An approximate causal Green’s function for the Stokes wave equation.

Abstract

A causal analytical approximation is derived for the time‐domain Green’s function of the Stokes wave equation. This causal approximation, which contains a modified Bessel function of the first kind, is the exact analytical time‐domain Green’s function for a related wave equation that approaches the Stokes wave equation in the low frequency limit. An asymptotic expression is also obtained for this approximate Green’s function. The causal approximation and the asymptotic expression are evaluated and compared to the numerically computed time‐domain Green’s function and the analytical frequency‐domain transfer function of the Stokes wave equation and to other causal and noncausal approximations. The results show that this approximate Green’s function closely matches the numerically computed Green’s function and the analytical frequency domain transfer function of the Stokes wave equation near the source, and at greater distances, the asymptotic expression accurately represents the time‐ and frequency‐domain behavior of the Stokes wave equation. Time‐domain analysis demonstrates that the analytical approximation and the asymptotic expression are dispersive, whereas causality is verified in the time‐domain and in the frequency‐domain. The differences and similarities between these causal approximations and other causal and noncausal approximations are also identified and discussed. [Work supported in part by NIH Grant No. EB012079.]