Sub-wavelength resonances in two-dimensional multi-layer elastic media

TL;DR

This work develops a rigorous sub-wavelength resonance theory for two-dimensional elastic media with high-contrast parameters, focusing on $N$-nested layered resonators. Using layer-potential techniques and Gohberg–Sigal spectral theory, it establishes the invertibility of the leading-order operator at low frequency and derives a determinant condition that yields $3N$ resonance frequencies, with an $\,\mathcal{O}(\omega^{-2})\,$-enhancement of the scattered field near resonance. For radial geometries, it provides explicit expressions via disk reductions and Bessel–Hankel representations, and validates the theory through numerical experiments that demonstrate resonance modes and the accuracy of a point-scatterer approximation. The results offer a multi-band, sub-wavelength control framework for elastic metamaterials and inform design of multi-layer resonators with potential cloaking and localization applications.

Abstract

In this paper, we focus on the sub-wavelength resonances in two-dimensional elastic media characterized by high contrasts in both Lamé parameters and density. Our contributions are fourfold. First, it is proved that the operator $\hat{\mathbf{S}}_{\partial D}^ω$, which serves as a leading order approximation to $\mathbf{S}_{\partial D}^ω$ as $ω\rightarrow0$, is invertible in the space $\mathcal{L}(L^{2}\left(\partial D)^{2},H^{1}(\partial D)^{2}\right)$. Second, based on layer potential techniques in combination with asymptotic analysis, we derive an original formula for the leading-order terms of sub-wavelength resonance frequencies, which are controlled by the determinant of the $3N \times 3N$ matrices. Specifically, there are $3N$ resonance frequencies within an $N$-nested layer structure. In addition, the scattering field exhibits an enhancement coefficient on the order of $\mathcal{O}(ω^{-2})$ as the incident frequency $ω$ approaches the resonance frequency. Third, by applying spectral properties to solve the corresponding eigenvalue problem, we compute the quantitative expressions for sub-wavelength resonance frequencies within a disk. Finally, some numerical experiments are provided to illustrate theoretical results and demonstrate the existence of the sub-wavelength resonance modes.

Figures (7)

• Figure 1: Single resonator.
• Figure 2: $N$-nested resonators.
• Figure 3: The normalized determinants of $q=1$ in the setup of \ref{['radii1']} with $N=4$.
• Figure 4: The normalized determinants of $q=2$ in the setup of \ref{['radii1']} with $N=4$.
• Figure 5: Norm of the displacement fields $\mathbf{u}_{S}$ in the setup of \ref{['radii1']} with $N=4$.

Theorems & Definitions (42)

• Definition 1.1
• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• proof
• Proposition 2.1
• proof
• Proposition 2.2
• proof
• Lemma 2.2