Surface Waves in Almost Incompressible Elastic Materials

A recent study shows that the classical theory concerning accuracy and points per wavelength is not valid for surface waves in almost incompressible elastic materials. The grid size must instead be proportional to $(\frac{\mu}{\lambda})^{(1/p)}$ to achieve a certain accuracy. Here $p$ is the order of accuracy the scheme and $\mu$ and $\lambda$ are the Lame parameters. This accuracy requirement becomes very restrictive close to the incompressible limit where $\frac{\mu}{\lambda} \ll 1$, especially for low order methods. We present results concerning how to choose the number of grid points for 4th, 6th and 8th order summation-by-parts finite difference schemes. The result is applied to Lambs problem in an almost incompressible material.