Systematic Approach to Higher-Order Parabolic Propagation in a Weakly Range-Dependent Duct

摘要

Energy-conserving transformations are exploited to split a monochromatic field in a weakly inhomogeneous waveguide into a pair of components that undergo uncoupled parabolic propagation in opposite directions along the waveguide axis. A systematic series of such transformations is developed to accomplish this splitting at increasing order of approximation while avoiding backscatter. In order to emphasize fundamentals, this technique is applied in what is arguably the simplest nontrivial case: waves of vertical displacement on a horizontal membrane with a smooth density inhomogeneity along one direction that forms a duct in the orthogonal direction. The evolution of these 'drumhead' vibrations is governed by a single continuous environmental variable, the density. This work is meant to serve as a link between an earlier treatment of the simpler, purely one-dimensional case (waves on a string) and the analyses of Wurmser and coworkers of considerably more complex cases featuring multiple environmental variables and/or ducts with discontinuities.