response functions, which describe the inertial-viscous and thermal interactions between the fluid and the solid, respectively. This novel modeling approach enables a more comprehensive study of high attenuating porous media, which are crucial for effective noise reduction. Additionally, it opens up new possibilities for characterization beyond the capabilities of current models.

Inverse identification of a higher order viscous parameter of rigid porous media in the high frequency domain.

In this paper, a modeling extension for the description of wave propagation in rigid porous media at high frequencies is used. To better characterize the visco-inertial and thermal interactions between the fluid and the structure in this regime, two additional characteristic viscous and thermal surfaces Σ and Σ' are taken into account, as initially introduced in Kergomard, Lafarge, and Gilbert [Acta Acust. Acust. 99(4), 557-571 (2013)]. This extends the modeling order of the dynamic tortuosity and compressibility. A sensitivity analysis is performed on the additional parameters, showing that only the viscous surface Σ has an influence on transmitted waves in the high frequency regime, for materials having a low viscous characteristic length. A general Bayesian inference is then conducted to infer simultaneously the posterior probability densities of the parameters associated with the visco-inertial effects, i.e., the porosity, tortuosity, the viscous characteristic length, and the viscous characteristic surface. The proposed method is based on the measurement of waves transmitted by a slab of rigid porous material in the time domain. Bayesian inference results obtained on three different porous materials are presented.

Bayesian inference for the ultrasonic characterization of rigid porous materials using reflected waves by the first interface.

The purpose of this paper is to present a method for the ultrasonic characterization of air-saturated porous media, by solving the inverse problem using only the reflected waves from the first interface to infer the porosity, the tortuosity, and the viscous and thermal characteristic lengths. The solution of the inverse problem relies on the use of different reflected pressure signals obtained under multiple obliquely incident waves, in the time domain. In this paper, the authors propose to solve the inverse problem numerically with a first level Bayesian inference method, summarizing the authors' knowledge on the inferred parameters in the form of posterior probability densities, exploring these densities using a Markov-Chain Monte-Carlo approach. Despite their low sensitivity to the reflection coefficient, it is still possible to extract the knowledge of the viscous and thermal characteristic lengths, allowing the simultaneous determination of all the physical parameters involved in the expression of the reflection operator. To further constrain the problem and guide the inference, the knowledge of a particular incident angle is used at one's advantage in order to more precisely define the thermal length, by effectively yielding a statistical relationship between tortuosity and characteristic length ratio.

Acoustical modeling and Bayesian inference for rigid porous media in the low-mid frequency regime.

In this article, a modeling extension for the description of wave propagation in porous media at low-mid frequencies is introduced. To better characterize the viscous and inertial interactions between the fluid and the structure in this regime, two additional terms described by two parameters α 1 and α 2 are taken into account in the representation of the dynamic tortuosity in a Laurent-series on frequency. The model limitations are discussed. A sensitivity analysis is performed, showing that the influence of α 1 and α 2 on the acoustic response of porous media is significant. A general Bayesian inference is then conducted to infer, simultaneously, the posterior probability densities of the model parameters. The proposed method is based on the measurement of waves transmitted by a slab of rigid porous material, using a temporal model for the direct and inverse transmission problem. Bayesian inference results obtained on three different porous materials are presented, which suggests that the two additional parameters are accessible and help reduce systematic errors in the identification of other parameters: porosity, static viscous permeability, static viscous tortuosity, static thermal permeability, and static thermal tortuosity.

Measuring static viscous permeability of porous absorbing materials.

Conventional acoustical methods for measuring the permeability or flow resistivity of a porous material require a priori estimation of the porosity. In this work, an acoustical method is presented in which a simplified expression (independent of both the frequency and porosity) for the transmitted waves at the Darcy's regime (low frequency range) is derived, and used for the inverse determination of both the viscous static permeability (or flow resistivity) and the thickness of air-saturated porous materials. The inverse problem is solved based on the least-square numerical method using experimental transmitted waves in time domain. Tests are performed using industrial plastic foams. Experimental and numerical validation results of this method are presented, which show the advantage of measuring the viscous permeability and thickness of a porous slab, without the required prior knowledge of the porosity, but by simply using the transmitted waves.

Transient ultrasound propagation in porous media using Biot theory and fractional calculus: application to human cancellous bone.

A temporal model based on the Biot theory is developed to describe the transient ultrasonic propagation in porous media with elastic structure, in which the viscous exchange between fluid and structure are described by fractional derivatives. The fast and slow waves obey a fractional wave equation in the time domain. The solution of Biot's equations in time depends on the Green functions of each of the waves (fast and slow), and their fractional derivatives. The reflection and transmission operators for a slab of porous materials are derived in the time domain, using calculations in the Laplace domain. Their analytical expressions, depend on Green's function of fast and slow waves. Experimental results for slow and fast waves transmitted through human cancellous bone samples are given and compared with theoretical predictions.

Generalized equation for transient-wave propagation in continuous inhomogeneous rigid-frame porous materials at low frequencies.

This paper provides a temporal model for the propagation of transient acoustic waves in continuous inhomogeneous isotropic porous material having a rigid frame at low frequency range. A temporal equivalent fluid model, in which the acoustic wave propagates only in the fluid saturating the material, is considered. In this model, the inertial effects are described by the inhomogeneous inertial factor [A. N. Norris, J. Wave Mat. Interact. 1, 365-380 (1986)]. The viscous and thermal losses of the medium are described by two inhomogeneous susceptibility kernels which depend on the viscous and thermal permeabilities. The medium is one-dimensional and its physical parameters (porosity, inertial factor, viscous, and thermal permeabilities) are depth dependent. A generalized wave propagation equation in continuous inhomogeneous material is established and discussed.

Scattering of acoustic waves by macroscopically inhomogeneous poroelastic tubes.

Wave propagation in macroscopically inhomogeneous porous materials has received much attention in recent years. For planar configurations, the wave equation, derived from the alternative formulation of Biot's theory of 1962, was reduced and solved recently: first in the case of rigid frame inhomogeneous porous materials and then in the case of inhomogeneous poroelastic materials in the framework of Biot's theory. This paper focuses on the solution of the full wave equation in cylindrical coordinates for poroelastic tubes in which the acoustic and elastic properties of the poroelastic tube vary in the radial direction. The reflection coefficient is obtained numerically using the state vector (or the so-called Stroh) formalism and Peano series. This coefficient can then be used to straightforwardly calculate the scattered field. To validate the method of resolution, results obtained by the present method are compared to those calculated by the classical transfer matrix method in the case of a two-layer poroelastic tube. As an example, a long bone excited in the sagittal plane is considered. Finally, a discussion is given of ultrasonic time domain scattered field for various inhomogeneity profiles, which could lead to the prospect of long bone characterization.

Measuring static thermal permeability and inertial factor of rigid porous materials (L).

An acoustic method based on sound transmission is proposed for deducing the static thermal permeability and the inertial factor of porous materials having a rigid frame at low frequencies. The static thermal permeability of porous material is a geometrical parameter equal to the inverse trapping constant of the solid frame [Lafarge et al., J. Acoust. Soc. Am. 102, 1995 (1997)] and is an important characteristic of the porous material. The inertial factor [Norris., J. Wave Mat. Interact. 1, 365 (1986)] describes the fluid structure interactions in the low frequency range (1-3 kHz). The proposed method is based on a temporal model of the direct and inverse scattering problems for the propagation of transient audible frequency waves in a homogeneous isotropic slab of porous material having a rigid frame. The static thermal permeability and the inertial factor are determined from the solution of the inverse problem. The minimization between experiment and theory is made in the time domain. Tests are performed using industrial plastic foams. Experimental and theoretical data are in good agreement. Furthermore, the prospects are discussed. This method has the advantage of being simple, rapid, and efficient.

Reconstruction of material properties profiles in one-dimensional macroscopically inhomogeneous rigid frame porous media in the frequency domain.

The present paper deals with the inverse scattering problem involving macroscopically inhomogeneous rigid frame porous media. It consists of the recovery, from acoustic measurements, of the profiles of spatially varying material parameters by means of an optimization approach. The resolution is based on the modeling of acoustic wave propagation in macroscopically inhomogeneous rigid frame porous materials, which was recently derived from the generalized Biot's theory. In practice, the inverse problem is solved by minimizing an objective function defined in the least-square sense by the comparison of the calculated reflection (and transmission) coefficient(s) with the measured or synthetic one(s), affected or not by additive Gaussian noise. From an initial guess, the profiles of the x-dependent material parameters are reconstructed iteratively with the help of a standard conjugate gradient method. The convergence rate of the latter and the accuracy of the reconstructions are improved by the availability of an analytical gradient.

Application of the biot model to ultrasound in bone: direct problem.

Ultrasonic wave propagation in human cancellous bone is considered using Biot's theory modified by the Johnson-Koplik-Dashen model for viscous exchange between fluid and structure. The transmission coefficient is derived for a slab of porous material. Experimental results for fast and slow waves transmitted through human cancellous bone samples are given and compared with theoretical predictions.

Application of the Biot model to ultrasound in bone: inverse problem.

This paper concerns the ultrasonic characterization of human cancellous bone samples by solving the inverse problem using experimentally measured signals. The inverse problem is solved numerically by the least squares method. Five parameters are inverted: porosity, tortuosity, viscous characteristic length, Young modulus, and Poisson ratio of the skeletal frame. The minimization of the discrepancy between experiment and theory is made in the time domain. The ultrasonic propagation in cancellous bone is modelled using the Biot theory modified by the Johnson-Koplik-Dashen model for viscous exchange between fluid and structure. The sensitivity of the Young modulus and the Poisson ratio of the skeletal frame is studied showing their effect on the fast and slow waveforms. The inverse problem is shown to be well posed, and its solution to be unique. Experimental results for slow and fast waves transmitted through human cancellous bone samples are given and compared with theoretical predictions.

Generalized hyperbolic fractional equation for transient-wave propagation in layered rigid-frame porous materials.

This paper provides a temporal model for the propagation of transient ultrasonic waves in a layered isotropic porous material having a rigid frame. A temporal equivalent fluid model is considered, in which the acoustic wave propagates only in the fluid saturating the material. In this model, the inertial effects are described by the layered tortuosity and the viscous and thermal losses of the medium are described by two layered susceptibility kernels which depend on the viscous and thermal characteristic lengths. The medium is one dimensional and its physical parameters (porosity, tortuosity, and characteristics lengths) are depth dependent. A generalized hyperbolic fractional equation for transient sound wave propagation in layered material is established.

Measuring permeability of porous materials at low frequency range via acoustic transmitted waves.

An acoustical transmission method is proposed for measuring permeability of porous materials having rigid frame. Permeability is one of the several parameters required by acoustical theory to characterize porous materials such as plastic foams and fibrous or granular materials. The proposed method is based on a temporal model of the direct and inverse scattering problem for the diffusion of transient low frequency waves in a homogeneous isotropic slab of porous material having a rigid frame. This time domain model of wave propagation was initially introduced by the authors [Z.E.A Fellah and C. Depollier, J. Acoust. Soc. Am. 107, 683 (2000)]. The viscous losses of the medium are described by the model devised by Johnson et al. [J. Fluid. Mech. 176, 379 (1987)]. Reflection and transmission scattering operators for a slab of porous material are derived from the responses of the medium to an incident acoustic pulse. The permeability is determined from the expressions of these operators. Experimental and numerical validation results of this method are presented. This method has the advantage of being simple, rapid, and efficient.

Ultrasonic characterization of human cancellous bone using the Biot theory: inverse problem.

This paper concerns the ultrasonic characterization of human cancellous bone samples by solving the inverse problem using experimental transmitted signals. The ultrasonic propagation in cancellous bone is modeled using the Biot theory modified by the Johnson et al. model for viscous exchange between fluid and structure. The sensitivity of the Young modulus and the Poisson ratio of the skeletal frame is studied showing their effect on the fast and slow wave forms. The inverse problem is solved numerically by the least squares method. Five parameters are inverted: the porosity, tortuosity, viscous characteristic length, Young modulus, and Poisson ratio of the skeletal frame. The minimization of the discrepancy between experiment and theory is made in the time domain. The inverse problem is shown to be well posed, and its solution to be unique. Experimental results for slow and fast waves transmitted through human cancellous bone samples are given and compared with theoretical predictions.

Propagation of mechanical waves in a one-dimensional nonlinear disordered lattice.

The propagation of transverse waves along a string loaded by masses, each of them being fixed to a spring with a quadratic nonlinearity, is studied. After presenting the nonlinear model and stating the equation of propagation into a lattice with discrete nonlinearities and disorder, we propose a perturbation approach to wave propagation in a nonlinear lattice using the Green's function formalism. We show how the nonlinearity acts on the propagation into a disordered lattice. In the low-frequency approximation, an analytical expression of the boundary between the propagative regime and the evanescent one is found. Numerical results are compared to the analytical results and phase diagrams are proposed in the ordered and disordered cases. A behavior of the transmission coefficient is found, on an empirical basis, as a function of the length of the lattice and the localization length in the nonlinear case. Finally, a dynamic approach is developed and the ordered and disordered cases are addressed. This method is based on a finite difference equation and allows the construction of the Poincaré section describing the propagation of the wave into the lattice. This approach distinguishes between the properties of propagation in the lattice in a propagative regime and in an evanescent one.

Measuring flow resistivity of porous materials at low frequencies range via acoustic transmitted waves.

An acoustic transmissivity method is proposed for measuring flow resistivity of porous materials having rigid frame. 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 proposed method is based on a temporal model of the direct and inverse scattering problem for the diffusion of transient low-frequency waves in a homogeneous isotropic slab of porous material having a rigid frame. The transmission scattering operator for a slab of porous material is derived from the response of the medium to an incident acoustic pulse. The flow resistivity is determined from the solution of the inverse problem. The minimization between experiment and theory is made in the time domain. Tests are performed using industrial plastic foams. Experimental and numerical results, and prospects are discussed.

Phase-space analysis of acoustics fields and its application to waveguide.

A description of two-dimensional acoustic fields by means of a joint "space-wave number" representation is discussed. A function defined in the phase-space domain (x,y,k(x),k(y)) is associated with a signal which is a function of spatial coordinates (x,y). This paper presents two methods to realize it. The first is to associate with each point (x,y) of the wave field a two-dimensional wave number spectrum (k(x),k(y)), called local spectrum. The second is to process by other coordinates the wave field along an arbitrary direction, introduced in quantum mechanics for the study of classical billiards, and provided by the Birkhoff variables (s,cos phi). Phase-space diagrams are given by quadratic phase-space distributions. Simulations are presented for wave fields in a 2D planar waveguide for a pedagogical point of view with Gaussian beam or point-source excitation, and nonuniform waveguides as a sudden area expansion chamber and an open billiard with a single incoming mode at the entrance of each of them. In these problems, local spectrum and Birkhoff analysis are used in order to identify the structures hidden in the field. The result is the contribution of different wave vectors which contribute to the field value at the analysis point or at a certain section of the boundary, and show complicated structure of the acoustic field like whispering gallery or diffracted waves.

Ultrasonic wave propagation in human cancellous bone: application of Biot theory.

Ultrasonic wave propagation in human cancellous bone is considered. Reflection and transmission coefficients are derived for a slab of cancellous bone having an elastic frame using Biot's theory modified by the model of Johnson et al. [J. Fluid Mech. 176, 379-402 (1987)] for viscous exchange between fluid and structure. Numerical simulations of transmitted waves in the time domain are worked out by varying the modified Biot parameters. The variation is applied to the governing parameters and is about 20%. From this study, we can gain an insight into the sensitivity of each physical parameter used in this theory. Some parameters play an important role in slow-wave wave form, such as the viscous characteristic length lambda and pore fluid bulk modulus Kf. However, other parameters play an important role in the fast-wave wave form, such as solid density rhos and shear modulus N. We also note from these simulations that some parameters such as porosity phi, tortuosity alpha(infinty), thickness, solid bulk modulus Ks, and skeletal compressibility frame Kb, play an important role simultaneously in both fast and slow wave forms compared to other parameters which act on the wave form of just one of the two waves. The sensitivity of the modified Biot parameters with respect to the transmitted wave depends strongly on the coupling between the solid and fluid phases of the cancellous bone. Experimental results for slow and fast waves transmitted through human cancellous bone samples are given and compared with theoretical predictions.

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

An ultrasonic reflectivity method of evaluating the acoustic parameters of porous materials saturated by air (or any other gas) is discussed. The method is based on experimental detection of waves reflected at normal incidence by the first and second interface of the material. This method is based on a temporal model of direct and inverse scattering problems for the propagation of transient ultrasonic waves in a homogeneous isotropic slab of porous material with a rigid frame [Fellah et al., J. Acoust. Soc. Am. 113, 61-73 (2003)]. Generally, the conventional ultrasonic approach can be used to determine tortuosity, and viscous and thermal characteristic lengths via transmitted waves. Porosity cannot be estimated in transmitted mode because of its very weak sensitivity. First interface use of the reflected wave at oblique incidence leads to the determination of porosity and tortuosity [Fellah et al., J. Acoust. Soc. Am. 113, 2424-2433 (2003)] but this is not possible at normal incidence. Using experimental data of reflected waves by the first and second interface at normal incidence simultaneously leads to the determination of porosity, tortuosity, viscous and thermal characteristic lengths. As with the classic ultrasonic approach for characterizing porous material saturated with one gas, both characteristic lengths are estimated individually by assuming a given ratio between them. Tests are performed using weakly resistive industrial plastic foams. Experimental and numerical results, and prospects are discussed.