  We present a solution method which combines the method of matched asymptotics
with the method of multipole expansions to determine the band structure of
cylindrical Helmholtz resonators arrays in two dimensions. The resonator
geometry is considered in the limit as the wall thickness becomes very large
compared with the aperture width (the specially-scaled limit). In this regime,
the existing treatment in Part I, with updated parameters, is found to return
spurious spectral behaviour. We derive a regularised system which overcomes
this issue and also derive compact asymptotic descriptions for the
low-frequency dispersion equation in this setting. In the specially-scaled
limit, our asymptotic dispersion equation not only recovers the first band
surface but also extends to high, but still subwavelength, frequencies. A
homogenisation treatment is outlined for describing the effective bulk modulus
and effective density tensor of the resonator array for all wall thicknesses.
We demonstrate that specially-scaled resonators are able to achieve
exceptionally low Helmholtz resonant frequencies, and present closed-form
expressions for determining these explicitly. We anticipate that the analytical
expressions and the formulation outlined here may prove useful in industrial
and other applications.
