Long time approximations for solutions of wave equations associated with the Steklov spectral homogenization problems

The interest in the use of quasimodes , or almost frequencies and almost eigenfunctions , to describe asymptotics for low‐frequency and high‐frequency vibrations in certain singularly perturbed spectral problems, which depend on a small parameter ε, has been recently highlighted in many papers. In this paper we deal with the low frequencies for a Steklov‐type eigenvalue homogenization problem: we consider harmonic functions in a bounded domain of ℝ 2 , and strongly alternating boundary conditions of the Dirichlet and Steklov type on a part of the boundary. The spectral parameter appears in the boundary condition on small segments T ε of size O (ε) periodically distributed along the boundary; ε also measures the periodicity of the structure. We consider associated second‐order evolution problems on spaces of traces that depend on ε, and we provide estimates for the time t in which standing waves , constructed from quasimodes, approach their solutions u ε ( t ) as ε→0. Copyright © 2009 John Wiley & Sons, Ltd.