Scattering of sound waves by an cylinder with an radial non-uniform elastic coating in a planar waveguide

In paper the problem of sound wave scattering by absolutely rigid cylinder with radially inhomogeneous isotropic elastic coating in a planar waveguide is considered. It is believed that a waveguide filled with a homogeneous ideal fluid, one of its borders is absolutely rigid and the other — acoustically soft, heterogeneity laws of a coating material are described by differentiable functions, harmonic sound wave excited by a given distribution of sources in the section waveguide.
In the case of steady state oscillations the propagation of small perturbations in ideal fluid is described by the Helmholtz's equation. The oscillations of an inhomogeneous isotropic elastic cylindrical layer described by general motion equations of the continuous medium.
The boundary-value problem for the system of ordinary second order differential equations is constructed for determination of the displacement field in inhomogeneous coating.
The primary field of disturbances is represented by a set of its own waveguide waves. The pressure of the field scattered by the cylindrical body is sought as potential of a simple layer.
The Green function for the Helmholtz equation that satisfies the given boundary conditions on the waveguide walls and conditions of radiation at infinity is constructed. The function of distribution density of sources are sought as a Fourier series expansion. The infinite linear system of equations is obtained for determination of the coefficients of this decomposition. The solution of truncated infinite system is found by the inverse matrix method.
Analytical expressions for the scattered acoustic field in different areas of the waveguide are obtained.