Longitudinal normals and the existence of acoustic axes in crystals

We obtain three conditions on the phase speeds of the longitudinal and transverse waves propagating along the longitudinal normals in a crystal so that each of these conditions guarantees existence of acoustic axes in this crystal. The result is based on the properties of the rational-valued topological degree and of the index of a singular point for some classes of discontinuous mappings. In addition, we give an upper estimate of the number of acoustic axes in a crystal and show some interrelation between their indices.