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.

The motion of subharmonic resonant modes of surface waves in a rectangular container subjected to vertical periodic oscillation is studied based on the weakly nonlinear model equations derived by both the average Lagrangian and the two-timescale method. Explicit estimates of the nonlinearity of some specific modes are given. The bifurcations of stationary states including a Hopf bifurcation are examined. Numerical calculations of the dissipative dynamical equations show periodic and chaotic attractors. Theoretical parameter-space diagrams and numerical results are compared in detail with Simonelli & Gollub's (1989) surface-wave modecompetition experiments. It is shown that the average Hamiltonian system for the present 2:1:1 external-internal resonance with suitable coefficients has homoclinic chaos, which was mathematically proven by Holmes (1986) for the specific case of 2:1:2 external-internal resonance.

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