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 weakly anisotropic media. The formulae for quasi-transverse waves derived from first-order perturbation theory possess the property of rotational invariance 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 relationships between the first-order formulae and other related existing approximate expressions are clarified. For a symmetry plane of orthorhombic media, we extended Gassmann's formula for the QL-wave phase velocity in transversely isotropic media to QL and QT waves. Three forms of phase velocity expressions are discussed. For non-symmetry planes of weakly orthorhombic media, we propose a unified form for the QL-wave phase velocity and in a very simple way derive the linearized Thomsen- and Gassmannn-like formulae. Furthermore we 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 performs best for weakly anisotropic materials.