Wavepacket Solutions of the Time-Dependent Wave Equation in Homogeneous and in Homogeneous Media

Abstract

Pulsed beam (PB) are highly localized space-time wavepackets solutions of the time-dependent wave equation that propagate along ray trajectories. Because they have these properties, PB may be useful in various applications including modeling of highly focused energy transfer, ultra wide band Radars and local interrogation of the propagation environment. Several classes of wavepacket solutions of the homogeneous wave-equation in free space have been introduced recently. They include the focus wave mode (FWM), 1-4 the “bullets,”5-6 and the complex source pulsed beam (CSPB)7-9 solutions. The FWM-type solutions are slowly diffracting wavepackets. Their special structure is a consequence of a balance between forward and backward propagating waves.4 Removing the backward propagating part that cannot be excited by an antenna causes a diffraction of the remaining wavepacket at about the classical Fresnel distance. The “bullets” are wavepackets whose far zone radiation patterns vanish identically outside a specified cone.5-6 Their near zone structure, however, does not have a well confined wavepacket shape.6 The CSPB are exact solutions that are modeled by radiation from a pulsed source located at a complex coordinate point. Physically, they are generated by radiation from real time-dependent source distributions of finite support so that the complex source model is just a mathematical trick to derive simple field solutions. These wavepackets stay collimated up to a certain distance from the sources and thereafter spread along a constant diffraction angle and decay like r -1. Except for the far zone spreading, the wavepacket structure remains essentially unchanged all the way from the sources to the far zone. Other important features of the CSPB will be discussed in Sec. 2.7.