Summary: Surface waves can be recorded in two kinds of ways, either with a fixed (Eulerian) probe or with a free-floating (Lagrangian) buoy. In steep waves, the differences between corresponding properties can be very marked.
By a simple physical model and by accurate calculation it is shown that the Lagrangian wave period may differ from the Eulerian wave period by as much as 38%. The Lagrangian mean level is also higher than the Eulerian mean, leading to possible discrepancies in remote sensing of the ocean from satellites.
Surface accelerations are of interest in relation to the incidence of breaking waves, and for interactions between short (gravity or capillary) waves and longer gravity waves. Eulerian accelerations tend to be very non-sinusoidal, with large downwards peaks, sometimes exceeding -g in magnitude, near to sharp wave crests. Lagrangian accelerations are much smoother; for uniform gravity waves they lie between -0.388g and +0.315g. These values are verified by laboratory experiments. In wind-generated waves the limits are probably wider.
In progressive gravity waves in deep water the horizontal accelerations generally exceed the vertical accelerations. In steep waves, the subsurface accelerations can slightly exceed those at the free surface.
A novel application is made to the rolling motion of ships. In very steep, irrotational waves it is shown theoretically that the flow near the wave crest can lead to the rotation of the hull through angles up to by a single wave, even if the wave is not breaking. This is confirmed by simple experiments. The efficiency of the keel appears to promote capsizing.