On subwavelength guided acoustic waves in multi‐layered high‐contrast resonators with a ring defect

Multi‐layered structures are ubiquitous in the construction of metamaterial devices to enable advanced waveguiding functionalities. This paper presents several contributions to the mathematical understanding of subwavelength resonances in multi‐layered nested structures with arbitrarily shaped resonators. First, by employing the Dirichlet‐to‐Neumann map and a variational framework, we derive an original formula for the subwavelength resonance frequencies in terms of the eigenvalues of a new type of capacitance matrix, thereby extending the celebrated Minnaert resonance from single‐layered to general nested configurations. We then introduce a finite‐layered dimer‐type metamaterial in which the geometric symmetry is deliberately broken to form a ring defect, and rigorously demonstrate that such a defect acts as an effective radial laminate waveguide, supporting wave localization and propagation along the defect. Furthermore, for structures with a sufficiently large number of layers, we prove the existence and uniqueness of an eigenvalue in the spectral gap of the defect structure, establishing the existence of a unique localized defect mode. Finally, numerical simulations are provided to support and illustrate the theoretical results.