Eigenwavelets of the Wave Equation

Abstract

We study a class of localized solutions of the wave equation, called eigenwavelets, obtained by extending its fundamental solutions to complex spacetime in the sense of hyper-functions. The imaginary spacetime variables y, which form a timelike vector, act as scale parameters generalizing the scale variable of wavelets in one dimension. They determine the shape of the wavelets in spacetime, making them pulsed beams that can be focused as tightly as desired around a single ray by letting y approach the light cone. Furthermore, the absence of any sidelobes makes them especially attractive for communications, remote sensing and other applications using acoustic waves. (A similar set of “electromagnetic eigenwavelets” exists for Maxwell’s equations.) I review the basic ideas in Minkowski space ℝ^3,1, then compute sources whose realization should make it possible to radiate and absorb such wavelets. This motivates an extension of Huygens’ principle allowing equivalent sources to be represented on shells instead of surfaces surrounding a bounded source.

To Carlos Berenstein on his 60th birthday.