A Second Order Homogenized Dispersive Wave Equation in a Quasiperiodic Medium

Wave propagation in a quasiperiodic medium is modeled by a second order homogenized scalar wave equation. The multiscale asymptotic expansion method for high-order homogenization of periodic structures in [1] is adapted to the quasiperiodic (cut-and-projection) setting. A periodic medium in a higher spatial dimension is used to model a quasiperiodic material by applying the cut-and-project procedure [2]. The partial differential operators (gradient and divergence) in the higher dimensional space are projected onto operators acting on quasiperiodic functions in a lower dimensional physical space. The second-order homogenized wave equation is dispersive, which is reflected by a fourth-order Burnett tensor for quasiperiodic structures.