Diffraction of a spherical sound wave by an elastic cylinder with an non-uniform anisotropic coating

In paper the problem of the diffraction of a spherical monochromatic sound wave on a homogeneous isotropic elastic cylinder with a radially non-uniform anisotropic elastic coating. It is assumed that the body is located in a free space filled with an ideal liquid. The analytical solution of the problem is obtained.
In the case of steady-state oscillations, the propagation of small perturbations in an ideal fluid is described by the Helmholtz scalar equation. The radiation field of a spherical source is written as a series expansion in cylindrical wave functions. The propagation of elastic waves in an isotropic elastic cylinder is described by the scalar and vector Helmholtz equations. The oscillations of an non-uniform anisotropic elastic cylindrical layer are described by the general equations of motion of a continuous medium.
The asymptotic formula for the far field zone is obtained by the steepest descent method.
Numerical calculations of the frequency characteristics of the scattered field are carried out for elastic cylinders with homogeneous and inhomogeneous transversally isotropic coatings, as well as for the case of a homogeneous isotropic coating. A significant influence and mutual influence of the inhomogeneity and anisotropy of the coating material on the acoustic properties of the scattering cylindrical body is revealed.