A path‐integral approach to the parabolic approximation. I

We use Feynman’s theory of path integrals to study the parabolic approximation. With the aid of path‐integral representations for the solutions to the parabolic equation and the Helmholtz equation, we show the parabolic approximation, for either the two‐dimensional or the three‐dimensional parabolic equation, may be decomposed into two, simpler approximations: a type of geometric optics or eikonal approximation and a stationary‐phase approximation. We then relax the geometric optics approximation by assuming, instead, a variant of the Rytov approximation. This development leads to a two‐dimensional parabolic equation which models a three‐dimensional ocean. Finally we discuss the validity of the stationary‐phase approximation.