Wave packets and localized pulses - A dual approach

It is shown that an interesting duality exists for wave packets and localized pulses as their representations in (x→,t) and (k→,ω) spaces are compared. A wave packet in (x→,t) space has a slowly varying envelope. This corresponds to a narrow spectrum in (k→,ω) space. On the other hand, a localized pulse in (x→,t) space corresponds to a slowly tapering spectrum in (k→,ω) space. The analysis of wave-packet propagation is usually carried out by means of ray theory. It is well known that ray tracing, in spite of its limitations, provides a powerful tool for the analysis of wave-packet propagation in dispersive, weakly inhomogeneous media. Similarly, it is shown here that localized pulses in inhomogeneous, weakly dispersive media, can be analyzed, using the concepts of dual-dispersion equation, dual-ray tracing, and group slowness. Hamilton's equations of geometrical optics are the Euler-Lagrange equations of the variational form known as Fermat's principle. In an analogous manner the dual Fermat principle is introduced here, being equivalent to the dual-ray equations. The method proposed here facilitates the analysis of localized pulses in inhomogeneous, weakly dispersive media. The dual-ray tracing provides a clue to the way in which the spectrum of a pulse in (k→,ω) space, hence its shape in (x→,t) space, change in the course of propagation.