Linearized approximations for phase velocities of elastic waves in weaklyanisotropic media

Abstract

This paper focuses on the development of linearized formulae for the phasevelocities of quasi-longitudinal (QL) and quasi-transverse (QT) waves in weaklyanisotropic media. The formulae for quasi-transverse waves derived fromfirst-order perturbation theory possess the property of rotational invariance in aplane perpendicular to the wave normal. Based on this property, a planarcoordinate system, describing the possible polarization of the two degeneratetransverse waves, can be conveniently selected. The relationships between thefirst-order formulae and other related existing approximate expressions areclarified. For a symmetry plane of orthorhombic media, we extended Gassmann'sformula for the QL-wave phase velocity in transversely isotropic media to QLand QT waves. Three forms of phase velocity expressions are discussed. Fornon-symmetry planes of weakly orthorhombic media, we propose a unified formfor the QL-wave phase velocity and in a very simple way derive the linearizedThomsen- and Gassmannn-like formulae. Furthermore we suggest QT-wave phasevelocity expressions which are linear functions of the elastic constantsassuming that wave propagation has a slight deviation from a symmetry plane oforthorhombic media. Finally, numerical results in a symmetry plane show thatthe set of first-order approximate formulae performs best for weaklyanisotropic materials.