A convergent low-wavenumber, high-frequency homogenization of the wave equation in periodic media with a source term

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 ${\mathbb{R}}^d$ ($d⩾1$), 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.

Keywords:

• Waves in periodic media
• dynamic homogenization
• finite frequency
• band gap
• (Floquet-)Bloch transform
• variational formulation

2010 Mathematics Subject Classifications:

• 74J20
• 78M40

Disclosure statement

No potential conflict of interest was reported by the author(s).