The homogenizations of initial boundary value problems for wave equations with asymptotically degenerate rapidly oscillating periodic coefficients are considered. Such problems model wave processes in heterogeneous periodic media like photonic crystals. The homogenized problems have the form of initial boundary value problems for integro-differential equations in convolutions and with external forces, even if in the original problem the forces were zero. The appearance of problems with convolutions indicates the possible presence of gaps in the spectrum of the relevant model, as is the case with semiconductors. Statements on the approximation of solutions to the original problem by solutions of the homogenized problem in various cases are discussed.