Two-dimensional Waves Around Almost Periodic Arrangements of Scatterers∗

Acoustic waves around an infinite configuration of identical circular scatterers are considered. Each scatterer is close to a node of a regular lattice: the geometrical configuration is almost periodic. Analytical estimates for the average field in such a random medium are obtained. Introduction Consider waves in a two-dimensional periodic structure, defined by a lattice Λ: each cell in the lattice is a parallelogram, each node in the lattice is a scatterer location. Let d be the shortest distance between nodes. For scalar waves governed by the Helmholtz equation, (∇2 +k2)u = 0, it is known how to calculate the dispersion relation, connecting the wavenumber k to the Bloch vector Q: solutions satisfy the Bloch condition u(r+rj) = u(r) exp (iQ · rj), for every lattice node rj . The periodic problems outlined above have been studied extensively. One important application concerns photonic crystals [1]. Fabrication of such structures inevitably introduces imperfections, leading to nearly periodic geometries or other forms of disorder. What are the effects of the disorder? There are publications on this question; the main result is that the band-gap phenomena seen with periodic structures are robust to small amounts of random disorder. Representative publications include [2], [3], [4], [5], [6]. All these papers include results from numerical simulations. Some [4], [6] use a ‘supercell’ method, which means that a periodic medium is constructed in which each period contains the same disordered arrangement of circular scatterers; evidently, such a periodic medium is not a random medium, so it is unclear how to interpret the results. The other papers [2], [3], [5] use a finite number of circular scatterers, 1152 in [2], 38 in [3] and 169 in [5]. 1 The periodic problem We consider identical circular scatterers of radius a. For simplicity, we suppose that ka 1 and that each scatterer is sound soft. This permits the use of Foldy’s (deterministic) model for the scattering: