On operator error estimates for homogenization of hyperbolic systems with periodic coefficients

In $L_2(\mathbb{R}^d;\mathbb{C}^n)$, we consider a selfadjoint matrix strongly elliptic second order differential operator $\mathcal{A}_\varepsilon$, $\varepsilon >0$. The coefficients of the $\mathcal{A}_\varepsilon$ are periodic and depend on $\mathbf{x}/\varepsilon$. We study behavior $\mathcal{A}_\varepsilon ^{-1/2}\sin (\tau \mathcal{A}_\varepsilon ^{1/2})$, $\tau\in\mathbb{R}$, in small period limit. principal term approximation $(H^1\rightarrow L_2)$-norm for this is found. Approximation $(H^2\rightarrow H^1)$-operator norm with correction taken into account also established. results applied to homogenization solutions nonhomogeneous hyperbolic equation $\partial ^2_\tau \mathbf{u}_\varepsilon =-\mathcal{A}_\varepsilon +\mathbf{F}$.