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The trajectories of particles in steep, symmetric gravity waves

To gain insight into the orbital motion in waves on the point of breaking, we first study the trajectories of particles in some ideal irrotational flows, including Stokes’ 120° corner-flow, the motion in an almost-highest wave, in periodic deep-water waves of maximum height, and in steep, solitary waves. In Stokes’ corner-flow the particles move as though under the action of a constant force directed away from the crest. The orbits are expressible in terms of an elliptic integral. The trajectory has a loop or not according as q [sqcup ] c where q is the particle speed at the summit of each trajectory, in a reference frame moving with speed c. When q = c, the trajectory has a cusp. For particles near the free surface there is a sharp vertical gradient of the horizontal displacement. The trajectories of particles in almost-highest waves are generally similar to those in the Stokes corner-flow, except that the sharp drift gradient at the free surface is now absent. In deep-water irrotational waves of maximum steepness, it is shown that the surface particles advance at a mean speed U equal to 0·274c, where c is the phase-speed. In solitary waves of maximum amplitude, a particle at the surface advances a total distance 4·23 times the depth h during the passage of each wave. The initial angle α which the trajectory makes with the horizontal is close to 60°. The orbits of subsurface particles are calculated using the ‘hexagon’ approximation for deep-water waves. Near the free surface the drift has the appearance of a thin forwards jet, arising mainly from the flow near the wave crest. The vertical gradient is so sharp, however, that at a mean depth of only 0.01L below the surface (where L is the wavelength) the forwards drift is reduced to less than half its surface value. Under the action of viscosity and turbulence, this sharp gradient will be modified. Nevertheless the orbital motion may contribute appreciably to the observed ‘winddrift current’. Implications for the drift motions of buoys and other floating bodies are also discussed.