Effective acoustic properties of a meta-material consisting of small Helmholtz resonators

Abstract

We investigate the acoustic properties of meta-materials that are inspired by sound-absorbing structures. We show that it is possible to construct meta-materials with frequency-dependent effective properties, with large and/or negative permittivities. Mathematically, we investigate solutions $u^{\epsilon }:{\Omega }_{\epsilon }\to \mathbb{R}$ to a Helmholtz equation in the limit $\epsilon \to 0$ with the help of two-scale convergence. The domain ${\Omega }_{\epsilon }$ is obtained by removing from an open set $\Omega \subset {\mathbb{R}}^n$ in a periodic fashion a large number (order ${\epsilon }^{-n}$) of small resonators (order $\epsilon$). The special properties of the meta-material are obtained through sub-scale structures in the perforations.