On homogenization of the first initial-boundary value problem for periodic hyperbolic systems

Let O subset of R-d be a bounded domain of class C-3,C-1. In L-2(O;C-n), we consider a self-adjoint matrix strongly elliptic second-order differential operator B-D,B-epsilon,0 < epsilon <= 1, with the Dirichlet boundary condition. The coefficients of the operator B-D,B-epsilon are periodic and depend on x/epsilon. We are interested in the behavior of the operators cos (tB(D,epsilon)(1/2)) and B-D,epsilon(-1/2) sin(t(BD,epsilon)(1/2)), t is an element of R, in the small period limit. For these operators, approximations in the norm of operators acting from a certain subspaceHof the Sobolev space H-4(O;C-n) to L-2(O;C-n)are found. Moreover, for B-D,epsilon(-1/2) sin(t(BD,epsilon)(1/2)), the approximation with the corrector in the norm of operators acting from H subset of H-4 (O;C-n) to H-1(O;C-n)is obtained. The results are applied to homogenization for the solution of the first initial-boundary value problem for the hyperbolic equation partial differential partial derivative(2)(t)u(epsilon) = -B(D,epsilon)u(epsilon).