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 \begin{document}$u^\varepsilon : \Omega_\varepsilon \to \mathbb{R}$\end{document} to a Helmholtz equation in the limit \begin{document}$\varepsilon \to 0$\end{document} with the help of two-scale convergence. The domain \begin{document}$\Omega_\varepsilon $\end{document} is obtained by removing from an open set \begin{document}$\Omega\subset \mathbb{R}^n$\end{document} in a periodic fashion a large number (order \begin{document}$\varepsilon ^{-n}$\end{document} ) of small resonators (order \begin{document}$\varepsilon $\end{document} ). The special properties of the meta-material are obtained through sub-scale structures in the perforations.