An approximation method for dispersive wave propagation

Abstract

A phase space approximation method that extends a previous single-mode approximation [Loughlin, Cohen, J. Acoust. Soc. Am. 118, 1268 (2005)] for linear dispersive wave propagation is developed. We show that each mode is governed by a Schrodinger-type equation, where the corresponding Hamiltonian operator is non-Hermitian if the dispersion relation is complex, for which case there is absorption. The propagated wave is obtained by evolving each mode according to its respective Schrodinger equation. We show how to obtain the initial modes from the initial conditions on the wave. We then formulate the propagation problem in phase space and obtain the exact equation of motion for the phase space function. We also obtain an approximate solution for the phase space evolution of the wave, which involves a simple substitution into the initial phase space function. Examples are given for a parallel plate wave guide and the beam equation. [Work supported by ONR, code 321US.]