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Effective medium theory aims to describe a complex inhomogeneous material interms of a few important macroscopic parameters. To characterise wavepropagation through an inhomogeneous material, the most crucial parameter isthe effective wavenumber. For this reason, there are many published studies onhow to calculate a single effective wavenumber. Here we present a proof thatthere does not exist a unique effective wavenumber; instead, there are aninfinite number of such (complex) wavenumbers. We show that in most parameterregimes only a small number of these effective wavenumbers make a significantcontribution to the wave field. However, to accurately calculate the reflectionand 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) withrandomly distributed circular cylindrical inclusions. We calculate theeffective medium by ensemble averaging over all possible inhomogeneities. Theproof is based on the application of the Wiener-Hopf technique and makes noassumption on the wavelength, particle boundary conditions/size, or volumefraction. This technique provides a simple formula for the reflectioncoefficient, which can be explicitly evaluated for monopole scatterers. Wecompare results with an alternative numerical matching method.

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