RAYLEIGH-WAVES FOR A DISCRETE ELASTIC PARAXIAL EQUATION

We investigate the effects of a free surface on the 45 degrees paraxial equation for Linear elasticity in a half space. We show that the 45 degrees equation has no Rayleigh wave and that it has an exponentially growing instability. We also examine a standard finite-difference approximation to the elastic paraxial equation. We show that the discrete equation also has an exponential instability, but this may be removed by projection. We also find that the discrete elastic paraxial equation has no Rayleigh waves when the mesh size is small, but a coarse mesh gives two Rayleigh waves. Moreover, by a proper choice of the mesh sizes, we may select a discretization so that one of these discrete Rayleigh waves has the speed of a physical Rayleigh wave, and we may project out the nonphysical discrete Rayleigh wave.