Diffraction of sound from a point source on an elastic cylinder with an inhomogeneous coating located near the elastic boundary

In paper the problem of the diffraction of a spherical monochromatic sound wave on a homogeneous isotropic elastic cylinder with a radially inhomogeneous elastic coating located near the boundary of half-spaces. It is assumed that the cylinder is located in the upper half-space filled with an ideal homogeneous liquid bordering on a homogeneous elastic half-space.
To represent a scattered field in an ideal fluid, a representation in the form of the Helmholtz-Kirchhoff integral is used, which is subsequently reduced to a system of linear algebraic equations with respect to the Fourier coefficients of the corresponding expansions of the total potential of the field and its normal derivative in a liquid half-space.
The oscillations of an inhomogeneous isotropic elastic layer are described by the general equations of motion of a continuous medium. To find the displacement field in an inhomogeneous coating, a boundary value problem for a system of second-order ordinary differential equations is constructed.
The asymptotic formula for the far field zone is obtained by the steepest descent method.
Numerical calculations of the angular characteristics of the scattered field are presented. A significant influence of continuously inhomogeneous coatings, as well as the presence of a plane near a cylindrical diffuser, on the diffraction pattern of the scattered field is revealed.