Two-dimensional Helmholtz resonator arrays. Part II. Matched asymptotic expansions for specially-scaled resonators

TL;DR

This paper develops a regularized asymptotic method combining matched asymptotics and multipole expansions to analyze the band structure of two-dimensional Helmholtz resonator arrays in a specially-scaled limit, extending understanding to high subwavelength frequencies.

Contribution

It introduces a regularized asymptotic dispersion equation for specially-scaled resonators, overcoming previous spurious spectral behavior and providing explicit formulas for low-frequency and resonant properties.

Findings

Derived a regularized system for accurate spectral analysis.

Extended the dispersion equation to high subwavelength frequencies.

Provided explicit formulas for low-frequency resonant behavior.

Abstract

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.