Rayleigh wave at the boundary of an inhomogeneous nonlinear viscoelastic half-space
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In the approximation of weak nonlinearity and weak viscosity of the medium, we obtain an equation describing the spectral density of the particle horizontal velocity for a Rayleigh wave propagating along the boundary of a half-space. The coefficients of nonlinear interaction between the wave harmonics are found on the assumption that the third-order elastic moduli arbitrarily depend on the depth. We find expressions for the complex correction to the wave frequency due to small relaxation corrections to the elastic moduli and small variations in the medium density, which arbitrarily depend on the depth as well. The imaginary part of this correction to the frequency determines the decay of the linear Rayleigh wave due to small relaxation corrections to the elastic moduli arbitrarily dependent on the depth. Using numerical simulation (with allowance for the interaction of 500 harmonics), we study distortions of an initially harmonic Rayleigh wave for a particular dependence of variations in the nonlinear moduli on the depth. An integral equation is derived for the nonlinear elastic moduli as functions of the depth. It is shown that for independent spatio-temporal distributions of the viscous moduli, functions determining the dependence of the viscosity on the depth are described by an analogous integral equation.
KeywordsSurface Wave Rayleigh Wave Wave Spectrum Nonlinear Distortion Relaxation Modulus
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