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Homogenization of sound-absorbing and high-contrast acoustic metamaterials in subcritical regimes

by F. Feppon and H. Ammari

(Report number 2021-35)

Abstract
We propose a quantitative effective medium theory for two types of acoustic metamaterials constituted of a large number \(N\) of small heterogeneities of characteristic size \(s\), randomly and independently distributed in a bounded domain. We first consider a ``sound-absorbing'' material, in which the total wave field satisfies a Dirichlet boundary condition on the acoustic obstacles. In the ``sub-critical'' regime \(sN=O(1)\), we obtain that the effective medium is governed by a dissipative Lippmann-Schwinger equation which approximates the total field with a relative mean-square error of order \(O(max((sN)^{2}N^{-frac{1}{3}}, N^{-frac{1}{2}}))\). We retrieve the critical size \(ssim 1/N\) of the literature at which the effects of the obstacles can be modelled by a ``strange term'' added to the Helmholtz equation. Second, we consider high-contrast acoustic metamaterials, in which each of the \(N\) heterogeneities are packets of \(K\) inclusions filled with a material of density much lower than the one of the background medium. As the contrast parameter vanishes, \(deltarightarrow 0\), the effective medium admits \(K\) resonant characteristic sizes \((s_i(delta))_{1

Keywords: Non-periodic homogenization, effective medium theory, strange term, subwavelength resonance, high-contrast medium, holomorphic integral operators.

@Techreport{FA21_977,
  author = {F. Feppon and H. Ammari},
  title = {Homogenization of sound-absorbing and high-contrast acoustic metamaterials in subcritical regimes },
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2021-35},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2021/2021-35.pdf },
  year = {2021}
}

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