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

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, va...

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Bibliographic Details
Published in:arXiv.org 2022-04
Main Authors: Smith, M J A, Cotterill, P A, Nigro, D, Parnell, W J, Abrahams, I D
Format: Article
Language:English
Subjects:
Absorption cross sections
Asymptotic methods
Asymptotic properties
Coupling
Extinction
Helmholtz resonators
Multipoles
Plane waves
Resonance scattering
Scattering coefficient
Thick walls
Wave scattering
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Summary: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.
ISSN:2331-8422