A convergent low-wavenumber, high-frequency homogenization of the wave equation in periodic media with a source term. Issue 18 (12th December 2022) - Meng, Shixu; Oudghiri-Idrissi, Othman; Guzina, Bojan B.
Abstract : We pursue a low-wavenumber, second-order homogenized solution of the time-harmonic wave equation at both low and high frequency in periodic media with a source term whose frequency resides inside a band gap. Considering the wave motion in an unbounded medium R d ( d ⩾ 1 ), we first use the (Floquet-)Bloch transform to formulate an equivalent variational problem in a bounded domain. By investigating the source term's projection onto certain periodic functions, the second-order model can then be derived via asymptotic expansion of the Bloch eigenfunction and the germane dispersion relationship. We establish the convergence of the second-order homogenized solution, and we include numerical examples to illustrate the convergence result.