versions of the differential equations. The resulting expressions have a counterpart in physics that is found in the pertaining reciprocity theorems of the Rayleigh (acoustic waves in fluids), Betti-Rayleigh (elastic waves in solids), or H.A. Lorentz (electromagnetic waves) types. This observation has led to the approach presented in the present paper, where the relevant reciprocity theorems are taken as points of departure. Through them, a computational scheme is conceptually taken to describe the interaction between the actual wavefield state to be computed and a suitably chosen "computational state" that is representative for the method at hand, just as in physics the reciprocity theorems describe the interaction between observing state and observed state, or quantify the reciprocity between transmitting and receiving properties of any device or system (transducer in acoustics and elastodynamics, antenna in electromagnetics, electromagnetic compatibility of interfering electromagnetic systems or devices). It is also believed that through this point of view one is guided to developing computational algorithms for each of the three types of wave fields in a manner that expresses the structures common to all of them. Background literature on reciprocity can be found in some papers by A.T. de Hoop (1987, 1988, 1989, 1990, 1991, 1992) and in a forthcoming book (de Hoop, 1995).

We consider linear acoustic, elastic, or electromagnetic wave motion in some subdomain of three-dimensional Euclidean space . The configuration in which the wave motion is considered to be present is assumed to be time invariant and linear in its physical behavior. The wave quantities involved are found to satisfy certain reciprocity properties which will be taken as the point of departure for our further considerations. Now, for the indicated type of configuration, there prove to be two kinds of reciprocity theorem: one of the time-convolution type, the other of the time-correlation type. Several operations on the wave quantities will occur throughout the paper. First, we shall introduce their notation.

Cartesian coordinates x={x₁,x₂,x₃} are used to specify position; t is the time coordinate. Differentiation with respect to x_p is denoted by is a reserved symbol for differentiation with respect to t. The subscript notation for the vectorial and tensorial quantities occurring in the wave motion will be used whenever appropriate; the subscripts are to be assigned the values 1, 2, and 3.

The characteristic function of the domain is denoted by χ_D and is given by

where is the boundary of , and is the complement of in

Let F = F(x,t) denote any space-time function. Then, the time reversal operator T is defined by

It has the property