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A proof that multiple waves propagate in ensemble-averaged particulate materials.

Published version
Peer-reviewed

Type

Article

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Authors

Abrahams, I David 

Abstract

Effective medium theory aims to describe a complex inhomogeneous material in terms of a few important macroscopic parameters. To characterize 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.

Description

Keywords

Wiener–Hopf, backscattering, ensemble averaging, multiple scattering, random media, wave propagation

Journal Title

Proc Math Phys Eng Sci

Conference Name

Journal ISSN

1364-5021
1471-2946

Volume Title

475

Publisher

The Royal Society
Sponsorship
Engineering and Physical Sciences Research Council (EP/K032208/1)
Engineering and Physical Sciences Research Council (EP/R014604/1)
Engineering and Physical Sciences Research Council (EP/M026205/1)