Transparent Boundary Conditions for Wave Propagation on Unbounded Domains

Abstract. The numerical solution of the time dependent wave equation in an unbounded domain generally leads to a truncation of this domain, which requires the introduction of an arti¯cial boundary with associated boundary conditions. Such nonre°ecting conditions ensure the equiva-lence between the solution of the original problem in the unbounded region and the solution inside the arti¯cial boundary. We consider the acoustic wave equation and derive exact transparent boundary conditions that are local in time and can be directly used in explicit methods. These conditions annihilate wave harmonics up to a given order on a spherical arti¯cial boundary, and we show how to combine the derived boundary condition with a ¯nite di®erence method. The analysis is complemented by a numerical example in two spatial dimensions that illustrates the usefulness and accuracy of transparent boundary conditions.