Time-reversal focusing works in three dimensions. Record the wavefield from a source, reverse it in time, play it back — the reversed wave retraces its path and refocuses at the original source location. This is the basis of time-reversal mirrors in acoustics and ultrasound: the technique works because the 3D wave equation is invariant under time reversal. What goes forward can come back.

In two dimensions, it cannot (arXiv:2603.17688). The 2D wave equation supports a wake tail — a long-lasting oscillation that trails behind the main wavefront. When a pulse propagates in 2D, it does not cleanly pass a point and leave silence behind. It leaves a residual oscillation, decaying but persistent, stretching behind the wavefront indefinitely.

Time-reverse this wave, and the wake tail reverses too — but it arrives at the focal point before the main wavefront. The tail, which was a trailing residual, becomes a leading precursor. The refocused pulse is contaminated by energy that arrives too early, blurring the focus and preventing perfect reconstruction. The refocusing is approximate, not exact, and the approximation is intrinsic to the dimensionality of the space.

This is not a practical limitation to be engineered around. It is a mathematical property of the 2D Green’s function, which has a fundamentally different structure from the 3D Green’s function. In 3D, the Green’s function is a delta function — a sharp pulse that exists at one instant. In 2D, it has a step-function tail — energy that persists after the pulse passes. This tail breaks the time-reversal symmetry that the equation itself possesses.

The wave equation is time-symmetric. The solutions are not, because the dimensionality introduces a structural asymmetry that the equation does not. Perfect refocusing is a property of three dimensions, not of wave physics in general.