The wave velocity analysis of rock medium is the main method used to explore the internal compositions in the crust and research seismic. In this paper, a compression-shear coupled nonlinear elastic constitutive relation is established, which is consistent with the mechanical properties of rock and mineral medium under high pressure. On this basis, numerical solutions of the wave equation and plane wave analytical solutions for the primary and secondary wave velocities are obtained. As is indicated by the comparison with the linear elastic constitutive theory, the results reflect the compression-shear coupling characteristics of the rock, including the stress path effect and the compression-shear coupling wave effect. With different parameter values, the velocity of the secondary wave changes from lower than that of the elastic shear wave, to higher than that of the elastic shear wave. The research results are expected to provide meaningful explanations for the physical mechanisms of the supershear wave and sub-Rayleigh wave, and guidance for the detection of rock and soil composition and the observation of seismic waves.

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