Scattering of dislocated wave fronts by vertical vorticity and the Aharonov-Bohm effect. I. Shallow water

When a surface wave interacts with a vertical vortex in shallow water the latter induces a dislocation in the incident wave fronts that is analogous to what happens in the Aharonov-Bohm effect for the scattering of electrons by a confined magnetic field. In addition to this global similarity between these two physical systems there is scattering. This paper reports a detailed calculation of this scattering, which is quantitatively different from the electronic case in that a surface wave penetrates the inside of a vortex while electrons do not penetrate a solenoid. This difference, together with an additional difference in the equations that govern both physical systems, lead to a quite different scattering in the case of surface waves, whose main characteristic is a strong asymmetry in the scattering cross section. The assumptions and approximations under which these effects happen are carefully considered, and their applicability to the case of the scattering of acoustic waves by vorticity is noted.