Wave field factorization for broadband acoustic signals

Abstract

A framework has been demonstrated for factoring the Helmholtz equation, governing for narrow-band acoustic signals, into a pair of one-way equations [L. Fishman and J. J. McCoy, J. Math. Phys. 25, 285–296 (1984)]. This factoring requires a range-independent propagation environment. The factored Helmholtz equation has been demonstrated to provide a marching algorithm [L. Fishman and J. J. McCoy, Geophys. J. R. Astron. Soc. 80, 439–461 (1985)], which is a generalization of the split-step algorithm, the latter being frequently used for marching the solution of the ordinary parabolic wave equation. The extension of these results to broadband acoustic signals is considered. Two factorizations are considered and shown to apply to different initial/boundary value problems. The first factorization is based on a Fourier synthesis of that obtained for narrow-band signals. This applies for a forcing problem as a broadband time series acting across a source range plane. The second factorization obtains for the time coordinate of the wave equation. This applies for an infinite spatial domain problem for specified initial time data.