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