An explicit relation for the apparent phase velocity of Rayleigh waves in a vertically heterogeneous elastic half-space

This paper presents the mathematical derivation of an explicit relation for the apparent (or effective) phase velocity of Rayleigh waves in a vertically heterogeneous, isotropic elastic half-space for harmonic excitation. As a kinematical feature, the apparent phase velocity captures the superposition, in a spatial Fourier series, of the individual modes of propagation of Rayleigh waves and describes the speed of propagation of a composite waveform generated by a vertically oscillating point load. The relation, which is a function of the distance from the source, frequency and depth, depends explicitly on the modal phase and group velocities of Rayleigh waves, and their corresponding wavenumbers and eigenfunctions, which can be computed directly from the solution of the Rayleigh-wave eigenproblem. A practical scenario for the application of the notion of apparent Rayleigh-wave phase velocity is the modelling of the dispersion curve in the well-known surface wave measurement methods 'spectral analysis of surface waves' (SASW) and 'multichannel analysis of surface waves' (MASW). Apart from a theoretical motivation, the availability in surface wave testing of an explicit formula for the calculation of the apparent Rayleigh-wave phase velocity may lead to the development of a new class of inversion algorithms capable of taking into account the influence of all the modes of surface wave propagation. To demonstrate the exactness of the explicit relation, the predicted values of apparent phase velocity are compared to those computed synthetically from a numerical simulation of SASW and MASW testing for three case studies, which show both single as well as multiple mode dominance effects.