We study the stability of the perfectly matched layer (PML) for symmetric second order hyperbolic partial differential equations on the upper half plane, with boundary conditions at y = 0. Using a mode analysis, we develop a technique to analyze the stability of the corresponding initial boundary value problem on the half plane. We apply our technique to the PML for the elastic wave equations subject to free surface and homogenous Dirichlet boundary conditions, and to the PML for the curl-curl Maxwell's equation subject to insulated walls and perfectly conducting walls boundary conditions. The conclusion is that these half-plane problems do not support temporally modes.
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