# American Institute of Mathematical Sciences

August  2017, 10(4): 815-835. doi: 10.3934/dcdss.2017041

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

 Technische Universität Dortmund, Fakultät für Mathematik, Vogelpothsweg 87,44227 Dortmund, Germany

* Corresponding author: Ben Schweizer

Received  March 2016 Revised  October 2016 Published  April 2017

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

Citation: Agnes Lamacz, Ben Schweizer. Effective acoustic properties of a meta-material consisting of small Helmholtz resonators. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 815-835. doi: 10.3934/dcdss.2017041
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Sketch of the scattering problem. Left: The sub-region $D\subset \Omega$ contains the small Helmholtz resonators, given by $\Sigma_\varepsilon \subset D$. The number of resonators in the region $D$ is of order $\varepsilon ^{-n}$. We are interested in the effective properties of the meta-material in $D$. Right: The microscopic geometry with the single resonator $R_Y$. The channel width inside $Y$ is of the order $\varepsilon ^p$
Sketch of the geometry around the end-point of the channel
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