The transmission of sound through an elastic plate with an inhomogeneous coating adjoining viscous liquids

In paper the problem of the reflection and transmission of a plane sound wave through a homogeneous elastic plate with an continuously inhomogeneous plate elastic coating adjoining viscous liquids is considered. It is believed that heterogeneity laws of a coating are described by differentiable functions.
The propagation of small perturbations in a viscous fluid in the case of steady state oscillations is described by scalar and vector Helmholtz's equations. The propagation of elastic waves in a uniform isotropic elastic plate is described by two wave equations for longitudinal and transverse waves. Oscillations of an inhomogeneous isotropic 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 coating.
An analytical description of the reflected and transmitted through the plate acoustic fields is obtained.
The results of numerical calculations of dependences of coefficients of reflection and transmission of longitudinal waves from the angle of incidence of plane wave are presented.