Multiple waves propagate in random particulate materials

For over 70 years it has been assumed that scalar wave propagation in (ensemble-averaged) random particulate materials can be characterized by a single effective wavenumber. Here, however, we show that there exist many effective wavenumbers, each contributing to the effective transmitted wave field. Most of these contributions rapidly attenuate away from boundaries, but they make a significant contribution to the reflected and total transmitted field beyond the low-frequency regime. In some cases at least two effective wavenumbers have the same order of attenuation. In these cases a single effective wavenumber does not accurately describe wave propagation even far away from boundaries. We develop an efficient method to calculate all of the contributions to the wave field for the scalar wave equation in two spatial dimensions, and then compare results with numerical finite-difference calculations. This new method is, to the best of the authors' knowledge, the first of its kind to give such accurate predictions across a broad frequency range and for general particle volume fractions.