Nonlinear Waves in Circular Basins

Summary

Waves generated near resonance in bays or harbours, by storms or earthquakes, may rise to large wave heights. This investigation examines waves progressing around a circular basin of finite uniform depth, with particular attention being given to those waves near resonance. Multiple families of free steady waves are associated with each of the depths at which resonance occurs, with different families having different orderings of the wave components composing them. Linear stability calculations indicate that those free waves dominated by resonating wave components are linearly unstable, but calculations of the nonlinear time evolution over many wave periods do not confirm the instability in the weakly unstable examples. When the model is made more realistic by including periodic forcing and wavenumber dependent damping, some of these weakly unstable examples are found to be stable, while others are unstable but evolve slowly in time into periodically modulated waves. Although the occurrence of multiple families of free waves is interesting for theoretical reasons, practical calculations of waves near resonance should include both forcing and damping. For any given depth ratio, forcing amplitude, and forcing period, there may be no, one, or multiple steady progressive waves.