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

By means of an continuous-non-uniform elastic coatings it is possible to change effectively scattering performances of bodies in determinate directions if to pick up corresponding the inhomogeneity laws for mechanical parametres of a coating. In paper the problem of diffraction of a spherical sound wave by a homogeneous isotropic elastic cylinder with radially non-uniform elastic coating is considered. It is believed that an infinite circular cylinder with a coating is placed in an ideal unlimited fluid, heterogeneity laws of a coating material are described by differentiable functions, on the body falls а harmonic spherical sound wave emitted by a point source.
In the case of steady state oscillations the propagation of small perturbations in ideal fluid is described by the scalar Helmholtz's equation, and in elastic homogeneous isotropic cylinder — scalar and vector Helmholtz's equations. The oscillations of an inhomogeneous isotropic elastic cylindrical layer described by general motion equations of the continuous medium.
The analytical solution of the viewed problem was obtained on the basis of the known solution for a similar problem of the diffraction of a plane wave. The velocity potential of a spherical wave is represented in integral form as a decomposition on wave cylindrical functions. The integrand turns out to be similar in form to the expression of the velocity potential of a plane wave. The velocity potential of the scattered wave in the case of a falling of a spherical wave on a cylinder with a coating is written as an integral, the integrand of which is similar in form to the expression of the potential of the scattered wave when a plane wave falls on the body. It is necessary to determine the displacement field in a non-uniform coating to calculate the integrand. For this the built boundary-value problem for the system of ordinary differential equations of the second order must be solved. The computational aspects of integral evaluation are considered.