Diffraction of sound from a point source on a cylinder with an elastic coating surrounded by an inhomogeneous liquid layer

The article considers the problem of diffraction of a spherical sound wave by an absolutely rigid cylinder with a coating in the form of a homogeneous isotropic elastic layer with an adjacent inhomogeneous liquid layer. It is assumed that a cylinder with a homogeneous coating is surrounded by a continuously inhomogeneous 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 acoustic pressure in a spherical wave is represented in an integral form as a decomposition in cylindrical wave functions. Wave processes in an elastic layer are described by a system of equations of the linear theory of elasticity of an isotropic body. To determine the wave field in an inhomogeneous liquid layer, a boundary value problem for an ordinary differential equation of the second order is constructed.