The velocity potential for water waves in water of finite depth, h, can be written
From this we obtain
and
Integrating these equations with respect to
gives
and 
then
This is the equation of an ellipse. For large
and
close to 0, far from the bottom where
so the ellipses are nearly circular. As
gets smaller – as the water becomes shallower – or asbecomes more negative – increasing depth - then
and
both decrease so the ellipses get smaller. Becausedecreases faster thanthe eccentricity increases as shown below.
