The asymptotic behaviour as $t \to \infty $, $|z| < a < \infty $ of solutions of exterior mixed problems for hyperbolic equations and systems is obtained when the boundary of a domain and coefficients of the equations depend periodically on time. It is supposed that the coefficients are constant in a neigborhood of infinity and that the non-trapping condition is fulfilled. The method of the research is based on using a special parametrix, Fourier-Bloch transform and analytical properties of an integral equation which arises. This method can be regarded as an alternative one to the Lax-Phillips scattering theory. Then the asymptotic behavior of the solutions is used to prove existence of the wave operators and of the scattering operator, if the general energy of any solution is uniformly bounded for $t \geq 0$ provided that it is bounded at $t = 0$.