  We present a novel multipole formulation for computing the band structures of
two-dimensional arrays of cylindrical Helmholtz resonators. This formulation is
derived by combining existing multipole methods for arrays of ideal cylinders
with the method of matched asymptotic expansions. We construct asymptotically
close representations for the dispersion equations of the first band surface,
correcting and extending an established lowest-order (isotropic) result in the
literature for thin-walled resonator arrays. The descriptions we obtain for the
first band are accurate over a relatively broad frequency and Bloch vector
range and not simply in the long-wavelength and low-frequency regime, as is the
case in many classical treatments. Crucially, we are able to capture features
of the first band, such as low-frequency anisotropy, over a broad range of
filling fractions, wall thicknesses, and aperture angles. In addition to
describing the first band we use our formulation to compute the first band gap
for both thick- and thin-walled resonators, and find that thicker resonator
walls correspond to both a narrowing of the first band gap and an increase in
the central band gap frequency.
