Validity of the parabolic equation

Abstract

Much of the discussion of the validity and the physical interpretation of the parabolic equation is borrowed from the literature on electromagnetic propagation in turbulent media and simply does not apply to acoustic propagation in the ocean. Here we start with the inhomogeneous Helmholtz equation characterized by a sound speed which depends on all three spatial coordinates and use standard techniques from scattering theory to develope the parabolic equation. We find two assumptions are required: a forward-scattering or eikonal approximation in the horizontal plane and a degenerate spectrum assumption. The degeneracy assumption is reflected in a stationary-phase approximation and is intimately related to the definition of the wavenumber ko. We derive improved equations by relaxing these assumptions. A weaker stationary-phase approximation reduces the dependence of the solution on the choice of ko and the forward-scattering approximation is relaxed by assuming the “supereikonal” approximation.