Spectral approach to homogenization of hyperbolic equations with periodic coefficients

In L2(Rd;Cn), we consider selfadjoint strongly elliptic second order differential operators Aε with periodic coefficients depending on x/ε, ε>0. We study the behavior of the operators cos⁡(Aε1/2τ) and Aε−1/2sin⁡(Aε1/2τ), τ∈R, for small ε. Approximations for these operators in the (Hs→L2)-operator norm with a suitable s are obtained. The results are used to study the behavior of the solution vε of the Cauchy problem for the hyperbolic equation ∂τ2vε=−Aεvε+F. General results are applied to the acoustics equation and the system of elasticity theory.