Acoustic and Electromagnetic Bullets: Derivation of New Exact Solutions of the Acoustic and Maxwell’s Equations

Abstract

Previously it has been shown that all finite energy, causal solutions of the time-dependent three-dimensional acoustic equation and Maxwell’s equations have forms for large radius that, except for a factor $1/r$, represent one-dimensional wave motions along straight lines through the origin. The asymptotic region is called the wave zone. Conversely, it also has been shown how the exact finite energy causal solutions can be obtained from the asymptotic solutions in the wave zone through the use of a refined Radon transform. Thus a way of finding exact, causal three-dimensional solutions from the essentially one-dimensional solutions in the wave zone has been obtained. When the asymptotic solutions are confined to a finite radial interval within a cone, the exact solutions are termed “bullets” and asymptotically represent packets of acoustic and electromagnetic energy “shot” through the cone, which is a kind of “rifle.” No reflectors are needed. In the present paper explicit examples of such exact solutions are derived.