Honeycomb-Lattice Minnaert Bubbles

Abstract

The ability to manipulate the propagation of waves on subwavelength scales is important for many different physical applications. In this paper, we consider a honeycomb-lattice of subwavelength resonators and prove, for the first time, the existence of a Dirac dispersion cone at subwavelength scales. As shown in [Ammari, Hiltunen, and Yu, Arch. Ration. Mech. Anal., 238 (2020), pp. 1559--1583], near the Dirac points, the use of honeycomb crystals of subwavelength resonators as near-zero materials has great potential. Here, we perform the analysis for the example of bubbly crystals, which is a classic example of subwavelength resonance, where the resonant frequency of a single bubble is known as the Minnaert resonance. Our first result is to derive an asymptotic formula for the quasi-periodic Minnaert resonance frequencies close to the symmetry points K in the Brilloun zone. Then we obtain the linear dispersion relation of a Dirac cone. Our findings in this paper are illustrated in the case of circular bubbles, where the multipole expansion method provides an efficient technique for computing the band structure. © 2020, Society for Industrial and Applied Mathematics.