Bloch waves in bubbly crystal near the first band gap: a high-frequency homogenization approach

TL;DR

This paper develops a high-frequency homogenization method to analyze Bloch waves in bubbly phononic crystals, explaining super-focusing phenomena and band gap opening near the first band gap.

Contribution

It proves the maximum of the first Bloch eigenvalue at the Brillouin zone corner and decomposes eigenfunctions into homogenized and periodic parts, advancing understanding of wave behavior in bubbly media.

Findings

First Bloch eigenvalue maximized at Brillouin zone corner

Eigenfunctions decompose into homogenized and periodic components

Confirmed band gap opening near critical frequency

Abstract

This paper is concerned with the high-frequency homogenization of bubbly phononic crystals. It is a follow-up of the works [H. Ammari et al., Sub-wavelength phononic bandgap opening in bubbly media, J. Diff. Eq., 263 (2017), 5610--5629] which shows the existence of a sub-wavelength band gap. This phenomena can be explained by the periodic inference of cell resonance which is due to the high contrast in both the density and bulk modulus between the bubbles and the surrounding medium. In this paper, we prove that the first Bloch eigenvalue achieves its maximum at the corner of the Brillouin zone. Moreover, by computing the asymptotic of the Bloch eigenfunctions in the periodic structure near that critical frequency, we demonstrate that these eigenfunctions can be decomposed into two parts: one part is slowly varying and satisfies a homogenized equation, while the other is periodic across each elementary crystal cell and is varying. They rigorously justify, in the nondilute case, the observed super-focusing of acoustic waves in bubbly crystals near and below the maximum of the first Bloch eigenvalue and confirm the band gap opening near and above this critical frequency.