Boundary Waves and Stability of the Perfectly Matched Layer for the Two Space Dimensional Elastic Wave Equation in Second Order Form

We study the stability of the perfectly matched layer (PML) as a closure for the system of second order elastic wave equations in two space dimensions on the upper half–plane, −∞ < x < ∞, y ≥ 0. Using mode analysis, we analyze the stability of the PML initial boundary value problem on the half–plane. We consider in particular the free surface and Dirichlet boundary conditions, but the analysis is valid for many other boundary conditions. The result is that if the corresponding PML Cauchy problem does not support temporally growing modes, then the PML introduces damping to all boundary wave modes except temporally constant modes, which remain temporally constant. Numerical experiments are presented verifying the stability of the PML. In particular we show that surface waves (Rayleigh waves) are damped as they move into the PML.