Fast Evaluation of Two-Dimensional Transient Wave Fields

This paper presents a novel scheme to efficiently evaluate transient linear wave fields that are generated by two-dimensional (2D) source configurations. The scheme, termed the plane wave time domain algorithm (PWTD), realizes a diagonal translation operator for 2D transient wave fields through their representation in terms of Hilbert transformed plane wave expansions. Numerical results are presented that validate the algorithm and demonstrate its convergence properties. The proposed PWTD algorithm can be coupled to classical 2D time domain integral equation solvers in a two-level and multilevel setting. It is shown that analysis of a 2D surface scattering phenomenon, in which sources are represented in terms of Ns spatial and Nt temporal samples, based on two-level and multilevel PWTD augmented integral equation solvers, requires O(N1.5sNt log Nt) and O(NsNt log Ns log Nt) computational resources, respectively (as opposed to O(N2sNt2) for a classical solver). Therefore, these PWTD schemes render feasible the rapid integral equation based analysis of 2D transient scattering phenomena involving large surfaces.