Diffraction of plane ellastic waves of the vertical polarization on a small inhomogeneity inside an ellastic layer

The problem of diffraction of the ellastic plane wave of the vertical polarization on a small inhomogeneity lying in a layer is investigated. The layer is situated on the ellastic half-space. We consider three-layer model of isotropic ellastic theory. The inhomogeneity is a circular cylinder, radius $a$ of which is small in comparision with the length of the falling wave. The insident wave is supposed to be polarized ortogonal to the axis of the cylinder.
Diffraction addition from the small inhomogeneity in the wave, reflected from the elastic layer, is proved to have the more order than the first with respect to parameter $(ka)^2/\sqrt{kr}$, $kr\gg 1$, $ka\ll 1$, where $k$ is the wave number, $r$ is distance between the inhomogeneity and observer point. The small inhomogeneity generates the cyliner wave, intensifity of which is proportional to the area of the inhomogeneity cross-section, to the jumps of the square velocities in the layer and in the inhomogeneity.
The diffraction coefficients, determining the radiation pattern of the scattering wave are obtained. The scattering of vertical polarised field by the inhomogeneity is behaved as scattering by a point source for $kr\gg 1$. The power of the scattering is proportional to the area of the inhomogeneous cross-section, the jumps of the densities $\rho_i$, $i=0,1,2,3$ and the jumps of Lame parameters $\mu_i$, $\lambda_i$ of the media and the inhomogeneity.