The general solution of the three-dimensional acoustic equation and of Maxwell’s equations in the infinite domain in terms of the asymptotic solution in the wave zone

Abstract

The general solution of the three-dimensional scalar wave equation (or acoustic equation) and of Maxwell’s equations in the infinite spatial domain is given in terms of the asymptotic forms for large times in the future and in the past, or, equivalently, in terms of the fields in the wave zone. One is therby able to obtain the exact solutions from arbitrary solutions in the wave zone. It is shown that the exact fields computed from an arbitrary wave zone solution always satisfy an initial value problem, and that, therefore, they are always physical. In contrast to earlier derivations of related results which required the use of Radon transforms and the introduction of somewhat sophisticated geometrical concepts, the derivations are simple and use only elementary properties of the Fourier transform.