Willis couplings in continuously varying cross-sectional area duct

Acoustic wave propagation in a one-dimensional periodic and asymmetric duct is studied theoretically and numerically to derive the effective properties. Closed form expressions for these effective properties, including the asymmetric Willis coupling, are derived through truncation of the Peano-Baker series expansion of the matricant (which links the state vectors at the two sides of the unit-cell) and Padé's approximation of the matrix exponential. The results of the first-order and second-order homogenization (with Willis coupling) procedures are compared with the numerical results. The second-order homogenization procedure provides scattering coefficients that are valid over a much larger frequency range than the usual first-order procedure. The frequency well below which the effective description is valid is compared with the lower bound of the first Bragg bandgap when the profile is approximated by a two-step function of identical indicator function, i.e., two different cross-sectional areas over the same length. This validity limit is then questioned, particularly with a focus on impedance modeling. This article attempts to facilitate the engineering use of Willis materials.

3D-printed sound absorbing metafluid inspired by cereal straws

Used as building biomaterials for centuries, cereal straws are known for their remarkable acoustic performances in sound absorption. Yet, their use as fibrous media disregards their internal structure made of nodes partitioning stems. Here, we show that such nodes can impart negative acoustic bulk modulus to straw balls when straws are cut on either side of a node. Such metafluid inspired by cereal straws combines visco-thermal diffusion with strong wave dispersion arising from quarter-wavelength resonances within straws. Large spectral bandgaps and slow sound regimes are theoretically predicted and experimental data from impedance tube measurements on an idealised 3D-printed sample layer are in good agreement with the theoretical model. Perfect absorption is achieved at wavelengths 13 times larger than the thickness of the metafluid layer, and slow sound entails an increased density of states causing a cascade of high absorption peaks. Such features could lead cereal straws to serve as cheap acoustic bio-metamaterials.

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.

Use of slow sound to design perfect and broadband passive sound absorbing materials

Perfect (100%) absorption by thin structures consisting of a periodic arrangement of rectangular quarter-wavelength channels with side detuned quarter-wavelength resonators is demonstrated. The thickness of these structures is 13-17 times thinner than the acoustic wavelength. This low frequency absorption is due to a slow sound wave propagating in the main rectangular channel. A theoretical model is proposed to predict the complex wavenumber in this channel. It is shown that the speed of sound in the channel is much lower than in the air, almost independent of the frequency in the low frequency range, and it is dispersive inside the induced transparency band which is observed. The perfect absorption condition is found to be caused by a critical coupling between the rectangular channel (sub-wavelength resonators) and the incoming wave. It is shown that the width of a large absorption peak in the frequency spectrum can be broadened if several rectangular channels in the unit cell are detuned. The detuning is achieved by varying the length of the side resonators for each channel. The predicted absorption coefficients are validated experimentally. Two resonant cells were produced with stereolithography which enabled the authors to incorporate curved side resonators.

Surface waves in granular phononic crystals

The existence of surface elastic waves at a mechanically free surface of granular phononic crystals is studied. The granular phononic crystals are made of spherical particles distributed periodically on a simple cubic lattice. It is assumed that the particles are interacting by means of normal, shear, and bending contact rigidities. First, Rayleigh-type surface acoustic waves, where the displacement of the particles takes place in the sagittal plane while the particles possess one rotational and two translational degrees of freedom, are analyzed. Second, shear-horizontal-type waves, where the displacement of the particles is normal to the sagittal plane while the particles possess one translational and two rotational degrees of freedom are studied. The existence of zero-group-velocity surface acoustic waves of Rayleigh type is theoretically predicted and interpreted. A comparison with surface waves predicted by the reduced Cosserat theory is performed, and some limitations of the latter are established.

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.

Localized transversal-rotational modes in linear chains of equal masses

The propagation and localization of transversal-rotational waves in a two-dimensional granular chain of equal masses are analyzed in this study. The masses are infinitely long cylinders possessing one translational and one rotational degree of freedom. Two dispersive propagating modes are predicted in this granular crystal. By considering the semi-infinite chain with a boundary condition applied at its beginning, the analytical study demonstrates the existence of localized modes, each mode composed of two evanescent modes. Their existence, position (either in the gap between the propagating modes or in the gap above the upper propagating mode), and structure of spatial localization are analyzed as a function of the relative strength of the shear and bending interparticle interactions and for different boundary conditions. This demonstrates the existence of a localized mode in a semi-infinite monatomic chain when transversal-rotational waves are considered, while it is well known that these types of modes do not exist when longitudinal waves are considered.

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.

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.

Sustainable sonic crystal made of resonating bamboo rods

The acoustic transmission coefficient of a resonant sonic crystal made of hollow bamboo rods is studied experimentally and theoretically. The plane wave expansion and multiple scattering theory (MST) are used to predict the bandgap in transmission coefficient of a non-resonant sonic crystal composed of rods without holes. The predicted results are validated against experimental data for the acoustic transmission coefficient. It is shown that a sonic crystal made from a natural material with some irregularities can exhibit a clear transmission bandgap. Then, the hollow bamboo rods are drilled between each node to create an array of Helmholtz resonators. It is shown that the presence of Helmholtz resonators leads to an additional bandgap in the low-frequency part of the transmission coefficient. The MST is modified in order to account for the resonance effect of the holes in the drilled bamboo rods. This resonant multiple scattering theory is validated experimentally and could be further used for the description and optimization of more complex resonant sonic crystals.

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.

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.

Total absorption peak by use of a rigid frame porous layer backed by a rigid multi-irregularities grating

The acoustic properties of a low resistivity porous layer backed by a rigid plate containing periodic rectangular irregularities, creating a multicomponent diffraction gratings, are investigated. Numerical and experimental results show that the structure possesses a total absorption peak at the frequency of the modified mode of the layer, when designed as proposed in the article. These results are explained by an analysis of the acoustic response of the whole structure and especially by the modal analysis of the configuration. When more than one irregularity per spatial period is considered, additional higher frequency peaks are observed.

Analytical method for the ultrasonic characterization of homogeneous rigid porous materials from transmitted and reflected coefficients

A frequency domain method dedicated to the analytic recovery of the four relevant parameters of macroscopically homogeneous rigid frame porous materials, e.g., plastic foams, at the high frequency range of the Johnson-Champoux-Allard model is developed and presented. The reconstructions appeal to experimental data concerning time domain measurements of the ultrasonic fields reflected and transmitted by a plate of the material at normal incidence. The effective density and bulk modulus of the material are first reconstructed from the frequency domain reflection and transmission coefficients. From the latter, the porosity, tortuosity, and thermal and viscous characteristic lengths are recovered. In a sense, the method presented herein is quite similar in the ultrasonic range, but also quite complementary, to the method developed by Panneton and Olny [J. Acoust. Soc. Am. 119, 2027-2040 (2006); 123, 814-824 (2008)] at low frequency, which appeal to experimental data measured in an impedance tube.

Acoustic response of a rigid-frame porous medium plate with a periodic set of inclusions

The acoustic response of a rigid-frame porous plate with a periodic set of inclusions is investigated by a multipole method. The acoustic properties, in particular, the absorption, of such a structure are then derived and studied. Numerical results together with a modal analysis show that the addition of a periodic set of high-contrast inclusions leads to the excitation of the modes of the plate and to a large increase in the acoustic absorption.

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.