Finite three-dimensional surface waves in circular channels

This paper is concerned with finite surface waves in incompressible fluid in circular channels or bowls. Fluid circulates through these channels and is contained in them by the action of an axial gravitational force. Under certain velocity conditions permanent waves of finite magnitude may be solutions to the differential equations of motion. These waves are related to the cnoidal and solitary waves of the infinite straight channel. Using stretching techniques on the nonlinear differential equations, we derive expressions for the wave shape, the critical velocity condition, and relations to waves in straight channels. The irrotational case is thoroughly treated and a short discussion of the rotational case is included. The fact that these waves deform continuously into those already derived for straight channels is also discussed. The most important result, however, is that stretching techniques have been shown to describe finite waves progressing in curved trajectories. While it is true that the trajectories are dictated by the curvature of the containing vessels, it is apparent that this technique may very well be useful in describing finite wave shape and trajectories for waves near beaches and breakwaters.