Equivalence between three scattering formulations for ultrasonic wave propagation in particulate mixtures

Abstract

The aim of this paper is to present a unified approach to the calculation of the complex wavenumber for a randomly distributed ensemble of homogeneous isotropic spheres suspended in a homogeneous isotropic continuum. Three classical formulations of the diffraction problem for a compression wave incident on a single particle are reviewed; the first is for liquid particles in a liquid continuum (Epstein and Carhart), the second for solid or liquid particles in a liquid continuum (Allegra and Hawley), and the third for solid particles in a solid continuum (Ying and Truell). Equivalences between these formulations are demonstrated and it is shown that the Allegra and Hawley formulation can be adapted to provide a basis for calculation in all three regimes. The complex wavenumber that results from an ensemble of such scatterers is treated using the formulations of Foldy (simple forward scattering), Waterman and Truell, and Lloyd and Berry (multiple scattering). The analysis is extended to provide an approximation for the case of a distribution of particle sizes in the mixture. A number of experimental measurements using a broadband spectrometric technique (reported elsewhere) to obtain the attenuation coefficient and phase velocity as functions of frequency are presented for various mixtures of differing contrasts in physical properties between phases in order to provide a comparison with theory. The materials used were aqueous suspensions of polystyrene spheres, silica spheres, iron spheres, pigment (AHR), droplets of 1-bromohexadecane, and a suspension of talc particles in a cured epoxy resin.

A comparison of stochastic and effective medium approaches to the backscattered signal from a porous layer in a solid matrix

“…We adopt the Rayleigh method to obtain the scattered wave potential for the propagational mode resulting from a planar incident wave in the z direction; it is given by a sum over all scattered partial wave modes, [38][39][40][41][42] thus…”

“…The coefficients can be derived from the generalized matrix equation set out in Ref. 42 41 results for the zero, first and second order coefficients have been derived in updated notation using the Maple algebraic software 44 [and correcting a factor of À 1 in the original paper, Eq. (34) For the effective medium model (bottom half, below dashed line) a homogeneous solid material is present in the region z min < z < z max .…”

“…41 The three formulations pertained to different physical states of the scattering object and surrounding media, namely, fluid in fluid, solid in fluid and any material in solid (neglecting thermal effects) respectively. The relationships between them have been discussed by Challis et al, 42 see also Challis et al 43 Our numerical models use these scattered fields, either for individual scatterers summed or averaged as appropriate, or to obtain the ensemble-averaged effective properties of the layer.…”

Iterative simulation of elastic wave scattering in arbitrary dispersions of spherical particles

“…This approach has been used extensively for evaluating classical formulations where multiple scattering is treated approximately. [32][33] Analogous experimental validation of the iterative multipole method is therefore an appropriate test for the model.…”

Acoustic scattering in dispersions: Improvements in the calculation of single particle scattering coefficients

“…This set of models includes viscous and thermal contributions and covers a range of values of the wavenumber-radius parameter ka ͑where k is the wave number and a the particle radius͒. The equivalence between the solutions for liquid/liquid ͑Epstein and Carhart 3 ͒, solid/liquid ͑Allegra and Hawley 4 ͒, and solid/solid systems ͑Ying and Truell 5 ͒ has been established ͑Challis et al 6 ͒, implying that the formulation is of general applicability.…”