  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.
