Multiple scattering of elastic waves by cylinders of arbitrary cross section. I. SH waves

Abstract

A scattering matrix approach, that involves only the transition matrix of a single obstacle, is proposed for studying the multiple scattering of elastic waves in a medium (matrix) containing identical, long, parallel, randomly distributed cylinders of arbitrary cross section. The elastic properties of the cylinders are assumed to be different from those of the matrix. A statistical approach in conjunction with Lax’s ’’quasicrystalline’’ approximation is employed to obtain equations for the average amplitudes of the scattered and exciting fields which may then be solved to yield the dispersion relations of the composite medium. Dynamic elastic properties of the composite medium containing circular and elliptical cylinders are found in the Rayleigh or low-frequency limit. Numerical results displaying phase velocity and damping effect of the composite medium are presented for a wide range of frequencies.

Elasticity Theory of Composites

“…These are ensemble averaged (essentially as in (6.12)) and the system is closed by making the quasicrystalline approximation of Lax (1952) (essentially as in (6.14)). Particular examples of this approach are provided in the papers of Mal (1973, 1974), Datta (1977Datta ( , 1978, Twersky (1977), Varadan, Varadan and Pao (1978) and Varadan and Varadan (1979).…”

Thermal transport in 2- and 3-dimensional periodic “holey” nanostructures

“…The coefficient A im is calculated by considering the acoustic impedance at hole-silicon interface and the multiply scattered wavefronts from all the other cylinders j i. Specifically, in order to obtain A im , we solve a hierarchy of equations 64 A im = iC m F im…”

A new integral equation approach to elastodynamic homogenization

“…These schemes are often difficult to justify mathematically or physically although steps to address this have recently been taken by Markov (2001) and Kanaun & Levin (2003). The method of multiple scattering initiated by Foldy (1945) and developed by Waterman & Truell (1961) and Fikioris & Waterman (1964) has also been successfully applied in elastodynamic and acoustic contexts (see Bose &Mal 1974 andLinton &Martin 2005, respectively) and by employing the T-matrix scattering theory by Varadan et al (1978). The multiple-scattering formulation allows correlation functions to be introduced into the analysis, although it requires a closure condition (the quasi-crystalline approximation being the usual choice), which itself is difficult to justify rigorously.…”

“…The multiple-scattering formulation allows correlation functions to be introduced into the analysis, although it requires a closure condition (the quasi-crystalline approximation being the usual choice), which itself is difficult to justify rigorously. The quasi-static limit of the multiple-scattering prediction in the well-stirred limit (see Varadan et al 1978) corresponds exactly to the special composite cylinders result of Hashin & Rosen (1964), when a particular limit corresponding to no gaps between composite cylinders, is taken. However, although multiple-scattering theory is well suited to isotropic phases, it becomes extremely complicated for anisotropic phases.…”