Effective wave numbers for thermo-viscoelastic media containing random configurations of spherical scatterers

The dispersion relation is derived for the coherent waves in fluid or elastic media supporting viscous and thermal effects and containing randomly distributed spherical scatterers. The formula obtained is the generalization of Lloyd and Berry's [Proc. Phys. Soc. Lond. 91, 678-688, 1067], the latter being limited to fluid host media, and it is the three-dimensional counterpart of that derived by Conoir and Norris [Wave Motion 47, 183-197, 2010] for cylindrical scatterers in an elastic host medium.Comment: 11 page

Dynamic effective properties of a random configuration of cylinders in a fluid

International audienceThe dynamic effective properties of a random medium consisting in a uniform concentration of cylindrical scatterers in an ideal fluid are looked for, with special focus on low frequencies. The effective medium is described as an isotropic viscous fluid whose mass density and dilation viscosity depend on frequency, and whose shear viscosity is nil. An explicit expression of the reflection coefficient of a harmonic plane wave incident upon the interface between the ideal fluid and the random medium may be obtained at low frequency, using the Fikioris and Waterman's approach, in two ways. In the first one, the low frequency assumption is introduced from the very beginning, while in the second one, the same hypotheses than those used by Linton et al. [J. Acoust. Soc. Am. 117 6, 2005] to calculate the effective wavenumber are used first, and, then, the low frequency assumption. In both cases, comparison of this reflection coefficient with that at the interface between the ideal fluid and the effective viscous fluid provides the effective density, which, coupled to the effective wavenumber, provides the effective dilatation viscosity. In the first case, the effective parameters found are identical to those found by Mei et al. [Phys. Rev. B 76, 2007] in a different way, while in the second case they are expressed in terms of form functions of the cylinders that reduce at low frequency to those found by Martin et al. [J. Acous. Soc. Am. 128, 2010]. PACS no. 43.20.Fn, 43.35.B

Effective wavenumbers and reflection coefficients for an elastic medium containing random configurations of cylindrical scatterers

Propagation of P and SV waves in an elastic solid containing randomly distributed inclusions in a half-space is investigated. The approach is based on a multiple scattering analysis similar to the one proposed by Fikioris and Waterman for scalar waves. The characteristic equation, the solution of which yields the effective wave numbers of coherent elastic waves, is obtained in an explicit form without the use of any renormalisation methods. Two approximations are considered. First, formulae are derived for the effective wave numbers in a dilute random distribution of identical scatterers. These equations generalize the formula obtained by Linton and Martin for scalar coherent waves. Second, the high frequency approximation is compared with the Waterman and Truell approach derived here for elastic waves. The Fikioris and Waterman approach, in contrast with Waterman and Truell's method, shows that P and SV waves are coupled even at relatively low concentration of scatterers. Simple expressions for the reflected coefficients of P and SV waves incident on the interface of the half space containing randomly distributed inclusions are also derived. These expressions depend on frequency, concentration of scatterers, and the two effective wave numbers of the coherent waves propagating in the elastic multiple scattering medium.Comment: 24 page

Multiple scattering by cylinders randomly located in a fluid: Effective properties

International audienceWe consider the interface between a homogeneous fluid and that same fluid with a random distribution of n0 cylinders per square meter inside. A harmonic plane wave, frequency v and wavenumber k, is incident upon that boundary under incidence angle α. The reflection coefficient obtained with the Fikioris and Waterman approach is expanded into powers of n0/K2 up to order 2, using Linton and Martin's expansion of the wavenumber of the coherent wave. This coefficient is then compared to that obtained when a homogeneous viscous fluid replaces the random medium. When the two reflection coefficients are equal, the random fluid is acoustically equivalent to the viscous one, which is called in that case the effective fluid. The coherent wave in the random medium is thus described as the acoustic mode in the effective fluid. Equating the two reflection coefficients provides expressions for the effective properties of the random medium: mass density ρeff, and coefficient of dilatation viscosity ρeff, as the shear viscosity is set to zero. Both depend on α and v, unless low frequencies only are considered, in which case the dependence on α vanishes

Ultrasound propagation in concentrated suspensions: shear- mediated contributions to multiple scattering

International audienceWe report analytical and numerical results of a multiple scattering model applied to silica-in-water suspensions. We investigate the shear-mediated effects due to mode conversion between compressional and shear wave modes, not included in standard multiple scattering models. We identify the dominant scattering contributions and develop analytical forms for them. Numerical calculations demonstrate the contribution of the additional shear-mediated effects to the compressional wave speed and attenuation through the suspension. As concentration is increased, we incorporate third order terms in concentration to the expansion of the effective wavenumber of the compressional wave. The calculations are compared with previously published experimental data

Generalization of the Waterman and Truell formula for an elastic medium containing random configurations of cylindrical scatterers

National audiencePropagation of P and SV waves in an elastic solid containing randomly distributed inclusions in a half-space is investigated. The approach is based on a multiple scattering analysis similar to the one proposed by Waterman and Truell for scalar waves. The characteristic equation, the solution of which yields the effective wave numbers of coherent elastic waves, is obtained. Formulae are derived for the effective wave numbers in a dilute random distribution of identical scatterers. They generalize the formula obtained by Waterman and Truell for scalar coherent waves. It is shown that P and SV waves are coupled for non circular cylinders but uncoupled for circular cylinders

Effective wave numbers for media sustaining the propagation of three types of bulk waves and hosting a random configuration of scatterers

International audienceWave propagation through an isotropic host medium containing a large number of randomly and uniformly located scatterers is considered at low frequency and for low concentrations of spheres, and the dispersion relation of the coherent waves is obtained. The same problem had been addressed by Lloyd and Berry for spheres in an ideal fluid, and more recently by Linton and Martin for cylinders in an ideal fluid, and by Conoir and Norris for cylinders in an elastic solid. Here, the dispersion relation is derived in the case of spheres, and extended to that of cylinders, from the comparison of the 3d and 2d cases in an elastic solid. The host medium considered may support the propagation of P different types of bulk waves, as for example a thermo-visco-elastic medium or a poro-elastic medium (P=3). As in the previous works mentioned above, the hole correction of Fikioris and Waterman is taken into account, along with the quasi-crystalline approximation

Ultrasound propagation in concentrated suspensions: shear- mediated contributions to multiple scattering

International audienceWe report analytical and numerical results of a multiple scattering model applied to silica-in-water suspensions. We investigate the shear-mediated effects due to mode conversion between compressional and shear wave modes, not included in standard multiple scattering models. We identify the dominant scattering contributions and develop analytical forms for them. Numerical calculations demonstrate the contribution of the additional shear-mediated effects to the compressional wave speed and attenuation through the suspension. As concentration is increased, we incorporate third order terms in concentration to the expansion of the effective wavenumber of the compressional wave. The calculations are compared with previously published experimental data

Longitudinal and transverse coherent waves in media containing randomly distributed spheres

International audienceMultiple scattering effects due to a random distribution of identical spheres are investigated in the general case of elastic or poroelastic host media, where both longitudinal and transverse waves may co-exist. Propagation of plane coherent waves is assumed, and their dispersion equation looked for, as well as analytic approximations of those particular solutions that are close to the wavenumbers in the free host, when the product of the concentration with the scattering cross section of the spheres is low. Under this last condition, pair-correlation effects are seen to be of second order. Numerical studies are performed under the hole correction assumption, and compared to experimental data for tungsten carbide spheres in an epoxy matrix, which is a rather illustrative situation of how longitudinal and transverse waves participate to coherent propagation

Dynamic effective properties of a random configuration of cylinders in a fluid

International audienceThe dynamic effective properties of a random medium consisting in a uniform concentration of cylindrical scatterers in an ideal fluid are looked for, with special focus on low frequencies. The effective medium is described as an isotropic viscous fluid whose mass density and dilation viscosity depend on frequency, and whose shear viscosity is nil. An explicit expression of the reflection coefficient of a harmonic plane wave incident upon the interface between the ideal fluid and the random medium may be obtained at low frequency, using the Fikioris and Waterman's approach, in two ways. In the first one, the low frequency assumption is introduced from the very beginning, while in the second one, the same hypotheses than those used by Linton et al. [J. Acoust. Soc. Am. 117 6, 2005] to calculate the effective wavenumber are used first, and, then, the low frequency assumption. In both cases, comparison of this reflection coefficient with that at the interface between the ideal fluid and the effective viscous fluid provides the effective density, which, coupled to the effective wavenumber, provides the effective dilatation viscosity. In the first case, the effective parameters found are identical to those found by Mei et al. [Phys. Rev. B 76, 2007] in a different way, while in the second case they are expressed in terms of form functions of the cylinders that reduce at low frequency to those found by Martin et al. [J. Acous. Soc. Am. 128, 2010]. PACS no. 43.20.Fn, 43.35.B