Scattering of a plane sound wave by a sphere with an inhomogeneous anisotropic coating near the flat surface

In paper the problem of scattering of a plane sound wave by an absolutely solid sphere with a continuously inhomogeneous anisotropic elastic coating in the presence of a flat surface is considered. It is believed that the body is placed in an ideal fluid, the spreading flat surface is absolutely rigid and absolutely soft, the laws of heterogeneity of the coating material are described by differentiable radial coordinate functions.
The approximate analytical solution to the problem is obtained for the case when the material of the sphere coating is radially inhomogeneous and transversally isotropic. In this case the reflection from the plane of the waves scattered by the body is not taken into account, but scattering by the sphere of the wave arising from the reflection of the incident wave from the plane is taken into account.
By virtue of the linear formulation of the problem, the velocity potential of the total acoustic field is represented as the sum of the potentials of the incident plane wave; wave arising from the reflection of the incident plane waves from the plane; wave arising from the scattering of an incident plane wave by sphere; wave arising from scattering by a sphere reflected from plane of the wave.
Wave fields in a containing medium are described by expansions in spherical wave functions. A boundary value problem is constructed for a system of ordinary differential equations of the second order for finding displacement fields in an inhomogeneous anisotropic coating of sphere.