Mathematical theory on multi-layer high contrast acoustic subwavelength resonators

TL;DR

This paper develops a mathematical framework for analyzing multi-layer high-contrast acoustic resonators, revealing how nested structures influence subwavelength resonances and providing formulas for resonance frequencies.

Contribution

It introduces a general mathematical approach for multi-layer resonators, extending single-layer resonance theory to complex layered structures with new formulas and insights.

Findings

Existence of subwavelength resonances proven using Gohberg-Sigal theory

Resonance frequency formulas derived for concentric dual-resonators

Numerical results confirm theoretical predictions

Abstract

Subwavelength resonance is a vital acoustic phenomenon in contrasting media. The narrow bandgap width of single-layer resonator has prompted the exploration of multi-layer metamaterials as an effective alternative, which consist of alternating nests of high-contrast materials, called ``resonators'', and a background media. In this paper, we develop a general mathematical framework for studying acoustics within multi-layer high-contrast structures. Firstly, by using layer potential techniques, we establish the representation formula in terms of a matrix type operator with a block tridiagonal form for multi-layer structures within general geometry. Then we prove the existence of subwavelength resonances via Gohberg-Sigal theory, which generalizes the celebrated Minnaert resonances in single-layer structures. Intriguingly, we find that the primary contribution to mode splitting lies in the fact that as the number of nested resonators increases, the degree of the corresponding characteristic polynomial also increases, while the type of resonance (consists solely of monopolar resonances) remains unchanged. Furthermore, we derive original formulas for the subwavelength resonance frequencies of concentric dual-resonator. Numerical results associated with different nested resonators are presented to corroborate the theoretical findings.