The linearized (approximate) expressions for the quasi-longitudinal (QL) phase and group velocities for an orthorhombic medium, which appeared in a recent work, are discussed. The equivalence of two apparently different phase velocity approximations is established. Using the method of characteristics, the exact group velocity is derived for the degenerate elliptical transversely isotropic (TI) case from the eikonal equation. The exact eikonal, if it can be obtained in an analytic form, is homogeneous in powers of 2 in the components of the (phase) slowness vector. The linearization process introduces quartic terms into the eikonal equation. The combination of quadratic and quartic terms in an eikonal presents difficulties that require estimations to reduce the quartic terms to quadratic. These are introduced to obtain an expression for the perturbed TI case. Next, an anisotropic medium displaying orthorhombic symmetry is considered. As in the TI problem, the degenerate ellipsoidal case is treated first to obtain an insight into the more complex perturbed ellipsoidal problem. The 3 symmetry plane perturbation terms are treated collectively to obtain an approximate group velocity or group slowness for an orthorhombic medium.
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