Local gaps in three-dimensional periodic media

TL;DR

This paper investigates acoustic wave propagation in 3D periodic media with small inclusions, revealing the absence of global gaps for small inclusion sizes and introducing the concept of local gaps dependent on wave vectors.

Contribution

It analytically characterizes the location of local spectral gaps in 3D periodic media with small inclusions, a novel approach in the study of wave propagation in such structures.

Findings

Global gaps do not exist if inclusions are sufficiently small.

Introduces and studies the concept of local gaps depending on wave vector.

Provides analytical determination of local gap locations for Dirichlet and transmission problems.

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.