Acoustic time-reversal mirrors in the framework of one-way wave theories

We investigate the implications of directional wavefield decomposition with a view to time reversibility. In particular, we discuss how wavefield decomposition preserves the reciprocity theorem of time-convolution type but looses the reciprocity theorem of time-correlation type. As a consequence, a perfect ‘time-reversal mirror’ in the framework of one-way wave theory does not exist: We find that on the wavefront set (‘classical limit’) a time-reversal mirror can retrofocus the wavefield to its originating source, but that non-perfectly retrofocusing lower-order distributions contribute to the process as well. These distributions can be attributed to ‘evanescent’ wave constituents but are not negligible; we will study them explicitly. As a peculiarity, we discuss how a Schrödinger-like equation can be obtained out of the (exact) frequency-domain oneway wave equation. This involves an approximation – known in ocean acoustics and exploration seismology as the ‘parabolic equation’ approximation – that restores timereversibility.