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We present a novel multipole formulation for computing the band structures oftwo-dimensional arrays of cylindrical Helmholtz resonators. This formulation isderived by combining existing multipole methods for arrays of ideal cylinderswith the method of matched asymptotic expansions. We construct asymptoticallyclose representations for the dispersion equations of the first band surface,correcting and extending an established lowest-order (isotropic) result in theliterature for thin-walled resonator arrays. The descriptions we obtain for thefirst band are accurate over a relatively broad frequency and Bloch vectorrange and not simply in the long-wavelength and low-frequency regime, as is thecase in many classical treatments. Crucially, we are able to capture featuresof the first band, such as low-frequency anisotropy, over a broad range offilling fractions, wall thicknesses, and aperture angles. In addition todescribing the first band we use our formulation to compute the first band gapfor both thick- and thin-walled resonators, and find that thicker resonatorwalls correspond to both a narrowing of the first band gap and an increase inthe central band gap frequency.

Two-dimensional Helmholtz resonator arrays. Part I. Matched asymptotic expansions for thick- and thin-walled resonators