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.
Highlights
Materials comprising particles or inclusions that are randomly distributed inside a uniform host medium occur frequently in the world around us
We focus on ensemble-averaged waves, satisfying the scalar wave equation in two dimensions, reflecting from, and propagating in, a half-space particulate material
We prove this by employing the Wiener–Hopf technique and for clarity, illustrate the solution for particles that scatter only in their monopole mode
Summary
Materials comprising particles or inclusions that are randomly distributed inside a uniform host medium occur frequently in the world around us. We focus on ensemble-averaged waves, satisfying the scalar wave equation in two dimensions, reflecting from, and propagating in, a half-space particulate material In certain scenarios, such as light scattering, it is easier to measure the average intensity of the wave. We illustrate that the Matching Method is robust, because combining many effective wavenumbers is not just a good approximation, it is an analytical solution to the integral equation governing the ensemble-averaged wave field. We prove this by employing the Wiener–Hopf technique and for clarity, illustrate the solution for particles that scatter only in their monopole mode.
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