Dispersion of Rayleigh waves in vertically-inhomogeneous prestressed elastic media

Herein we consider Rayleigh waves propagating in a given direction along the traction-free boundary of an anisotropic and prestressed elastic half-space, the effective incremental elasticity tensor and mass density of which are smooth functions of depth from the traction-free surface. We derive a high-frequency asymptotic formula which, for a large wave number, expresses the phase velocity of the Rayleigh waves in question in terms of the values of the effective incremental elasticity tensor, the mass density, and their first and higher-order normal derivatives at the traction-free boundary. We develop a procedure which, through an asymptotic expansion of the surface impedance matrix, can deliver an expression for each term of the asymptotic expansion of the Rayleigh-wave velocity. Our asymptotic formula thus gives a characterization of the frequency dependence of Rayleigh waves as caused by the vertical inhomogeneity of the medium, thereby providing a mathematical foundation for work on non-destructive evaluation of the depth dependence of near-surface stress and/or material parameters through their effect on the dispersion of Rayleigh waves.