Quasi-Minnaert Resonances in 2D Elastic Wave Scattering with Applications

TL;DR

This paper extends the concept of quasi-Minnaert resonances from 3D to 2D elastic wave scattering, addressing theoretical challenges and demonstrating boundary-localized resonances with potential applications in cloaking.

Contribution

It provides the first rigorous analysis of quasi-Minnaert resonances in 2D elastic scattering, overcoming coupling and spectral challenges using layer potential and asymptotic techniques.

Findings

Established quasi-Minnaert resonances in 2D elastic scattering

Quantified stress concentration effects

Identified boundary-localized scattered fields for cloaking

Abstract

In our earlier work [13], we introduced a novel quasi-Minnaert resonance for three-dimensional elastic wave scattering in the sub-wavelength regime. Therein, we provided a rigorous analysis of the boundary localization and surface resonance phenomena for both the total and scattered waves, achieved through carefully selected incident waves and tailored physical parameters of the elastic medium. In the present study, we focus on quasi-Minnaert resonances in the context of two-dimensional elastic wave scattering. Unlike the 3D case [13], the 2D setting introduces fundamental theoretical challenges stemming from (i) the intrinsic coupling between shear and compressional waves, and (ii) the complex spectral properties of the associated layer potential operators. By combining layer potential techniques with refined asymptotic analysis and strategically designed incident waves, we rigorously establish quasi-Minnaert resonances in both the internal and scattered fields. In addition, the associated stress concentration effects are quantitatively characterized. Notably, the boundary-localized nature of the scattered field reveals potential applications in near-cloaking (via wave manipulation around boundaries). Our results contribute to a more comprehensive framework for studying resonance behaviors in high-contrast elastic systems.