Triple singularities of elastic wave propagation in anisotropic media

A typical singularity of elastic wave propagation, often termed a shear-wave singularity, takes place when the Christoffel equation has a double root or, equivalently, two out of three slowness or phase-velocity sheets share a common point. We examine triple singularities, corresponding to triple degeneracies of the Christoffel equation, and establish their two notable properties: (i) if multiple triple singularities are present, the phase velocities along all of them are exactly equal, and (ii) a triple singularity maps onto a finite-size planar patch shared by the group-velocity surfaces of the P-, S1-, and S2-waves. There are no other known mechanisms that create finite-size planar areas on group-velocity surfaces in homogeneous anisotropic media.