Asymptotics of the meta-atom: plane wave scattering by a single Helmholtz resonator

TL;DR

This paper develops a mathematical framework to analyze how a single Helmholtz resonator scatters plane acoustic waves, deriving explicit formulas and examining effects of wall thickness and dissipation on resonance behavior.

Contribution

It introduces a combined multipole and asymptotic method to derive closed-form scattering coefficients for a single resonator across different wall thickness regimes, including dissipation effects.

Findings

Strong field and cross section enhancement at resonance frequencies

Dissipation shifts resonance frequency and reduces scattering efficiency

Depolarisability effects do not guarantee Willis coupling in arrays

Abstract

Using a combination of multipole methods and the method of matched asymptotics, we present a solution procedure for acoustic plane wave scattering by a single Helmholtz resonator in two dimensions. Closed-form representations for the multipole scattering coefficients of the resonator are derived, valid at low frequencies, with three fundamental configurations examined in detail: the thin-walled, moderately thick-walled, and very thick-walled limits. Additionally, we examine the impact of dissipation for very thick-walled resonators, and also numerically evaluate the scattering, absorption, and extinction cross sections (efficiencies) for representative resonators in all three wall thickness regimes. In general, we observe strong enhancement in both the scattered fields and cross sections at the Helmholtz resonance frequencies. As expected, dissipation is shown to shift the resonance frequency, reduce the amplitude of the field, and reduce the extinction efficiency at the fundamental Helmholtz resonance. Finally, we confirm results in the literature on Willis-like coupling effects for this resonator design, and crucially, connect these findings to earlier works by the authors on two-dimensional arrays of resonators, deducing that depolarisability effects (off-diagonal terms) for a single resonator do not ensure the existence of Willis coupling effects (bianisotropy) in bulk.