Velocity circulations in free-surface flows

The equations of balance between horizontal and vertical accelerations on a free surface are derived from the Euler equations for vortical Rows with arbitrary distribution of density. From these equations it follows that the velocity circulations on a free surface are invariants of motion. Application of these invariants to the problem of persistence of ship wakes over long distances is discussed. The effects of surface tension and molecular viscosity on velocity circulations are also considered.