The time-dependent inverse source problem for the acoustic and electromagnetic equations in the one- and three-dimensional cases

Abstract

The object of the time-dependent inverse source problem of electromagnetic theory and acoustics is to find time-dependent sources and currents, which are turned on at a given time and then off to give rise to prescribed radiation fields. In an early paper for the three-dimensional electromagnetic case, the present writer showed that the sources and currents are not unique and gave conditions which make them so. The ideas of that paper are reformulated for the three-dimensional electromagnetic case and extended to the acoustical three-dimensional case and the one-dimensional electromagnetic and acoustic cases. The one-dimensional cases show very explicitly the nature of the ambiguity of the choice of sources and currents. This ambiguity is closely related to one which occurs in inverse scattering theory. The ambiguity in inverse scattering theory arises when one wishes to obtain the off-shell elements of the T matrix from some of the on-shell elements (i.e., from the corresponding elements of the scattering operator). In inverse scattering theory prescribing of the representation in which the potential is to be diagonal removes the ambiguity. For the inverse source problem a partial prescription of the time dependence of the sources and currents removes the ambiguity. The inverse source problem is then solved explicitly for this prescribed time dependence. The direct source problems for the one- and three-dimensional acoustic and electromagnetic cases are also given to provide a contrast with the inverse source problem and for use in later papers. Moreover, the present author’s earlier work on the eigenfunctions of the curl operator is reviewed and used to simplify drastically the three-dimensional direct and inverse source problems for electromagnetic theory by splitting off the radiation field and its currents from the longitudinal field and its sources and currents. Finally, for a prescribed time dependence, the inverse source problem is solved explicitly in closed form. Using methods developed for the inverse source problem for three-dimensional electromagnetic theory, we solve the following important direct problem. Consider a sphere of finite radius and uniform charge density rotating about a fixed axis. Assume that the angular velocity is zero before a certain time, varies in an arbitrary fashion, and then becomes zero again. What is the final electromagnetic field?