In L ₂(R ^d;C ⁿ), we consider selfadjoint strongly elliptic second order differential operators A _ε with periodic coefficients depending on x/ε ε>0. We study the behavior of the operators cos⁡(A _ε ^1/2τ) and A _ε ^−1/2sin⁡(A _ε ^1/2τ), τ∈R, for small ε. Approximations for these operators in the (H ^s→L ₂)-operator norm with a suitable s are obtained. The results are used to study the behavior of the solution v _ε of the Cauchy problem for the hyperbolic equation ∂ _τ ²v _ε=−A _εv _ε+F. General results are applied to the acoustics equation and the system of elasticity theory.