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Angular variations of seismic velocities have been observed in the Earth and attributed to some form of anisotropy caused by aligned crystals, orientated cracks and inclusions, and laminated strata. The exact analytical expressions for the velocities in each particular symmetry-system, derived from the Kelvin-Christoffel equations, are complicated functions of the elastic constants and cannot be easily manipulated. This paper examines the form of the velocity variations for the several systems of elastic symmetry; five of these seven symmetry-systems have been suggested for possible Earth structures. We shall demonstrate that the approximate equations of Backus (1965) and Crampin (1977 a) are good estimates for the velocity variations in symmetry planes of all symmetry systems, but not in general for off-symmetry planes. These equations are linear in the elastic constants, and provide a convenient link between velocity variations and elastic constants, if used judiciously. The behaviour of shear waves in off-symmetry directions is complicated by pinches, caused by the proximity of shear-wave singularities, where the two shear-waves exchange polarizations. Despite the restrictions to their use, the equations are the fundamental relationship for a number of modelling studies.
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