Materials comprising randomly distributed particles, or inclusions, occur frequently in the world around us. Common examples include composites, emulsions, suspensions, complex gases, etc… This paper is the first to clearly prove that, contrary to the belief of most in the community, multiple effective waves propagate in particulate materials (in an ensemble-average sense).

We show that in general there is not only one effective wave, but there is a series of effective waves in an ensemble averaged particulate material. By particulate material we mean a medium filled with randomly placed particles and consider waves governed by the scalar wave equation. Although most of these waves decay rapidly, they make a significant contribution to reflection and transmission beyond the low-frequency regime. In two spatial dimensions, we develop an efficient method to calculate all these waves, which gives highly accurate predictions when compared to a finite-difference scheme. This method is, to the authors knowledge, the first of its kind to give accurate predictions across a broad frequency range and particle volume fraction.

To what extent can particulate random media be characterised using direct wave backscattering from a single receiver/source? Here, in a two-dimensional setting, we show using a machine learning approach that both the particle radius and concentration can be accurately measured when the boundary condition on the particles is of Dirichlet type. Although the methods we introduce could be applied to any particle type. In general backscattering is challenging to interpret for a wide range of particle concentrations, because multiple scattering cannot be ignored, except in the very dilute range. Across the concentration range from 1% to 20% we find that the mean backscattered wave field is sufficient to accurately determine the concentration of particles. However, to accurately determine the particle radius, the second moment, or average intensity, of the backscattering is necessary. We are also able to determine what is the ideal frequency range to measure a broad range of particles sizes. To get rigorous results with supervised machine learning requires a large, highly precise, dataset of backscattered waves from an infinite half-space filled with particles. We are able to create this dataset by introducing a numerical approach which accurately approximates the backscattering from an infinite half-space.