We present here some corrector results for the homogenization of the wave equation in a two-component composite with $\varepsilon$-periodic connected inclusions which complete the homogenization results proved in [P. Donato, L. Faella, and S. Monsurro, J. Math. Pures Appl., 87 (2007), pp. 119–143] by the authors. On the interface separating the two components we prescribe a jump of the solution proportional to the conormal derivatives via a function of order $\varepsilon^\gamma$, with $-1<\gamma\leq1$. Due to different expressions of the energies of the limit problems, the cases $-1<\gamma<1$ and $\gamma=1$ need to be treated separately. The second one, where a memory effect appears in the homogenized problem, is the most interesting. For this critical case, displaying lack of compactness, we in particulur establish the central upper semicontinuity type inequalities by splitting a related energy term into a compact part and a part vanishing in appropriate norms.