Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Huaian Diao, Kaixin Lu, Ruixiang Tang, Weisheng Zhou

TL;DR

This paper develops a boundary-integral framework for 2D elastic wave scattering by a high-contrast hard inclusion in a soft medium under sub-wavelength excitation, focusing on a unit disk after rescaling. It proves boundary localization of both the interior total field and the exterior scattered field via carefully designed incident waves and small-$\omega$ asymptotics of layer potentials and the Neumann-Poincaré operator. The main contributions include (i) a rigorous demonstration of boundary localization leading to quasi-Minnaert resonance with a continuous spectrum in the sub-wavelength regime, (ii) precise conditions under which simultaneous boundary localization and surface resonance occur, and (iii) quantitative stress-concentration results near the inclusion boundary, plus implications for near-cloaking and high-contrast material design. Together, the results extend quasi-Minnaert resonance theory to 2D elasticity and provide analytic tools for engineering high-contrast metamaterials with boundary-localized fields.

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.

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

TL;DR

This paper develops a boundary-integral framework for 2D elastic wave scattering by a high-contrast hard inclusion in a soft medium under sub-wavelength excitation, focusing on a unit disk after rescaling. It proves boundary localization of both the interior total field and the exterior scattered field via carefully designed incident waves and small- asymptotics of layer potentials and the Neumann-Poincaré operator. The main contributions include (i) a rigorous demonstration of boundary localization leading to quasi-Minnaert resonance with a continuous spectrum in the sub-wavelength regime, (ii) precise conditions under which simultaneous boundary localization and surface resonance occur, and (iii) quantitative stress-concentration results near the inclusion boundary, plus implications for near-cloaking and high-contrast material design. Together, the results extend quasi-Minnaert resonance theory to 2D elasticity and provide analytic tools for engineering high-contrast metamaterials with boundary-localized fields.

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.

Paper Structure

This paper contains 4 sections, 11 theorems, 129 equations.

Table of Contents

  1. Introduction
  2. Mathematical setup
  3. Boundary localization of the interior total and exterior scattered wave fields
  4. Surface resonances and stress concentrations

Key Result

Lemma 3.1

The single-layer potentials $\mathbf{S}_{\partial D_{R}}^{\omega}[e^{\mathrm{i}n\theta}\bm{\upsilon}]$ and $\mathbf{S}_{\partial D_{R}}^{\omega}[e^{\mathrm{i}n\theta}\bm{t}]$ have the following expressions for $|\mathbf{x}|=R$: where The single-layer potentials $\mathbf{S}_{\partial D_{R}}^{\omega}[e^{\mathrm{i}n\theta}\bm{\upsilon}]$ and $\mathbf{S}_{\partial D_{R}}^{\omega}[e^{\mathrm{i}n\thet

Theorems & Definitions (28)

  • Definition 2.1
  • Remark 2.1
  • Definition 2.2
  • Remark 2.2
  • Definition 2.3
  • Remark 2.3
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • proof
  • ...and 18 more