Linearized approximations for phase velocities of elastic waves in weakly anisotropic media

This paper focuses on the development of linearized formulae for the phase velocities of quasi-longitudinal (QL) and quasi-transverse (QT) waves in we akly anisotropic media. The formulae for quasi-transverse waves derived fro m first-order perturbation theory possess the property of rotational invari ance in a plane perpendicular to the wave normal. Based on this property, a planar coordinate system, describing the possible polarization of the two degenerate transverse waves, can be conveniently selected. The relationship s between the first-order formulae and other related existing approximate e xpressions are clarified. For a symmetry plane of orthorhombic media, we ex tended Gassmann's formula for the QL-wave phase velocity in transversely is otropic media to QL and QT waves. Three forms of phase velocity expressions are discussed. For non-symmetry planes of weakly orthorhombic media, we pr opose a unified form for the QL-wave phase velocity and in a very simple wa y derive the linearized Thomsen- and Gassmannn-like formulae. Furthermore w e suggest QT-wave phase velocity expressions which are linear functions of the elastic constants assuming that wave propagation has a slight deviation from a symmetry plane of orthorhombic media. Finally, numerical results in a symmetry plane show that the set of first-order approximate formulae per forms best for weakly anisotropic materials.