On the computation of elastic wave group velocities for a general anisotropic medium

This paper deals exclusively with the computation of the group velocities for the three wave modes (qP, qS1, qS2) in a general anisotropic medium, which may involve up to 21 density-normalized elastic moduli. We tackled the shear-wave singularity problem through two independent approaches: (1) an eigenvalue method, and (2) an eigenvector method. In the former, we derive analytic formulae and introduce an approximation for the directional derivative of the phase velocity at the singularity points. In the latter, we develop two simple schemes to find the eigenvectors of the quasi-shear waves at the singularity points. Computational experiments have been conducted to show the merits and validity of both approaches. Furthermore, the numerical results demonstrate that both methods produce consistent and satisfactory results for any degree of anisotropic media, notwithstanding the possible discrepancy between the specific solution for a general TI medium and the general solution for arbitrary anisotropy. The former is more suitable than the general solutions for such media, because it achieves complete polarization discrimination of the qSV and qSH modes. For more complex forms on anisotropy, e.g. orthorhombic, the general solutions yield mixed versions of the two quasi-shear waves qS1 and qS2, which have many singularity points in the phase velocity space, but overall recover the true modes.