A proof that multiple waves propagate in ensemble-averaged particulate materials
A proof that multiple waves propagate in ensemble-averaged particulate materials
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DOI：
10.1098/rspa.2019.0344
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发表时间：
2019-09-01
影响因子：
3.5
通讯作者：
Parnell, William J.
中科院分区：
文献类型：
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作者：
Gower, Artur L.;Abrahams, I. David;Parnell, William J.
有效介质理论旨在用一些重要的宏观参数来描述复杂的非均匀材料。为了表征波在非均匀材料中的传播，最关键的参数是有效波数。因此，有许多关于如何计算单个有效波数的研究已发表。在这里，我们证明了不存在唯一的有效波数；相反，这样的（复数）波数有无限多个。我们表明，在大多数参数范围中，只有一小部分有效波数对波场做出了重大贡献。然而，为了准确计算反射和透射系数，需要大量（高度衰减的）有效波。为了清楚起见，我们给出了填充有随机分布的圆柱形夹杂物（在半空间上）的二维材料的标量（声学）波结果。我们通过对所有可能的不均匀性进行整体平均来计算有效介质。该证明基于 Wiener-Hopf 技术的应用，并且不对波长、颗粒边界条件/尺寸或体积分数做出假设。该技术提供了反射系数的简单公式，可以明确地评估单极散射体。我们将结果与替代的数值匹配方法进行比较。
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.