Elastodynamic multiple scattering: effective wavenumbers in three-dimensional elastic media

We derive expressions for the effective wavenumbers in a three-dimensional elastic medium with a low
number density of embedded identical scatterers of arbitrary shape and random orientation. We adopt a
classical approach addressing the half-space problem using standard vector spherical wavefunctions and their associated addition theorems. Both quasi-longitudinal and quasi-shear effective wavenumbers are obtained at first and second order in concentration by ensemble-averaging under the assumption of hard-sphere non-interacting scatterers, together with the quasi-crystalline approximation. We assume that the scatterer orientations are independent of each other and independent of position, and demonstrate that the ensemble averaging can be achieved by first taking an orientational average of the single-scatterer T -matrix before taking the positional average. The expressions for effective wavenumbers indicate the contributions of mode conversion (longitudinal to shear and vice versa) at second order in concentration.