Eigenwavelets of the Wave equation

TL;DR

This paper introduces eigenwavelets, a new class of localized wave solutions in spacetime derived from the wave equation, which can be focused tightly and have no sidelobes, making them ideal for communication and sensing applications.

Contribution

It extends fundamental solutions of the wave equation into complex spacetime to define eigenwavelets, providing a novel approach for focused, sidelobe-free wave solutions with potential practical applications.

Findings

Eigenwavelets can be focused around a single ray by adjusting imaginary spacetime variables.

They have no sidelobes, reducing interference in applications.

A method to generate sources for radiating and absorbing these wavelets is proposed.

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 hyperfunctions. 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, 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.