Clusters of Bloch waves in three-dimensional periodic media

TL;DR

This paper analyzes acoustic wave propagation in 3D periodic media, deriving dispersion relations and revealing exceptional frequencies where multiple wave clusters propagate simultaneously, with rigorous mathematical justification.

Contribution

It introduces a rigorous derivation of the dispersion relation for arbitrary-shaped inclusions and identifies exceptional frequencies with multi-directional wave clusters.

Findings

Dispersion relation derived for general frequencies

Existence of exceptional frequencies with multi-directional wave clusters

Examples provided for spherical inclusions

Abstract

We consider acoustic wave propagation through a periodic array of the inclusions of arbitrary shape. The inclusion size is much smaller than the array period while the wavelength is fixed. We derive and rigorously justify the dispersion relation for general frequencies and show that there are exceptional frequencies for which the solution is a cluster of waves propagating in different directions with different frequencies so that the dispersion relation cannot be defined uniquely. Examples are provided for the spherical inclusions.