On anisotropic elastic materials for which one sheet of the slowness surface is a sphere or a cross-section of a slowness sheet is a circle

It is shown that, among anisotropic elastic materials, certain orthotropic and hexagonal materials can have a sphere as a slowness surface branch that is associated with purely longitudinal or transverse waves. If it is associated with a longitudinal wave, it is the inner slowness sheet which is disjoint from the two outer slowness sheets. The two outer slowness sheets touch each other at certain points so that there are acoustic axes of double degeneracy. The waves associated with the two outer slowness sheets are necessarily transverse waves. Therefore any plane wave in these materials is either longitudinal or transverse. If the spherical slowness surface is associated with a transverse wave, it is the medial slowness sheet. The acoustic axes are at the crystallographic axes where the spherical slowness sheet touches the inner and/or outer sheets. All three sheets can touch each other so that the acoustic axis can be of triple degeneracy. In the case of special hexagonal materials with the x3-axis being the axis of symmetry, the x3-axis is an acoustic axis of double or triple degeneracy while any line on the x3 = 0 plane is an acoustic axis of double degeneracy. We also consider anisotropic elastic materials for which a cross-section of a slowness sheet is a circle. Anisotropic elastic materials that possess a circular cross-section in the slowness surface are not restricted to hexagonal or orthotropic materials.