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 / λ ̄ , where λ = 2 π λ ̄ 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 ≪ λ ̄ is 1 / l ( R ) ∼ ω 5 + 2 n for three- and 1 / ł ( R ) ∼ ω 4 + 2 n 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 ≫ λ ̄ for three-dimensional roughness 1 / l ( R ) ∼ c o n s t independently on the considered correlator form. The value of c o n s t depends on the form-factor. For two-dimensional roughness at a ≫ λ ̄ 1 / l ( R ) ∼ ω 4 + 2 N e − ( a ω / c R ) 2 ( c R 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 / λ ̄ 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 ≫ λ ̄ 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 / λ ̄ . So, a new form of Rayleigh wave diffraction is theoretically found.