Surface Waves in Anisotropic Elastic Materials for Which the Matrix N(v) is Extraordinary Degenerate, Degenerate, or Semisimple

The 6 × 6 real matrix N(v) for anisotropic elastic materials under a two-dimensional steady-state motion with speed v is extraordinary degenerate when N(v) has three identical complex eigenvalues p but has only one associated eigenvector. It has been an open question if such an N(v) exists for surface waves. In this paper we first modify the solution for ordinary surface waves to the case when N(v) is extraordinary degenerate. The displacement consists of three waves. One of which decays exponentially, as in the ordinary surface waves, according to e^{-λ x₂} where λ is a positive constant. The other two waves decay in proportion to x₂e^{-λ x₂} and x₂²e^{-λ x₂}. We then consider a class of materials that has seven independent elastic constants for two-dimensional deformations. For this class of materials it is shown that there is a five-parameter family of extraordinary degenerate N(v) for surface waves. Thus, not only an extraordinary degenerate N(v) for surface waves exists, its existence is not as rare as one might have expected. We also study surface waves for which N(v) is degenerate or semisimple. In the former, N(v) has two or three identical complex eigenvalues but has two independent eigenvectors. In the latter, two of the three eigenvalues are identical but N(v) has three independent eigenvectors. It is shown that there are families of degenerate and semisimple N(v) for surface waves.