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

Agnes Lamacz; Ben Schweizer

doi:10.3934/dcdss.2017041

Article Contents

2017, Volume 10, Issue 4: 815-835. Doi: 10.3934/dcdss.2017041

This issue Previous Article On the geometry of the p-Laplacian operator Next Article On the existence of weak solutions of an unsteady p-Laplace thermistor system with strictly monotone electrical conductivities

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

Agnes Lamacz and
Ben Schweizer^,

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

^* Corresponding author: Ben Schweizer
^* Corresponding author: Ben Schweizer

Received: February 29, 2016

Revised: September 30, 2016

Published: July 2017

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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 $u^\varepsilon : \Omega_\varepsilon \to \mathbb{R}$ 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.

Keywords:

Helmholtz equation,
homogenization,
resonance,
perforated domain,
frequency dependent effective properties.

Mathematics Subject Classification: 78M40, 35P25, 35J05.

Citation:

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Figure 1. 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 $\varepsilon ^p$

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Figure 2. Sketch of the geometry around the end-point of the channel

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