In an anisotropic medium there are only certain directions along which elastic waves can propagate in pure longitudinal and transverse modes. For the determination of third‐order elastic coefficients from sound speed measurements in stressed crystals it is desirable to know these modes. Using a method due to Borgnis the pure mode directions are determined for all crystal point groups belonging to the orthorhombic, tetragonal, cubic, rhombohedral, and hexagonal systems. The eigenvalue problem is solved for each of these directions, and the polarization vectors and the wave speeds are tabulated.