Statistics for velocities of Gaussian waves

The sea surface is best modeled as a random field evolving in time. Although a great deal of research has been done on statistical distributions of static characteristics of sea waves, little is known about statistical properties of wave kinematics. By studying distributions for velocities of sea surface motions me are making a step forward in this direction. We extend the approach from the pioneering work of Longuet-Higgins (1957) by taking into consideration the geometry of the sea, as well as its evolution in time. We discuss the following velocities: (1) the ratio of the wave length to the wave period; (2) velocity in the direction of the gradient of the sea surface; (3) velocity of upcrossings and local maxima; (4) velocity of crossing contours; and (5) velocity of wave groups. We derive intensity distributions for these quantities and discuss their interpretation. The results involve generalizations of Rice's formula. They are illustrated by computing these distributions for an example of directional Gaussian sea.