A proof that multiple waves propagate in ensemble-averaged particulate materials.
Proceedings. Mathematical, physical, and engineering sciences
The Royal Society
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Gower, A. L., Abrahams, I., & Parnell, W. J. (2019). A proof that multiple waves propagate in ensemble-averaged particulate materials.. Proceedings. Mathematical, physical, and engineering sciences, 475 (2229), 20190344. https://doi.org/10.1098/rspa.2019.0344
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.
EPSRC (via University of Manchester) (EP/M026205/1)
External DOI: https://doi.org/10.1098/rspa.2019.0344
This record's URL: https://www.repository.cam.ac.uk/handle/1810/297015
Attribution 4.0 International
Licence URL: https://creativecommons.org/licenses/by/4.0/