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

TL;DR

This paper investigates sub-wavelength resonances in 2D elastic media with high contrast, deriving formulas for resonance frequencies, proving operator invertibility, and validating results through numerical experiments.

Contribution

It introduces a new formula for sub-wavelength resonance frequencies in layered elastic media and proves the invertibility of a key operator as frequency approaches zero.

Findings

Resonance frequencies are determined by the determinant of a specific matrix.

The scattering field is enhanced by a factor of order ext{O}( ext{ extomega}^{-2}) near resonance.

Numerical results confirm the theoretical predictions and resonance mode existence.

Abstract

In this paper, we focus on the sub-wavelength resonances in two-dimensional elastic media characterized by high contrasts in both Lam\'e parameters and density. Our contributions are fourfold. First, it is proved that the operator $\hat{\mathbf{S}}_{\partial D}^{\omega}$, which serves as a leading order approximation to $\mathbf{S}_{\partial D}^{\omega}$ as $\omega\rightarrow0$, 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 $\mathcal{O}(\omega^{-2})$ as the incident frequency $\omega$ 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.