Abstract

In this talk I shall outline a framework for describing the effective propagation or scattering response of two-dimensional composite bodies, or surfaces, composed of small (subharmonic) inclusions.  We shall focus on acoustic waves, although the methodology is applicable to elastic, electromagnetic or other waves. The presentation will be split into two parts, the first will focus on periodic inclusions (cylinders) which have resonant properties, and the second will examine randomly distributed cylinders. Multipole methods will allow both problems to be formulated as systems of equations, which in the former case are simplified via the method of matched asymptotic expansions, and in the latter case via ensemble averaging and the quasi-crystalline approximation. In the low frequency limit, methods of complex analysis are shown to be at the heart of obtaining simplified expressions for the dispersion relation and effective material properties. Results will be presented as time allows. 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, 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, 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, 2022.4. Artur L. Gower, William J. Parnell and I. David Abrahams, Multiple Waves Propagate in Random Particulate Materials, SIAM Journal on Applied Mathematics, 79(6), 2569-2592, 2019.5. Artur L. Gower, I. David Abrahams and William J. Parnell, A proof that multiple waves propagate in ensemble-averaged particulate materials, Proceedings of the Royal Society A475 , 20190344, 2019.