Rayleigh wave scattering by statistical arbitrary form roughness

The problem of Rayleigh wave scattering by three- and two-dimensional statistical roughness of isotropic solid is solved in the Born (Rayleigh-Born) approximation of perturbation theory in roughness amplitude. Statistically homogeneous and isotropic roughness is described by a correlation function which has the form of an exponentially modulated Chebyshëv-Laguerre polynomials sum. This approximation of a correlator is new, but it agrees with experimental data well. Expressions are derived for the coefficient of scattering into a secondary Rayleigh wave and asymptotic expressions for it in different limits in a/λ̄, where λ=2πλ̄ is the wavelength, a is the correlation radius of roughness. It is shown, that the frequency dependence of the scattering coefficient in the Rayleigh limit a≪λ̄ is 1/l(R)∼ω5+2n for three- and 1/ł(R)∼ω4+2n for two-dimensional roughness, where n=0,1,2,3,… depending on the roughness form, i.e. violation of the Rayleigh law of scattering is established. It is found, that in short-wavelength limit a≫λ̄ for three-dimensional roughness 1/l(R)∼const independently on the considered correlator form. The value of const depends on the form-factor. For two-dimensional roughness at a≫λ̄1/l(R)∼ω4+2Ne−(aω/cR)2 (cR is the Rayleigh wave velocity, N=0,1,2,3,… and depends on roughness form). It is shown, that the structure of three-dimensional roughness strongly influences the scattering angular distribution in all ranges of a/λ̄ variation. It is a violation of the law about Rayleigh scattering isotropy (except forbidden angles); and of the law about the maximum of diffuse a≫λ̄ scattering in directions close to the forward one. The form of 1/l(R) strongly depends on the roughness form-factor in all ranges of a/λ̄. So, a new form of Rayleigh wave diffraction is theoretically found.