Transmission of spherical sound wave through an elastic plate with an inhomogeneous coating

In paper the problem of spherical sound wave reflection and transmission through a homogeneous isotropic elastic plate with a continuously inhomogeneous in thickness elastic coating is considered. It is believed that the plate is placed in an infinite ideal fluid, and the incident sound wave is harmonic and is generated by point source.
An analytical solution of the posed problem is obtained on the basis of the known solution of the problem about the passage of plane sound waves through plate with a continuously inhomogeneous coating and using integral representation of a spherical wave in the form of an expansion on flat waves.
Finding the displacement field in an inhomogeneous layer is reduced to solving boundary value problem for a system of ordinary differential equations of the second order.
The results of numerical calculations of frequency characteristics are presented for reflected and transmitted acoustic fields.