Abstract

In this talk I will outline a framework, developed with Dr Mike Smith, for describing the effective propagation or scattering response of two-dimensional composite bodies or surfaces composed of subharmonic resonator inclusions.  We shall focus on acoustic waves, and employ as a Helmholtz resonator element, a rigid cylinder with a narrow notch (or neck) connecting to an interior cavity. Multipole methods have been used to good effect to obtain the effective behaviour of cylindrical inclusions in the many areas of photonics and phononics. We examine this newer design element, the resonant cylinder, to create metamaterials and metasurfaces, via the combined methods of matched asymptotic expansions and multipoles. I shall commence the talk by briefly summarising the evaluation of the propagation characteristics of a single resonant scatterer via matched asymptotic expansions, and then use this to derive the low-frequency solution for one-dimensional and two-dimensional arrays of circular cylindrical resonators. I shall conclude by offering brief results and indicating the benefits of this approach to practical metamaterial design. References: 1. Michael J. A. Smith and I. David Abrahams, Tailored acoustic metamaterials. Part I. Thin- and thick-walled Helmholtz resonator arrays, Proceedings of the Royal Society A478 , 20220124, June 2022. 2. Michael J. A. Smith and I. David Abrahams, Tailored acoustic metamaterials. Part II. Extremely thick-walled Helmholtz resonator arrays, Proceedings of the Royal Society A478 , 20220125, June 2022. 3. Michael J. A. Smith, Philip A. Cotterill, David Nigro, William J. Parnell and I. David Abrahams, Asymptotics of the meta-atom: plane wave scattering by a single Helmholtz resonator, Philosophical Transactions of the Royal Society 380, 20210383, October 2022.