In (Formula presented.), we consider an elliptic differential operator (Formula presented.), (Formula presented.), of the form (Formula presented.) with periodic coefficients. For hyperbolic equations with the operator (Formula presented.), analogs of homogenization problems related to an arbitrary point of the dispersion relation of the operator (Formula presented.) are studied (the so-called high-frequency homogenization). For the solutions of the Cauchy problems for these equations with special initial data, approximations in (Formula presented.) -norm for small ε are obtained.