Diffraction of sound on an elastic sphere with a nonhomogeneous coating and cavity in semi-space

The problem for diffraction a plane harmonic sound wave on an elastic sphere $T$ with cavity near ideal plane $\Pi$ is considered. The outer ball layer is nonhomogeneous.
The solution is carried out by expanding the scope of the problem to the full space. In this case, an additional obstacle is introduced, which is a copy of $T$, which is located mirror with respect to the $\Pi$.
A second incident plane wave is also added. This wave ensures the fulfillment of that condition at the points of the plane $\Pi$, which corresponds to the type of the half-space boundary in the initial formulation of the problem. Thus, the problem is transformed into the problem of scattering of two plane sound waves on two inhomogeneous spheres in unbounded space.
The solution is based on the linear theory of elasticity and the model of propagation of small vibrations in an ideal fluid. In the outer part of the liquid, the solution is sought analytically in the form of an expansion in spherical harmonics and Bessel functions. In the spherical region, which includes two elastic balls and an adjacent layer of liquid, the finite element method (FEM) is used.
The results of the calculation of the directivity patterns of the scattered sound field in the far zone are presented. These dependencies show the influence of the geometric and material parameters of the obstacle on the diffraction of sound.