Let (Formula presented.) be a bounded domain of class (Formula presented.). In (Formula presented.), we consider a self-adjoint matrix strongly elliptic second-order differential operator (Formula presented.), (Formula presented.), with the Dirichlet boundary condition. The coefficients of the operator (Formula presented.) are periodic and depend on (Formula presented.). We are interested in the behavior of the operators (Formula presented.) and (Formula presented.), (Formula presented.), in the small period limit. For these operators, approximations in the norm of operators acting from a certain subspace (Formula presented.) of the Sobolev space (Formula presented.) to (Formula presented.) are found. Moreover, for (Formula presented.), the approximation with the corrector in the norm of operators acting from (Formula presented.) to (Formula presented.) is obtained. The results are applied to homogenization for the solution of the first initial-boundary value problem for the hyperbolic equation (Formula presented.).