Paths of Particles in Water Waves

The velocity potential for water waves in water of finite depth, h, can be writtenFrom this we obtain


Integrating these equations with respect togivesand


This is the equation of an ellipse. For largeandclose to 0, far from the bottom whereso the ellipses are nearly circular. Asgets smaller – as the water becomes shallower – or asbecomes more negative – increasing depth - thenandboth decrease so the ellipses get smaller. Becausedecreases faster thanthe eccentricity increases as shown below.