Solutions to wave propagation problems in linear elastic transversely isotropic media can be constructed by superposing solutions for the quasi-P wave, the quasi S-wave, and a S-wave polarized orthogonal to the quasi S-wave.
However, the equations for the angular variation of velocity are algebraically complex and the plane-wave velocities are functions of the propagation angle  are.  The direction dependent wave speeds for elastic waves through the material can be found by using the Christoffel equation and are given by
where  is the angle between the axis of symmetry and the wave propagation direction,  is mass density and the  are elements of the elastic stiffness matrix.  The Thomsen parameters are used to simplify these expressions and make them easier to understand.