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

F Feppon (1) , H Ammari (1)

1 D-MATH - Department of Mathematics [ETH Zurich]

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 $s\sim 1/N$ 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, $\delta\rightarrow 0$, the effective medium admits $K$ resonant characteristic sizes $(s_i(\delta))_{1\

Mots clés: Non-periodic homogenization; effective medium theory; strange term; subwavelength resonance; high-contrast medium; holomorphic integral operators

Domaines: Equations aux dérivées partielles [math.AP]; Variables complexes [math.CV]; Analyse fonctionnelle [math.FA]; Physique mathématique [math-ph]

F Feppon, H Ammari. Homogenization of sound-absorbing and high-contrast acoustic metamaterials in subcritical regimes. 2021. ⟨hal-03372593⟩