The statistics of particle velocities and accelerations under waves with directional spread

The statistics of the horizontal component of the water particle velocity vector under random waves are considered. A spread in the directions of travel of the component wave trains does not affect the rms value of the modulus r but it affects the shape of its probability distribution in such a way that the pr3bability of extreme values is reduced. For small and moderate directional spreads the modulus for a given probability of exceedance is reduced by a factor which tends to F_s as the probability of exceedance tends to zero, where F_s is the spread factor (which is typically 0.9 in temperate storms). However, the convergence on this asymptote is not complete for probabilities of engineering importance, particularly for wide directional spreads. The standard deviation of the probability distribution of the direction of the vector is reduced roughly in inverse proportion to r. The same conclusions apply to particle accelerations.