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.
Highlights
In this article, we are interested in the acoustic properties of a particular metamaterial, inspired by sound absorbing structures
The acoustic properties of the meta-material are determined by the Helmholtz equation since the acoustic pressure p of a time-harmonic sound wave of fixed frequency ω is of the form p(x, t) = u(x)eiωt, where u solves a Helmholtz equation
In order to analyze the effect of the resonator region D, we study solutions uε to the Helmholtz equation (1.1) and investigate their behavior inside and outside of D in the limit ε → 0
Summary
We are interested in the acoustic properties of a particular metamaterial, inspired by sound absorbing structures. In standard homogenization settings, nothing special can be expected concerning the acoustic properties of a meta-material (e.g. large or negative coefficients) Instead, in this contribution, we introduce a setting where the small inclusions. Schweizer are resonators and where the effective behavior of the meta-material introduces new features Let us describe these statements in a more mathematical language: We consider a domain Ωε ⊂ Rn, n = 2 or n = 3, which is obtained by removing small obstacles of typical size ε > 0 from a domain Ω ⊂ Rn. For a fixed frequency ω ∈ R, we study solutions uε ∈ H1(Ωε) to the Helmholtz equation. Neither A∗ nor λ are frequency dependent In contrast to such a standard approach we investigate (1.1) for a domain Ωε, where every single inclusion (perforation) has the shape of a small resonator.
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