Reflection of a spherical sound wave from an elastic half-space with an adjacent inhomogeneous liquid layer

In paper the problem of reflection of a spherical sound wave from an elastic half -space with an adjacent inhomogeneous liquid layer is considered. It is assumed that a homogeneous isotropic elastic half-space is covered by an continuously-inhomogeneous of a plane liquid layer with an arbitrary law of inhomogeneity. A point source of harmonic sound waves is placed in an ideal homogeneous liquid bordering an inhomogeneous layer.
The analytical solution of the viewed problem is obtained on the basis of the solution of a similar problem for the case of a plane incident wave.
The acoustic pressure in a spherical wave is represented in integral form as a decomposition on a plane waves. The integrand expression turns out to be similar in form to the expression for pressure in a plane incident wave. Therefore, the pressure in a scattered wave in the case of a spherical wave falling on a half-space with an inhomogeneous liquid layer is written as an integral, the integral expression of which is similar in form to the expression for the pressure in a scattered wave when a plane wave falls. For the determination of the wave field in an inhomogeneous liquid layer the boundary value problem for a system of ordinary differential equations of the second order is built, the approximate analytical solution of which is obtained by the power series method.