Consider the mixed problem with Dirichelet condition associated to the wave equation ∂ 2t  u − divx(ɑ(t, x)∇x u) = 0, where the scalar metric ɑ(t, x) is T-periodic in t and uniformly equal to 1 outside a compact set in x, on a T-periodic domain. Let 𝘜(t, 0) be the associated propagator. Assuming that the perturbations are non-trapping, we prove the meromorphic continuation of the cut-off resolvent of the Floquet operator 𝘜(T, 0) and we establish sufficient conditions for local energy decay.