Low frequency acoustic stop bands in cubic arrays of thick spherical shells with holes

TL;DR

This paper investigates low frequency acoustic stop bands in 3D arrays of thick spherical shells with holes, revealing how these structures support localized modes due to Helmholtz resonances, with potential for designing acoustic filters.

Contribution

It introduces a new analysis of low frequency stop bands in shell arrays with holes, combining band diagram calculations, a mechanical model, and a practical design for localized modes.

Findings

Identification of low frequency stop bands due to Helmholtz resonances.

Development of a mechanical model for estimating stop band frequencies.

Design of a macrocell with defect resonators supporting localized modes.

Abstract

We analyse the propagation of pressure waves within a fluid filled with a three-dimensional array of rigid coated spheres (shells). We first draw band diagrams for corresponding Floquet-Bloch waves. We then dig a channel terminated by a cavity within each rigid shell and observe the appearance of a low frequency stop band. The underlying mechanism is that each holey shell now acts as a Helmholtz resonator supporting a low frequency localized mode: Upon resonance, pressure waves propagate with fast oscillations in the thin water channel drilled in each shell and are localized in each fluid filled inner cavity. The array of fluid filled shells is approximated by a simple mechanical model of springs and masses allowing for asymptotic estimates of the low frequency stop band. We finally propose a realistic design of periodic macrocell with a large defect surrounded by 26 resonators connected by thin straight rigid wires, which supports a localized mode in the low frequency stop band.