Homogenization of hyperbolic equations with periodic coefficients in ${\mathbb{R}}^d$: sharpness of the results

In $L_2({\mathbb{R}}^d;{\mathbb{C}}^n)$, a selfadjoint strongly elliptic second order differential operator ${\mathcal{A}}_{\epsilon }$ is considered. It is assumed that the coefficients of the operator ${\mathcal{A}}_{\epsilon }$ are periodic and depend on $\mathbf{x}\mathrm{/}\epsilon$, where $\epsilon >0$ is a small parameter. We find approximations for the operators $\cos ( {\mathcal A}\varepsilon^{1/2}\tau)$and${\mathcal A}\varepsilon^{-1/2}\sin ( {\mathcal A}\varepsilon^{1/2}\tau)$ in the norm of operators acting from the Sobolev space $H^s({\mathbb{R}}^d)$ to $L_2({\mathbb{R}}^d)$ (with suitable $s$). We also find approximation with corrector for the operator ${\mathcal A}\varepsilon^{-1/2}\sin ( {\mathcal A}\varepsilon^{1/2}\tau)$inthe$(H^s \to H^1)$-norm. The question about the sharpness of the results with respect to the type of the operator norm and with respect to the dependence of estimates on $\tau$ is studied. The results are applied to study the behavior of the solutions of the Cauchy problem for the hyperbolic equation $\partial\tau^2 {\mathbf u}\varepsilon = - {\mathcal A}\varepsilon {\mathbf u}_\varepsilon + {\mathbf F}$.