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Local gaps in three-dimensional periodic media

Yuri A. Godin, Boris Vainberg

Abstract

We consider the propagation of acoustic waves in a medium with a periodic array of small inclusions of arbitrary shape. The inclusion size $a$ is much smaller than the array period. We show that global gaps do not exist if $a$ is small enough. The notion of local gaps which depends on the choice of the wave vector $\bk$ is introduced and studied. We determine analytically the location of local gaps for the Dirichlet and transmission problems.

Local gaps in three-dimensional periodic media

Abstract

We consider the propagation of acoustic waves in a medium with a periodic array of small inclusions of arbitrary shape. The inclusion size is much smaller than the array period. We show that global gaps do not exist if is small enough. The notion of local gaps which depends on the choice of the wave vector is introduced and studied. We determine analytically the location of local gaps for the Dirichlet and transmission problems.
Paper Structure (7 sections, 10 theorems, 56 equations, 3 figures)

This paper contains 7 sections, 10 theorems, 56 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Outline of the approach
  3. Necessary prerequisites
  4. Asymptotics of the DtN operators in the neighborhoods of exceptional points
  5. Local and global gaps in the Dirichlet problem
  6. Local and global gaps in the transmission problem
  7. Conclusions

Key Result

Lemma 2.1

Figures (3)

  • Figure 1: The cell of periodicity $\Pi$ with a small inclusion $\Omega$ of size $a$. A ball $B_R \subset \Pi$ of radius $R$ centered at the origin encloses the inclusion $\Omega$.
  • Figure 2: Several cones forming the dispersion surface of the unperturbed problem in dimension two.
  • Figure 3: Shaded unit discs on the faces of the first Brillouin zone show the location of the Bloch vector where waves cannot propagate.

Theorems & Definitions (13)

  • Lemma 2.1
  • Lemma 3.1
  • Lemma 4.1
  • Theorem 4.1
  • proof
  • proof
  • Theorem 5.1
  • Lemma 5.1
  • Theorem 5.2
  • Remark 1
  • ...and 3 more