Passive earplug including Helmholtz resonators arranged in series to achieve broadband near zero occlusion effect at low frequencies

The use of passive earplugs is often associated with the occlusion effect: a phenomenon described as the increased auditory perception of one's own physiological noise at low frequencies. As a notable acoustic discomfort, the occlusion effect penalizes the use and the efficiency of earplugs. This phenomenon is objectively characterized by the increase in sound pressure level in the occluded ear canal compared to the open ear canal. Taking inspiration from acoustic metamaterials, a new design of a three-dimensional printed "meta-earplug," made of four Helmholtz resonators arranged in series, is proposed for achieving near zero objective occlusion effect measured on artificial ear in a broadband frequency range (300 Hz to 1 kHz). For this purpose, the geometry of the meta-earplug is optimized to achieve a null occlusion effect target based on an analytical model of the phenomenon. It results from the optimization process that the input impedance of the meta-earplug medial surface approximately matches the input impedance of the open ear canal, weighted by the ratio of volume velocity imposed by the ear canal wall to the ear canal cavity between open and occluded cases. Acoustic properties of the meta-earplug are also shown to significantly improve its sound attenuation at the piston-like mode of the system.

Reduction of the occlusion effect induced by earplugs using quasi perfect broadband absorption

Passive earplugs are used to prevent workers from noise-induced hearing loss. However, earplugs often induce an acoustic discomfort known as the occlusion effect. This phenomenon corresponds to an increased auditory perception of the bone-conducted part of physiological noises at low-frequency and is associated with the augmentation of the acoustic pressure in the occluded earcanal. In this work, we report a new concept of passive earplugs for mitigating the occlusion effect between 100 Hz and 1 kHz. The strategy consists in reducing the input impedance of the earplug seen from the earcanal by using quasi-perfect broadband absorbers derived from the field of meta-materials. The proposed "meta-earplug" is made of 4 critically coupled Helmholtz resonators arranged in parallel. Their geometry is optimized using an evolutionary algorithm associated with a theoretical model of the meta-earplug input impedance. The latter is validated against a finite-element approach and impedance sensor measurements. The meta-earplug is manufactured by 3D printing. Artificial test fixtures are used to assess the occlusion effect and the insertion loss. Results show that the meta-earplug induces an occlusion effect approximately 10 dB lower than foam and silicone earplugs while it provides an insertion loss similar to the silicone earplug up to 5 kHz.

The ACOUCOU platform: Online acoustic education developed by an interdisciplinary team

The ACOUCOU platform is a web-based, interactive, acoustics training platform that includes a set of free educational materials in various technical fields of acoustics. Educational materials are designed to serve as a modern self-development tool for students and engineers, as well as a comprehensive solution for professional education in the work environment. On the other hand, the provided materials of the platform can be a useful tool, supporting teachers, company researchers, and academic lecturers in the process of teaching acoustics. The ACOUCOU platform is a part of a strategic plan for expanding and strengthening acoustic knowledge web-based tools and supporting the development of innovative teaching methods based on attractive and effective delivery of digital content, and best practices at national and international levels. It addresses the challenge of a lack of experts in the acoustics field and the growing needs of the market.

Estimating the material parameters of an inhomogeneous poroelastic plate from ultrasonic measurements in water

The estimation of poroelastic material parameters based on ultrasound measurements is considered. The acoustical characterisation of poroelastic materials based on various measurements is typically carried out by minimising a cost functional of model residuals, such as the least squares functional. With a limited number of unknown parameters, least squares type approaches can provide both reliable parameter and error estimates. With an increasing number of parameters, both the least squares parameter estimates and, in particular, the error estimates often become unreliable. In this paper, the estimation of the material parameters of an inhomogeneous poroelastic (Biot) plate in the Bayesian framework for inverse problems is considered. Reflection and transmission measurements are performed and 11 poroelastic parameters, as well as 4 measurement setup-related nuisance parameters, are estimated. A Markov chain Monte Carlo algorithm is employed for the computational inference to assess the actual uncertainty of the estimated parameters. The results suggest that the proposed approach for poroelastic material characterisation can reveal the heterogeneities in the object, and yield reliable parameter and uncertainty estimates.

Dynamic equations of a transversely isotropic, highly porous, fibrous material including oscillatory heat transfer effects

The dynamic equations of a transversely isotropic fibrous, highly porous material are presented in terms of microstructure-derived analytical expressions for viscous dissipation, and analytical expressions for the oscillatory heat transfer between the thermal fields of the solid cylindrical glassfibres and the surrounding viscous fluid. This represents the non-equilibrium thermal expansion of the fluid, occurring when waves propagate in the porous material, and results in a frequency-dependent scaling of the fluid dilatation term. A state-space transfer matrix solution of the governing equations has been introduced, allowing the numerical acoustical performance of the fibrous material to be investigated, including the acoustical effects of heat transfer. In order to understand the dissipation mechanisms of the viscous and thermal boundary layers on the surface of the fibres and the validity of the assumptions made in the current model, a thermoviscous acoustic fluid finite element procedure has also been introduced. The results from these simulations illustrate the frequency-dependent interaction of the boundary layers between neighbouring fibres in the porous material.

Erratum: A simplified model for thin acoustic screens [J. Acoust. Soc. Am. 144(1), EL76-EL81 (2018)]

The present erratum reports an error impacting the figures of a contribution published in 2018 about a simplified model for thin acoustic screens in a transfer matrix context [Gaborit, Dazel, and Göransson (2018). J. Acoust. Soc. Am. 144(1), EL76-EL81]. A mistake in the implementation of the rigid termination condition for the systems under study is identified and a correct version is proposed along with the corrected figures. It is shown that this error does not impact the conclusions of the original contribution and that the model proposed therein keeps its advantages as the approximation error remains very similar to the previously reported values.

A simplified model for thin acoustic screens

A generalization of the commonly used pressure jump modeling of thin porous layers is proposed. The starting point is a transfer matrix model of the layer derived using matrix exponentials. First order expansions of the propagating terms lead to a linear approximation of the associated phenomena and the resulting matrix is further simplified based on physical assumptions. As a consequence, the equivalent fluid parameters used in the model may be reduced to simpler expressions and the transfer matrix rendered sparser. The proposed model is validated for different backing conditions, from normal to grazing incidence and for a wide range of thin films. In the paper, the physical hypotheses are discussed, together with the origin of the field jumps.

Deterministic and statistical characterization of rigid frame porous materials from impedance tube measurements

A method to characterize macroscopically homogeneous rigid frame porous media from impedance tube measurements by deterministic and statistical inversion is presented. Equivalent density and bulk modulus of the samples are reconstructed with the scattering matrix formalism, and are then linked to its physical parameters via the Johnson-Champoux-Allard-Lafarge model. The model includes six parameters, namely the porosity, tortuosity, viscous and characteristic lengths, and static flow and thermal permeabilities. The parameters are estimated from the measurements in two ways. The first one is a deterministic procedure that finds the model parameters by minimizing a cost function in the least squares sense. The second approach is based on statistical inversion. It can be used to assess the validity of the least squares estimate, but also presents several advantages since it provides valuable information on the uncertainty and correlation between the parameters. Five porous samples with a range of pore properties are tested, and the pore parameter estimates given by the proposed inversion processes are compared to those given by other characterization methods. Joint parameter distributions are shown to demonstrate the correlations. Results show that the proposed methods find reliable parameter and uncertainty estimates to the six pore parameters quickly with minimal user input.

Derivation of the state matrix for dynamic analysis of linear homogeneous media

A method to obtain the state matrix of an arbitrary linear homogeneous medium excited by a plane wave is proposed. The approach is based on projections on the eigenspace of the governing equations matrix. It is an alternative to manually obtaining a linearly independent set of equations by combining the governing equations. The resulting matrix has been validated against previously published derivations for an anisotropic poroelastic medium.

Asymptotic limits of some models for sound propagation in porous media and the assignment of the pore characteristic lengths

Modeling of sound propagation in porous media requires the knowledge of several intrinsic material parameters, some of which are difficult or impossible to measure directly, particularly in the case of a porous medium which is composed of pores with a wide range of scales and random interconnections. Four particular parameters which are rarely measured non-acoustically, but used extensively in a number of acoustical models, are the viscous and thermal characteristic lengths, thermal permeability, and Pride parameter. The main purpose of this work is to show how these parameters relate to the pore size distribution which is a routine characteristic measured non-acoustically. This is achieved through the analysis of the asymptotic behavior of four analytical models which have been developed previously to predict the dynamic density and/or compressibility of the equivalent fluid in a porous medium. In this work the models proposed by Johnson, Koplik, and Dashn [J. Fluid Mech. 176, 379-402 (1987)], Champoux and Allard [J. Appl. Phys. 70(4), 1975-1979 (1991)], Pride, Morgan, and Gangi [Phys. Rev. B 47, 4964-4978 (1993)], and Horoshenkov, Attenborough, and Chandler-Wilde [J. Acoust. Soc. Am. 104, 1198-1209 (1998)] are compared. The findings are then used to compare the behavior of the complex dynamic density and compressibility of the fluid in a material pore with uniform and variable cross-sections.

Acoustic behavior of a rigidly backed poroelastic layer with periodic resonant inclusions by a multiple scattering approach

The acoustic response of a rigidly backed poroelastic layer with a periodic set of elastic cylindrical inclusions embedded is studied. A semi-analytical approach is presented, based on Biot's 1956 theory to account for the deformation of the skeleton, coupling mode matching technique, Bloch wave representation, and multiple scattering theory. This model is validated by comparing the derived absorption coefficients to finite element simulations. Numerical results are further exposed to investigate the influence of the properties of the inclusions (type, material properties, size) of this structure, while a modal analysis is performed to characterize the dynamic behaviors leading to high acoustic absorption. Particularly, in the case of thin viscoelastic membranes, an absorption coefficient larger than 0.8 is observed on a wide frequency band. This property is found to be due to the coupling between the first volume mode of the inclusion and the trapped mode induced by the periodic array and the rigid backing, for a wavelength in the air smaller than 11 times the material thickness.

Enhancing the absorption properties of acoustic porous plates by periodically embedding Helmholtz resonators

This paper studies the acoustical properties of hard-backed porous layers with periodically embedded air filled Helmholtz resonators. It is demonstrated that some enhancements in the acoustic absorption coefficient can be achieved in the viscous and inertial regimes at wavelengths much larger than the layer thickness. This enhancement is attributed to the excitation of two specific modes: Helmholtz resonance in the viscous regime and a trapped mode in the inertial regime. The enhancement in the absorption that is attributed to the Helmholtz resonance can be further improved when a small amount of porous material is removed from the resonator necks. In this way the frequency range in which these porous materials exhibit high values of the absorption coefficient can be extended by using Helmholtz resonators with a range of carefully tuned neck lengths.

Using simple shape three-dimensional rigid inclusions to enhance porous layer absorption

The absorption properties of a metaporous material made of non-resonant simple shape three-dimensional rigid inclusions (cube, cylinder, sphere, cone, and ring torus) embedded in a rigidly backed rigid-frame porous material are studied. A nearly total absorption can be obtained for a frequency lower than the quarter-wavelength resonance frequency due to the excitation of a trapped mode. To be correctly excited, this mode requires a filling fraction larger in three-dimensions than in two-dimensions for purely convex (cube, cylinder, sphere, and cone) shapes. At long wavelengths compared to the spatial period, a cube is found to be the best purely convex inclusion shape to embed in a cubic unit cell, while the embedment of a sphere or a cone cannot lead to an optimal absorption for some porous material properties and dimensions of the unit cell. At a fixed position of purely convex shape inclusion barycenter, the absorption coefficient only depends on the filling fraction and does not depend on the shape below the Bragg frequency arising from the interaction between the inclusion and its image with respect to the rigid backing. The influence of the incidence angle and of the material properties, namely, the flow resistivity is also shown. The results of the modeling are validated experimentally in the case of cubic and cylindrical inclusions.

Absorption of sound by porous layers with embedded periodic arrays of resonant inclusions

The aim of this work is to design a layer of porous material with a high value of the absorption coefficient in a wide range of frequencies. It is shown that low frequency performance can be significantly improved by embedding periodically arranged resonant inclusions (slotted cylinders) into the porous matrix. The dissipation of the acoustic energy in a porous material due to viscous and thermal losses inside the pores is enhanced by the low frequency resonances of the inclusions and energy trapping between the inclusion and the rigid backing. A parametric study is performed in order to determine the influence of the geometry and the arrangement of the inclusions embedded in a porous layer on the absorption coefficient. The experiments confirm that low frequency absorption coefficient of a composite material is significantly higher than that of the porous layer without the inclusions.

Vibroacoustic response sensitivity due to relative alignment of two anisotropic poro-elastic layers

The effects of relative alignment of two different types of anisotropic open cell porous materials are investigated in terms of the acoustic response of a multi-layered configuration. Numerical experiments, where gradient based optimization techniques were used, are conducted to find possible extremal values. It is shown that, depending on the degree of anisotropy of the porous material properties, their angular orientations have a significant and frequency dependent influence on the measured response. The results highlight the importance of further advancing the knowledge of anisotropic porous material behavior.

Enhancing rigid frame porous layer absorption with three-dimensional periodic irregularities

This papers reports a three-dimensional (3D) extension of the model proposed by Groby et al. [J. Acoust. Soc. Am. 127, 2865-2874 (2010)]. The acoustic properties of a porous layer backed by a rigid plate with periodic rectangular irregularities are investigated. The Johnson-Champoux-Allard model is used to predict the complex bulk modulus and density of the equivalent fluid in the porous material. The method of variable separation is used together with the radiation conditions and Floquet theorem to derive the analytical expression for the acoustic reflection coefficient from the porous layer with 3D inhomogeneities. Finite element method is also used to validate the proposed analytical solution. The theoretical and numerical predictions agree well with the experimental data obtained from an impedance tube experiment. It is shown that the measured acoustic absorption coefficient spectrum exhibits a quasi-total absorption peak at the predicted frequency of the mode trapped in the porous layer. When more than one irregularity per spatial period is considered, additional absorption peaks are observed.

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.

An application of the Peano series expansion to predict sound propagation in materials with continuous pore stratification

This work reports on an application of the state vector (Stroh) formalism and Peano series expansion to solve the problem of sound propagation in a material with continuous pore stratification. An alternative Biot formulation is used to link the equivalent velocity in the oscillatory flow in the material pores with the acoustic pressure gradient. In this formulation, the complex dynamic density and bulk modulus are predicted using the equivalent fluid flow model developed by Horoshenkov and Swift [J. Acoust. Soc. Am. 110(5), 2371-2378 (2001)] under the rigid frame approximation. This model is validated against experimental data obtained for a 140 mm thick material specimen with continuous pore size stratification and relatively constant porosity. This material has been produced from polyurethane binder solution placed in a container with a vented top and sealed bottom to achieve a gradient in the reaction time which caused a pore size stratification to develop as a function of depth [Mahasaranon et al., J. Appl. Phys. 111, 084901 (2012)]. It is shown that the acoustical properties of this class of materials can be accurately predicted with the adopted theoretical model.

Linear and nonlinear Biot waves in a noncohesive granular medium slab: transfer function, self-action, second harmonic generation

Experimental results are reported on second harmonic generation and self-action in a noncohesive granular medium supporting wave energy propagation both in the solid frame and in the saturating fluid. The acoustic transfer function of the probed granular slab can be separated into two main frequency regions: a low frequency region where the wave propagation is controlled by the solid skeleton elastic properties, and a higher frequency region where the behavior is dominantly due to the air saturating the beads. Experimental results agree well with a recently developed nonlinear Biot wave model applied to granular media. The linear transfer function, second harmonic generation, and self-action effect are studied as a function of bead diameter, compaction step, excitation amplitude, and frequency. This parametric study allows one to isolate different propagation regimes involving a range of described and interpreted linear and nonlinear processes that are encountered in granular media experiments. In particular, a theoretical interpretation is proposed for the observed strong self-action effect.

Enhancing the absorption coefficient of a backed rigid frame porous layer by embedding circular periodic inclusions

The acoustic properties of a porous sheet of medium static air flow resistivity (around 10,000 N m s(-4)), in which a periodic set of circular inclusions is embedded and which is backed by a rigid plate, are investigated. The inclusions and porous skeleton are assumed motionless. Such a structure behaves like a multi-component diffraction grating. Numerical results show that this structure presents a quasi-total (close to unity) absorption peak below the quarter-wavelength resonance of the porous sheet in absence of inclusions. This result is explained by the excitation of a complex trapped mode. When more than one inclusion per spatial period is considered, additional quasi-total absorption peaks are observed. The numerical results, as calculated with the help of the mode-matching method described in this paper, agree with those calculated using a finite element method.

Propagation of acoustic waves in a one-dimensional macroscopically inhomogeneous poroelastic material

Wave propagation in macroscopically inhomogeneous porous materials has received much attention in recent years. The wave equation, derived from the alternative formulation of Biot's theory of 1962, was reduced and solved recently in the case of rigid frame inhomogeneous porous materials. This paper focuses on the solution of the full wave equation in which the acoustic and the elastic properties of the poroelastic material vary in one-dimension. The reflection coefficient of a one-dimensional macroscopically inhomogeneous porous material on a rigid backing 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 at both normal and oblique incidence and to experimental measurements at normal incidence for a known two-layers porous material, considered as a single inhomogeneous layer. Finally, discussion about the absorption coefficient for various inhomogeneity profiles gives further perspectives.

Influence of static compression on mechanical parameters of acoustic foams

The modification of elastic properties of compressed acoustic foams is investigated. The porous sample is first submitted to a static compression and then to a dynamic excitation of smaller amplitude, corresponding to acoustical applications. The static compression induces the modification of the dynamic elastic parameters of the material. This work focuses on Young's modulus. The variation is measured with two different experimental methods: The classical rigidimeter and an absorption measurement. The effective Young's modulus is directly measured with the first method and is indirectly determined through the quarter-wave length resonance of the frame with the second one. The results of the two measurements are compared and give similar tendencies. The variation of the dynamic Young's modulus as a function of the degree of compression of the sample is shown to be separated in several zones. In the zones associated with weak compression (those usually zones encountered in practice), the variation of the effective Young's modulus can be approximated by a simple affine function. The results are compared for different foams. A simple model of the dependency of the Young's modulus with respect to the static degree of compression is finally proposed for weak compressions.

Absorption of a rigid frame porous layer with periodic circular inclusions backed by a periodic grating

The acoustic properties of a periodic rigid frame porous layer with multiple irregularities in the rigid backing and embedded rigid circular inclusions are investigated theoretically and numerically. The theoretical representation of the sound field in the structure is obtained using a combination of multipole method that accounts for the periodic inclusions and multi-modal method that accounts for the multiple irregularities of the rigid backing. The theoretical model is validated against a finite element method. The predictions show that the acoustic response of this structure exhibits quasi-total, high absorption peaks at low frequencies which are below the frequency of the quarter-wavelength resonance typical for a flat homogeneous porous layer backed by a rigid plate. This result is explained by excitation of additional modes in the porous layer and by a complex interaction between various acoustic modes. These modes relate to the resonances associated with the presence of a profiled rigid backing and rigid inclusions in the porous layer.

Prediction of sound reflection by corrugated porous surfaces

The coupled mode (CM) and finite-element methods (FEMs) are developed and used to predict the acoustic reflection coefficient of a semi-infinite porous medium with closely spaced two-dimensional (2D) periodical corrugations. These methods are also applied to predict the reflection coefficient of a periodic array of porous corrugations installed on an acoustically rigid surface. It is shown that the predictions by the both methods agree closely. The reflection coefficient and Brewster angle of total refraction for the corrugated semi-infinite medium predicted with these methods are compared against that predicted by the Biot/Tolstoy/Howe/Twersky and extended Twersky models. A similar analysis is carried out for porous corrugations set on a rigid backing. The behavior of the reflection coefficient and the pole in the expression for the reflection coefficient located close to grazing incidence is studied.

Nonlinear Biot waves in porous media with application to unconsolidated granular media

The nonlinear propagation through porous media is investigated in the framework of Biot theory. For illustration, and considering the current interest for the determination of the elastic properties of granular media, the case of nonlinear propagation in "model" granular media (disordered packings of noncohesive elastic beads of the same size embedded in a visco-thermal fluid) is considered. The solutions of linear Biot waves are first obtained, considering the appropriate geometrical and physical parameters of the medium. Then, making use of the method of successive approximations of nonlinear acoustics, the solutions for the second harmonic Biot waves are derived by considering a quadratic nonlinearity in the solid frame constitutive law (which takes its origin from the high nonlinearity of contacts between grains). The propagation in a semi-infinite medium with velocity dispersion, frequency dependent dissipation, and nonlinearity is first analyzed. The case of a granular medium slab with rigid boundaries, often considered in experiments, is then presented. Finally, the importance of mode coupling between solid and fluid waves is evaluated, depending on the actual fluid, the bead diameter, or the applied static stress on the beads. The application of these results to other media supporting Biot waves (porous ceramics, polymer foams, etc.) is straightforward.

A description of transversely isotropic sound absorbing porous materials by transfer matrices

A description of wave propagation in transversely isotropic porous materials saturated by air with a recent reformulation of the Biot theory is carried out. The description is performed in terms of a transfer matrix method (TMM). The anisotropy is taken into account in the mechanical parameters (elastic constants) and in the acoustical parameters (flow resistivity, tortuosity, and characteristic lengths). As an illustration, the normal surface impedance at normal and oblique incidences of transversely isotropic porous layers is predicted. Comparisons are performed with experimental results.

Expressions of dissipated powers and stored energies in poroelastic media modeled by {u,U} and {u,P} formulations

This paper is devoted to the rigorous obtention of the energy balance in porous materials. The wave propagation in the porous media is described by Biot-Allard's {u,U} and {u,P} formulations. The paper derives the expressions for stored kinetic and strain energies together with dissipated energies. It is shown that, in the case of mixed formulations, these expressions do not correspond to the real and imaginary parts of the variational formulations. A quantitative convergence analysis of finite element scheme is then undertaken with the help of these indicators. It is shown that the order of convergence of these indicators for linear finite-element is one and that they are then well fitted to check the validity of finite-element models.

Near field Rayleigh wave on soft porous layers

Simulations performed for a typical semi-infinite reticulated plastic foam saturated by air show that, at distances less than three Rayleigh wavelengths from the area of mechanical excitation by a circular source, the normal frame velocity is close to the Rayleigh pole contribution. Simulated measurements show that a good order of magnitude estimate of the phase speed and damping can be obtained at small distances from the source. Simulations are also performed for layers of finite thickness, where the phase velocity and damping depend on frequency. They indicate that the normal frame velocity at small distances from the source is always close to the Rayleigh pole contribution and that a good order of magnitude estimate of the phase speed of the Rayleigh wave can be obtained at small distances from the source. Furthermore, simulations show that precise measurements of the damping of the Rayleigh wave need larger distances. Measurements performed on a layer of finite thickness confirm these trends.

Validity of the limp model for porous materials: a criterion based on the Biot theory

The validity of using the limp model for porous materials is addressed in this paper. The limp model is derived from the poroelastic Biot model assuming that the frame has no bulk stiffness. Being an equivalent fluid model accounting for the motion of the frame, it has fewer limitations than the usual equivalent fluid model assuming a rigid frame. A criterion is proposed to identify the porous materials for which the limp model can be used. It relies on a new parameter, the frame stiffness influence (FSI), based on porous material properties. The critical values of FSI under which the limp model can be used are determined using a one-dimensional analytical modeling for two boundary sets: absorption of a porous layer backed by a rigid wall and radiation of a vibrating plate covered by a porous layer. Compared with other criteria, the criterion associated with FSI provides information in a wider frequency range and can be used for configurations that include vibrating plates.

An alternative Biot's displacement formulation for porous materials

This paper proposes an alternative displacement formulation of Biot's linear model for poroelastic materials. Its advantage is a simplification of the formalism without making any additional assumptions. The main difference between the method proposed in this paper and the original one is the choice of the generalized coordinates. In the present approach, the generalized coordinates are chosen in order to simplify the expression of the strain energy, which is expressed as the sum of two decoupled terms. Hence, new equations of motion are obtained whose elastic forces are decoupled. The simplification of the formalism is extended to Biot and Willis thought experiments, and simpler expressions of the parameters of the three Biot waves are also provided. A rigorous derivation of equivalent and limp models is then proposed. It is finally shown that, for the particular case of sound-absorbing materials, additional simplifications of the formalism can be obtained.