A bidirectional traveling plane wave representation of exact solutions of the scalar wave equation

Abstract

A new decomposition of exact solutions to the scalar wave equation into bidirectional, forward and backward, traveling plane wave solutions is described. The resulting representation is a natural basis for synthesizing pulse solutions that can be tailored to give directed energy transfer in space. The development of known free-space solutions, such as the focus wave modes, the electromagnetic directed energy pulse trains, the spinor splash pulses, and the Bessel beams, in terms of this decomposition will be given. The efficacy of this representation in geometries with boundaries, such as a propagation in a circular waveguide, will also be demonstrated.

A novel approach to the synthesis of nondispersive wave packet solutions to the Klein–Gordon and Dirac equations

“…It has been demonstrated by the authors 17 not dimensionless, we can use the more stringent condition {3 'a I < 1. A good estimate of the waist of such a pulse is (a l /{3')1/2; as a consequence, the condition (a l /{3') 112 < lI{3' has to be satisfied.…”

“…17, where it was applied to various classes of equations, e.g., the 3-D scalar wave equation, the 3-D Klein-Gordon equation, and the telegraph equation. As mentioned earlier, the resulting solutions had envelopes moving with the speed oflight, a property we would like to avoid in the next section.…”

“…The bidirectional representation 17 was originally developed in order to provide a natural basis for synthesizing Brittingham-like solutions. In this section, we shall outline the salient features of this technique and use it to derive the scalarFWMs.…”

“…The work is based on an embedding technique that has been utilized to derive a natural basis for the synthesis of Brittingham-like solutions. This novel basis has been termed the bidirectional representation 17 because it is a superposition of elementary solutions built up of a product of two plane waves, one traveling to the left and the other to the right. Our plan is to give a brief introduction to the bidirectional representation in the next section and use it to derive the scalar FWMs.…”

Wavelet-based integral representation for solutions of the wave equation

“…There are several practical methods that allow one to obtain an exact solution of the wave equation directly in the position space, without integrating over the whole R 3 . These methods were originated in papers [26]- [30] (see [31] for a review of such methods). The aim of this section is to look at some of these methods from the point of view of physical wavelets.…”