We present a simple method for simulating 2-D elastic waves in a model with free-surface topography of polygonal shape, i.e., a continuous but irregular surface composed of line segments. Our method requires special treatment for each of the six specific cases involving line segments of various slopes as well as transition points between the sloping segments. For brevity, only nonnegatively sloping segments are specifically included.On an inclined free surface, vanishing stress conditions are implemented using a rotated coordinate system parallel to the inclined boundary. At transition points on the topography between line segments, we use a first-order approximate boundary condition in a locally rotated coordinate system aligned with the bisector of the corner. As in the classical one-sided explicit approximation scheme widely used for the flat freesurface case, these extrapolation formulas are accurate to first order in spatial increment. Numerical tests indicate that the present scheme is stable over a range of Poisson's ratios of practical interest (v > 0.3) for fairly complicated geometric shapes consisting of ridges and valleys with both steep and gentle slopes. Stability for complicated shapes enables us to study realistic problems for which the topography plays a significant role in shaping the wave field and for which analytical solutions are not generally available.

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The main results of this article are (I) Let B be a homogeneous Banach algebra, A a closed subalgebra of B, and I the largest closed ideal of B contained in A. We assert that for some closed subalgebra J of B. Furthermore, under suitable conditions, we show that A is an R-subalgebra if and only if J is an R-subalgebra. A number of concrete closed subalgebras of a homogeneous Banach algebra therefore are R-subalgebras. For the definition of P(A) and that of an R-subalgebra, see the introduction in Section 1. (II) We give sufficient and necessary conditions for a closed subalgebra of Lp(G), 1 ≦ p ≦ ∞, to be an R-subalgebra.