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 operator $\mathcal{ A}_\varepsilon$ are periodic and depend on $\mathbf{ x}/\varepsilon$. We study the behavior of the operator $\mathcal{ A}_\varepsilon ^{ -1/2}\sin (\tau \mathcal{ A}_\varepsilon ^{ 1/2})$, $\tau\in\mathbb{ R}$, in the small period limit. The principal term of approximation in the $(H^1\rightarrow L_2)$-norm for this operator is found. Approximation in the $(H^2\rightarrow H^1)$-operator norm with the correction term taken into account is also established. The results are applied to homogenization for the solutions of the nonhomogeneous hyperbolic equation $\partial ^2_\tau \mathbf{ u}_\varepsilon =-\mathcal{ A}_\varepsilon \mathbf{ u}_\varepsilon +\mathbf{ F}$.
Categories: math.AP