On decomposition of the fundamental solution of the Helmholtz equation over solutions of iterative parabolic equations

摘要

Recently, it was shown that the solution of the Helmholtz equation can be approximated by a series over the solutions of iterative parabolic equations (IPEs). An expansion of the fundamental solution of the Helmholtz equation over solutions of IPEs is considered. It is shown that the resulting Taylor-like series can be easily transformed into a Pade-type approximation. In practical propagation problems such iterative Pade approximations exhibit improved wide-angle capabilities and faster convergence to the solution of the Helmholtz equation in comparison to Taylor-like expansion over IPEs solutions. A Gaussian smoothing of the expansion terms gives insight into the derivation of initial conditions consistent for IPEs, which can be used for point source simulation. A correct point source model consistent with the wide-angle one-way propagation equations is important in many practical applications of the parabolic equations theory.