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

TL;DR

This paper analyzes the effective acoustic properties of meta-materials made of small Helmholtz resonators, demonstrating their ability to exhibit frequency-dependent and negative permittivities through mathematical modeling.

Contribution

It introduces a mathematical framework using two-scale convergence to characterize the effective properties of meta-materials with sub-scale resonator structures.

Findings

Meta-materials can have frequency-dependent effective properties.

Large and negative permittivities are achievable.

Mathematical analysis confirms the design principles.

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 $u^\varepsilon: \Omega_\varepsilon \rightarrow \mathbb{R}$ to a Helmholtz equation in the limit $\varepsilon\rightarrow 0$ 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.