A convergent low-wavenumber, high-frequency homogenization of the wave equation in periodic media with a source term. Issue 18 (12th December 2022)

• Record Type: Journal Article
• Authors: Meng, Shixu; Oudghiri-Idrissi, Othman; Guzina, Bojan B.
• Is Part Of: Applicable analysis. Volume 101:Issue 18(2022)
• Page Start: 6451
• Page End: 6484
• Publication Date: 2022-12-12
• Subjects: Waves in periodic media -- dynamic homogenization -- finite frequency -- band gap -- (Floquet-)Bloch transform -- variational formulation; 74J20 -- 78M40; Mathematical analysis -- Periodicals; 515
• DOI: 10.1080/00036811.2021.1929932
• Languages: English
• ISSNs: 0003-6811

Abstract

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.