On the relation between time-reversed acoustics and Green's function retrieval in space-variant and in time-variant materials

The methods of time-reversed acoustics and Green's function retrieval are traditionally deployed for classical inhomogeneous, time-invariant materials. The mutual relation between these methods is well established. Recently, similar methods have been proposed for homogeneous, time-variant materials. Here, we investigate their mutual relation and their relation with the corresponding methods in classical materials. For this analysis, we make use of the fact that the wave equations for both classes of material are similar, with the roles of time and space interchanged. However, the principle of causality holds for both classes of material; hence, here the roles of time and space are not interchanged. We find that (1) whereas classical time-reversed acoustics involves emission of a time-reversed single-component wave field from a (ideally closed) boundary into the inhomogeneous material, its idealized counterpart involves emission of a sign-reversed two-component wave field, recorded in a time-reversed material, from a single time instant into the actual time-variant material; and (2) whereas classical Green's function retrieval involves temporal cross-correlation of wave fields at two space locations in response to single-component sources on a (ideally closed) boundary, its counterpart involves spatial cross-correlation of wave fields at two time instants in response to two-component sources at a single time instant.