# A Proof that Multiple Waves Propagate in Ensemble-Averaged Particulate Materials

## TL;DR

This paper proves that multiple effective wavenumbers exist in ensemble-averaged particulate materials, showing that wave propagation involves several waves and that accurate transmission calculations require considering many of these waves.

## Contribution

It provides a rigorous proof of the non-uniqueness of effective wavenumbers and analyzes their significance in wave propagation through particulate media.

## Key findings

- Multiple effective wavenumbers exist in such materials.
- Only a few dominate the wave field in most regimes.
- Accurate reflection and transmission calculations need many effective waves.

## Abstract

Effective medium theory aims to describe a complex inhomogeneous material in terms of a few important macroscopic parameters. To characterise wave propagation through an inhomogeneous material, the most crucial parameter is the effective wavenumber. For this reason, there are many published studies on how to calculate a single effective wavenumber. Here we present a proof that there does not exist a unique effective wavenumber; instead, there are an infinite number of such (complex) wavenumbers. We show that in most parameter regimes only a small number of these effective wavenumbers make a significant contribution to the wave field. However, to accurately calculate the reflection and transmission coefficients, a large number of the (highly attenuating) effective waves is required. For clarity, we present results for scalar (acoustic) waves for a two-dimensional material filled (over a half space) with randomly distributed circular cylindrical inclusions. We calculate the effective medium by ensemble averaging over all possible inhomogeneities. The proof is based on the application of the Wiener-Hopf technique and makes no assumption on the wavelength, particle boundary conditions/size, or volume fraction. This technique provides a simple formula for the reflection coefficient, which can be explicitly evaluated for monopole scatterers. We compare results with an alternative numerical matching method.