Classical standing-wave analog of the quantum Aharonov-Bohm effect: Theory

We present a theoretical study of the scattering of standing surface waves on a draining bathtub vortex. When the waves interact with the vortex, spinning propeller like lines are formed upon which the wave vanishes. These zero-lines run radially through the vortex and act as domain walls that separate regions where the waves have different phases. Their origin is similar to the wave front dislocations discussed in the fluid dynamic Aharonov-Bohm (AB) effect with traveling waves (Berry et al. European Journal of Physics, 1980) as an analog to the scattering of a beam of charged particles on a magnetic vector potential. In this classical analog, the height of the water-air interface corresponds to the real part of the quantum mechanical wave function, which is not a physical observable in the quantum case but can be observed in the classical setting. We consider the same analog quantum system using two oppositely directed beams to represent the standing wave. From the asymptotic wave function, one can predict the number of propellers (and their behavior), which is determined by a dimensionless parameter α. These theoretical results have received both experimental and numerical confirmation. Like the wave front dislocation of the classic AB effect, these propellers are only observable in the classical case.