Designing band gaps with randomly distributed sub-wavelength Helmholtz resonators

It is well-known that band gaps for wave propagation, in the frequency domain, can be achieved by using periodically-arranged inclusions in a host material. However it has been challenging to design materials with broad band gaps or that have multiple overlapping band gaps. For periodic composites this difficulty arises because many different length scales would have to be repeated periodically within the same structure to have multiple overlapping band gaps. Here we present an alternative: to design band gaps with disordered materials. We show how to tailor band gaps by choosing any combination of Helmholtz resonators that are positioned randomly within a host acoustic medium. One key result is that, via analytical (asymptotic) analysis, we are able to derive simple formulae for the effective material properties, which work over a broad frequency range. These can therefore be used to rapidly design tailored metamaterials. We show that these formulae are robust by comparing them with high-fidelity Monte-Carlo simulations over randomly positioned resonant scatterers.