We study the process of shock interaction of a solid body and elastic buffer, mounted on elastic orthotropic plate having cylindrical anisotropy, the dynamic behavior is described by wave equations such as Ufliand-Mindlin, take into account the rotational inertia of the cross-sections and shear deformations. We use the approach of the wave associated with the spread of the wave discontinuity surfaces in contacting the plate, and as a method of solving the ray method, and the method of matching asymptotic expansions obtained for short times in the contact zone and outside it. The interaction of the body with the plate in the latter begin to spread and quasilongitudinal quasitransverse waves representing the surface of strong discontinuity. Behind the front of the wave solution is constructed in the form of ray series whose coefficients are racing time derivatives of the unknown functions of various orders, and variable – the time elapsed since the arrival of the wave at a given point of the plate. The aim is to study the influence of the anisotropy of the plate material on dynamic contact force and the deflection occurring at the site of impact interaction.