Boundary estimates for the elastic wave equation in almost incompressible materials

We study the half-plane problem for the elastic wave equation subject to a free surface boundary condition, with particular emphasis on almost incompressible materials. A normal mode analysis is developed to estimate the solution in terms of the boundary data, showing that the problem is boundary stable. The dependence on the material properties, which is difficult to analyze by the energy method, is made transparent by our estimates. The normal mode technique is used to analyze the influence of truncation errors in a finite difference approximation. Our analysis explains why the number of grid points per wave length must be increased when the shear modulus ($\mu$) becomes small, that is, for almost incompressible materials. To obtain a fixed error in the phase velocity of Rayleigh surface waves as $\mu\to 0$, our analysis predicts that the grid size must be proportional to $\mu^{1/2}$ for a second order method. For a fourth order method, the grid size can be proportional to $\mu^{1/4}$. Numerical experiments confirm this scaling and illustrate the superior efficiency of the fourth order method.