Abstract: 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 ?, 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 ?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? of size O(?) periodically distributed along the boundary; ? also measures the periodicity of the structure. We consider associated second-order evolution problems on spaces of traces that depend on ?, and we provide estimates for the time t in which standing waves, constructed from quasimodes, approach their solutions u?(t) as ??0.

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