Shallow water gravity waves: A note on the particle orbits

When surface gravity waves of small amplitude progress in shallow water of constant mean depth, the fluid particle orbits are observed to be oval, where the longer axis of the oval is parallel to the flat bottom, and at the bottom the orbits are straight lines. Potential flow, upon which the standard wave theory is based, predicts that the oval orbits are ellipses, but by a rather lengthy mathematical procedure that is founded on the questionable assumption of irrotationality. Using a more elementary and physical method, that does not employ the irrotational assumption, the elliptical orbits can be understood much more easily. The elementary method features a balance of two oppositely directed forces on each fluid particle: the outward centrifugal force and the inward pressure force.