Scattering of sound waves by an elastic ellipsoid with an inhomogeneous coating in the half-space with ideal surface

The solution of the diffraction problem for a plane sound wave on an elastic ellipsoid $E$ with an outer inhomogeneous layer is presented. The ellipsoid is in a half-space filled with an ideal fluid. The boundary of a half-space $\Pi$ is an acoustically rigid or acoustically soft surface.
To obtain a solution, the area occupied by the liquid is expanded to full space. An additional scattering obstacle is introduced. This obstacle is a copy of $E$, located mirror-wise with respect to the plane $\Pi$. 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 ellipsoids 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 environment, the solution is sought analytically in the form of an expansion in spherical harmonics and Bessel functions. In the spherical region, which includes two ellipsoids 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 dependences show the influence of the geometric and material parameters of the ellipsoid on the diffraction of sound.