  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.
