Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Subwavelength resonances in two-dimensional elastic media with high contrast

Yuanchun Ren, Yixian Gao

TL;DR

This work develops a boundary-integral framework to understand subwavelength resonances in two-dimensional elastic media with high contrast in density and Lamé parameters. By exploiting kernel spaces of boundary operators and constructing an invertible leading-order operator, the authors derive leading-order equations that determine three subwavelength resonant frequencies for disk-shaped inclusions, and they characterize the interior and exterior fields near resonance, including explicit far-field patterns for longitudinal and transverse waves. Extending to a dilute phononic crystal, they locate subwavelength bandgaps using quasi-periodic Green’s functions and diagonalized operator structures, yielding practical insights for design of elastic metamaterials. The results provide a rigorous, geometry-specific toolset for predicting resonant behavior and band structure in 2D high-contrast elastic composites, with potential impact on subwavelength imaging, waveguiding, and cloaking applications in elastic media.

Abstract

This paper employs layer potential techniques to investigate wave scattering in two-dimensional elastic media exhibiting high contrasts in both Lamé parameters and density. Our contributions are fourfold. First, we construct an invertible operator based on the kernel spaces of boundary integral operators, which enables the characterization of resonant frequencies through an orthogonality condition. Second, we use asymptotic analysis to derive the equation governing the leading-order terms of these resonant frequencies. Third, we analyze the scattered field in the interior domain for incident frequencies across different regimes and characterize the longitudinal and transverse far-field patterns in the exterior domain. Finally, we examine the subwavelength bandgap in the phononic crystal with a dilute structure.

Subwavelength resonances in two-dimensional elastic media with high contrast

TL;DR

This work develops a boundary-integral framework to understand subwavelength resonances in two-dimensional elastic media with high contrast in density and Lamé parameters. By exploiting kernel spaces of boundary operators and constructing an invertible leading-order operator, the authors derive leading-order equations that determine three subwavelength resonant frequencies for disk-shaped inclusions, and they characterize the interior and exterior fields near resonance, including explicit far-field patterns for longitudinal and transverse waves. Extending to a dilute phononic crystal, they locate subwavelength bandgaps using quasi-periodic Green’s functions and diagonalized operator structures, yielding practical insights for design of elastic metamaterials. The results provide a rigorous, geometry-specific toolset for predicting resonant behavior and band structure in 2D high-contrast elastic composites, with potential impact on subwavelength imaging, waveguiding, and cloaking applications in elastic media.

Abstract

This paper employs layer potential techniques to investigate wave scattering in two-dimensional elastic media exhibiting high contrasts in both Lamé parameters and density. Our contributions are fourfold. First, we construct an invertible operator based on the kernel spaces of boundary integral operators, which enables the characterization of resonant frequencies through an orthogonality condition. Second, we use asymptotic analysis to derive the equation governing the leading-order terms of these resonant frequencies. Third, we analyze the scattered field in the interior domain for incident frequencies across different regimes and characterize the longitudinal and transverse far-field patterns in the exterior domain. Finally, we examine the subwavelength bandgap in the phononic crystal with a dilute structure.

Paper Structure

This paper contains 10 sections, 15 theorems, 153 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Mathematical setup
  3. Problem formulation
  4. Layer potential operators
  5. Subwavelength resonance
  6. The scattered fields
  7. Solution to the internal problem
  8. Far-field patterns
  9. Dilute Phononic Structure
  10. Concluding remarks

Key Result

Lemma 2.1

As $k\rightarrow 0$, the fundamental solution $\boldsymbol G^k$ admits the asymptotic expansion where $\boldsymbol A(\boldsymbol x)=(a_{ij}(\boldsymbol x))^2_{i,j=1},$$\boldsymbol B(\boldsymbol x)=(b_{ij}(\boldsymbol x))^2_{i,j=1},$ and Here, the matrix elements are given by with the constants given by

Figures (1)

  • Figure 1: Two-dimensional phononic crystal with a dilute structure.

Theorems & Definitions (27)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 17 more