Two S-wave eigenvectors of the Christoffel matrix need not exist in anisotropic viscoelastic media

Abstract

The 3×3×3×3 frequency-domain stiffness tensor is complex-valued in viscoelastic media. The 3 × 3 Christoffel matrix is then also complex-valued. Using a simple example, we demonstrate that a complex-valued Christoffel matrix need not have all three eigenvectors at an S-wave singularity, and we thus cannot apply the eigenvectors to calculating the phase-space derivatives of the Hamiltonian function.