Phase space approximations for dispersive propagation

Abstract

Sounds propagating in shallow water can undergo frequency-dependent changes, specifically damping and dispersion. Phase space analysis and approximations have yielded a clear physical picture of these frequency-dependent effects. In particular, in position-wavenumber phase space, each mode propagates, approximately, at constant velocity given by its group velocity. In time-frequency phase space, each mode undergoes a frequency-dependent time shift given, approximately, by its group slowness. The effects of damping (i.e., a complex dispersion relation) are also clearly manifest. We review and extend various phase space approximations for dispersive propagation, and explore their accuracy. We show that one can recover the exact spectral magnitude and group delay of the propagating pulse from the phase space approximations. One reason the approximations are generally very good is that one does not have to assume that the initial spectrum is slowly varying, as is done, for example, in the stationary phase approximation.