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Let $\mathcal{O}\subset \mathbb{R}^d$ be a bounded domain of class $C^{1,1}$.In $ L_2(\mathcal{O};\mathbb{C}^n)$, we consider a matrix elliptic second orderdifferential operator $A_{D,\varepsilon}$ with the Dirichlet boundarycondition. Here $\varepsilon >0$ is a small parameter. The coefficients of theoperator $A_{D,\varepsilon}$ are periodic and depend on$\mathbf{x}/\varepsilon$. The principal terms of approximations for theoperator cosine and sine functions are given in the $(H^2\rightarrow L_2)$- and$(H^1\rightarrow L_2)$-operator norms, respectively. The error estimates are ofthe precise order $O(\varepsilon)$ for a fixed time. The results in operatorterms are derived from the quantitative homogenization estimate forapproximation of the solution of the initial-boundary value problem for theequation $(\partial _t^2+A_{D,\varepsilon})\mathbf{u}_\varepsilon =\mathbf{F}$.

Homogenization of the first initial-boundary value problem for periodic hyperbolic systems. Principal term of approximation