Designing band gaps with randomly distributed sub-wavelength Helmholtz resonators
Erratum
Equation (12) for the T-matrix of a Helmholtz resonator has two sign errors. It should read exactly as Eq. (6): the term proportional to the resonance factor $z(\lambda)$ should have a minus sign, and its exponential should be $\mathrm e^{-\mathrm i(n-p)\theta_0}$. With the wrong sign in front of $z(\lambda)$, the T-matrix would not conserve energy. The wrong sign in the exponent only reverses the direction in which the aperture angle $\theta_0$ is measured. No result in the paper is affected, because the orientation-averaged T-matrix, Eq. (13), and our code in MultipleScattering.jl both use the correct form, Eq. (6).
In addition, the low-frequency expansion, Eq. (29), should read
$$h = \frac{1}{2} + \frac{2\mathrm{i}}{\pi (kb)^2} + \frac{4\mathrm{i}}{\pi}\left(\gamma_{\mathrm{e}} + \log\frac{k\ell}{4}\right) - \frac{3\mathrm{i}}{\pi}\left(\gamma_{\mathrm{e}} + \log\frac{kb}{2} + \frac{1}{4}\right) + o(1) \quad \text{as } kb \to 0,$$
so that $h \approx 1/2$, not $2$, at resonance; our numerical results use the full formula, Eq. (28), and are unaffected.
It is well-known that band gaps, in the frequency domain, can be achieved by using periodic metamaterials. However it has been challenging to design materials with broad band gaps or that have multiple overlapping band gaps. For periodic materials 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 we are able to reach simple formulae for the effective material properties, which work over a broad frequency range, and 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.
