A non-axisymmetric diffraction problem of cylindrical sound waves on an elastic cylinder with an inhomogeneous coating located near the boundary of an elastic half-space

The article considers the problem of diffraction of a cylindrical sound wave on a homogeneous isotropic elastic cylinder with a radially inhomogeneous elastic coating located near the boundary of half-spaces in the case when the linear source is in a plane parallel to the surface of the half-space and is not parallel to the axis of the cylinder. It is assumed that the cylinder is located in a half-space filled with an ideal homogeneous liquid bordering on a homogeneous elastic half-space.
To represent the scattered field in an ideal liquid, a representation in the form of the Helmholtz-Kirchhoff integral is used. The oscillations of an inhomogeneous isotropic elastic body are described by the equations of the linear theory of elasticity. To find the displacement field in an inhomogeneous coating, a boundary value problem for a system of second-order ordinary differential equations is constructed.
Based on the solution of the direct problem, the inverse problem of determining the laws of coating heterogeneity that provide the least sound reflection in a given frequency range is considered. A functional is constructed expressing the average intensity of sound scattering in a given frequency range. The constructed functional is written in the form of a double integral, which cannot be evaluated analytically. The resulting integral is calculated numerically using a quadrature formula based on a parallelepipedal Korobov grid.
Numerical calculations of the angular characteristics of the scattered field are presented. A significant effect of continuously inhomogeneous coatings on the diffraction pattern of the scattered field has been revealed.