Initial conditions, sources, and currents for prescribed time-dependent acoustic and electromagnetic fields in three dimensions, Part I: The inverse initial value problem. Acoustic and electromagnetic "bullets," expanding waves, and imploding waves

Abstract
In a previous paper one of us gave methods of constructing time-dependent pulses of sources and currents which give rise to prescribed time-dependent acoustic or electromagnetic fields. Not all of the prescribed field is required but only the field (and its time-derivative in the acoustic case) at the time that the sources and currents are switched off. Thus the prescribed field is described by its initial values at the time that the sources and currents are switched off. The usual (direct) initial value problem is well known: give conditions on the field at one time and find the field at later times. The inverse source problem indicates that an inverse initial value problem is of importance: give the field and find the initial values of the field which lead to this field. This inverse initial value problem is also clearly of interest in its own right. It will be shown that one does not need to know the entire field to construct the initial values but only a portion of the field corresponding to the wave zone. Moreover, it will be possible to give all acceptable forms for the fields in the wave zone. In many practical problems it is the field in the wave zone which is of importance. In communications, for example, the field in the wave zone the signal which is to be received. We can thus solve the inverse initial value problem given by the following statement: find the initial conditions which lead to a prescribed signal. Having solved the inverse initial value problem, we can combine our results with those of the inverse source problem and solve the inverse source problem given by the following statement: Find the currents and sources which will give rise to prescribed acoustic or electromagnetie asymptotic fields or signals. It will be shown that initial values can be chosen in such a way that signals can be as highly directed as one wishes. Such signals, when their extent in space is small, form "bullets" of energy. The possibility of having such signals leads to a host of applications which will be discussed briefly. Part I of the present paper concentrates on the inverse initial value problem. In the electromagnetic case we make use of eigenfunctions of the curl operator, which we introduced previously. In Part II we shall use our result to obtain examples of sources and currents which give rise to the prescribed field in the wave zone. In Part III we shall show how our currents and sources can be approximated by multipole fields and give estimates of the accuracy of the approximations. The exact solutions of the acoustic and electromagnetic wave equations which have bullets as asymptotic forms can be superposed so that the fields in the wave zone are expanding or imploding spherical waves with amplitudes which vary in a prescribed way with the direction of observation. These solutions, too, have their applications.