Mathematical Theory on Multi‐Layered High‐Contrast Acoustic Subwavelength Resonators

ABSTRACT Subwavelength resonance is a vital acoustic phenomenon in contrasting media. The narrow bandgap width of single‐layered resonator has prompted the exploration of multi‐layered 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‐layered high‐contrast structures. First, by using layer potential techniques, we establish the representation formula in terms of a matrix type operator with a block tridiagonal form for multi‐layered structures within general geometry. Then, we prove the existence of subwavelength resonances via the Gohberg–Sigal theory, which generalizes the celebrated Minnaert resonances in single‐layered 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 for different nested resonators are presented to corroborate the theoretical findings.