Localized Waves from Transform and Phase Space Points-of-View

Abstract

The transfer of energy and information is a basic tenet of acoustic, electromagnetic and quantum wave applications. Localized waves are nonseparable space-time solutions of the wave equations associated with these and other physical phenomena. They exhibit unusual propagation characteristics such as nondispersive propagation, i.e., their central region propagates without spreading. While many of these localized waves have been derived directly in space and time coordinates, they have also been represented and studied using transform and phase space techniques. The transform approach, which deals with algebraic equations in a general three-dimensional wavevector and one-dimensional frequency domain setting, is appealing because of their simplicity, but they must be handled with care in a generalized function manner. Nonetheless, interesting features of the LW solutions can be revealed by representing them geometrically with respect to the dispersion surfaces associated with their governing space-time wave equations. Phase space techniques, i.e., those dealing with a mixture of space-time and transform domain coordinates, further augment the understanding and derivation of the performance characteristics of these localized waves, e.g., determining if they are finite energy solutions. This paper briefly discusses and demonstrates various aspects of these approaches. Both transform and phase space representations of subluminal, luminal and superluminal localized wave solutions of several classes of wave equations (homogeneous wave equation in free space and in a lossy medium related to electromagnetic phenomena and Klein-Gordon equation associated with quantum systems) will also be discussed in my presentation.