Surface Waves in Anisotropic Elastic Materials for Which the Matrix N(v) is Extraordinary Degenerate, Degenerate, or Semisimple
Abstract
The 6 × 6 real matrix N(v) for anisotropic elastic materials under a twodimensional steadystate 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 x_{2}e^{λ x2} and x_{2}^{2}e^{λ x2}. We then consider a class of materials that has seven independent elastic constants for twodimensional deformations. For this class of materials it is shown that there is a fiveparameter 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.
 Publication:

Proceedings of the Royal Society of London Series A
 Pub Date:
 March 1997
 DOI:
 10.1098/rspa.1997.0026
 Bibcode:
 1997RSPSA.453..449T