Statistics for velocities of random waves

K. Podgórski, I. Rychlik and E. Sjö

Centre for Mathematical Sciences
Mathematical Statistics
Lund Institute of Technology,
Lund University,

ISSN 1403-9338

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 dynamics. By studying distributions for velocities of sea surface motions we 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 and 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, 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.
Key words:
directional spectrum, Gaussian sea, Rice's formula, velocity of local maxima, velocity of level-crossings, level crossing
contours, wave groups.