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Propagation and dispersion of Bloch waves in periodic media with soft inclusions

Yuri A. Godin, Boris Vainberg

Abstract

We investigate the behavior of waves in a periodic medium containing small soft inclusions or cavities of arbitrary shape, such that the homogeneous Dirichlet conditions are satisfied at the boundary. The leading terms of Bloch waves, their dispersion relations, and cutoff frequencies are rigorously derived. Our approach reveals the existence of exceptional wave vectors for which Bloch waves are comprised of clusters of perturbed plane waves that propagate in different directions. We demonstrate that for these exceptional wave vectors, no Bloch waves propagate in any one specific direction.

Propagation and dispersion of Bloch waves in periodic media with soft inclusions

Abstract

We investigate the behavior of waves in a periodic medium containing small soft inclusions or cavities of arbitrary shape, such that the homogeneous Dirichlet conditions are satisfied at the boundary. The leading terms of Bloch waves, their dispersion relations, and cutoff frequencies are rigorously derived. Our approach reveals the existence of exceptional wave vectors for which Bloch waves are comprised of clusters of perturbed plane waves that propagate in different directions. We demonstrate that for these exceptional wave vectors, no Bloch waves propagate in any one specific direction.
Paper Structure (8 sections, 12 theorems, 94 equations, 1 figure)

This paper contains 8 sections, 12 theorems, 94 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Formulation of the problem and main results
  3. Outline of the approach
  4. Expansion of the exterior DtN operator
  5. Asymptotics of the interior DtN operator
  6. Asymptotics of the interior DtN map in the basis $\psi_{\mathscr E}, \psi_{\perp}$
  7. Derivation of the dispersion relations and clusters of Bloch waves
  8. Conclusions

Key Result

Lemma 2.1

If ${\bm k}$ is a non-exceptional wave vector, then the solution space of the unperturbed problem (unp) consists of functions $C \mathrm{e}^{-\mathrm{i} {\bm k} \cdot {\bm x}}$. However, if ${\bm k}$ is an exceptional wave vector of order $n$, then the solution space is $n$-dimensional and can be ex

Figures (1)

  • Figure 1: The cell of periodicity $\Pi$ with a small cavity $\Omega$ inside. A ball $B_R \subset \Pi$ of radius $R$ centered at the origin encloses the cavity $\Omega$.

Theorems & Definitions (25)

  • Definition
  • Remark 1
  • Lemma 2.1
  • proof
  • Lemma 3.1
  • Lemma 3.2
  • proof
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • ...and 15 more