Reflection and refraction of a plane sound wave in an elastic plate with an inhomogeneous anisotropic coating

In paper the problem of the reflection and refraction of a plane harmonic sound wave by a homogeneous isotropic elastic plate with an continuously inhomogeneous anisotropic elastic coating is considered.
It is assumed that the plate is adjoin to ideal fluids.
The propagation of small perturbations in a ideal fluid in the case of steady state oscillations is described by Helmholtz's equation. The propagation of elastic waves in a uniform isotropic elastic plate is described by scalar and vector Helmholtz's equations for longitudinal and transverse waves. Oscillations of an inhomogeneous anisotropic elastic coating are described by general motion equations of the continuous medium.
The boundary-value problem for the system of ordinary second order differential equations is constructed for determination of the displacement field in inhomogeneous anisotropic coating.
An analytical description of the reflected and transmitted through the plate acoustic fields is obtained.
Particular cases when the material of an inhomogeneous coating is transversely isotropic and isotropic are considered.
The results of numerical calculations of dependence of reflection coefficient of a homogeneous isotropic plate with a transversely isotropic coating from the incidence angle of plane wave are presented.