Effect of forward speed on the level-crossing distribution of kinematic variables in multidirectional ocean waves

Abstract

The influence of forward speed on stochastic free-surface crossing, in a Gaussian wave field, is investigated. The case of a material point moving with a constant forward speed is considered; the wave field is assumed stationary in time, and homogeneous in space. The focus is on up-crossing events, which are defined as the material point crossing the free surface, into the water domain. The effect of the Doppler shift (induced by the forward speed) on the up-crossing frequency, and the related conditional joint distribution of wave kinematic variables is analytically investigated. Some general trends are illustrated through different examples, where three kinds of wave direction distribution are considered: unidirectional, short-crested anisotropic, and isotropic. The way the developed approach may be used in the context of slamming on marine structures is briefly discussed.

Level-crossing distributions of kinematic variables in multidirectional second-order ocean waves

“…Or alternatively, η ,x (1) |η (1) (t) ↑ may be viewed as resulting from the sum of two independent random variables, one being Gaussian and the other being Rayleigh-distributed (see e.g. [9,10]). The normal distribution is centered with a variance equal to [1 − ( ρ(1) ) 2 ]σ 2 η,x (1) , where ρ( 1) is the non-conditional correlation coefficient between η(1) and η ,x (1) , and σ η,x (1) is the nonconditional standard deviation of η ,x (1) .…”

“…subtracted from) the Gaussian variable if ρ(1) > 0 (resp. ρ(1) < 0) -see appendix A.1.2 in [10] for more details. Similarly to w (1) , the conditional distribution of η ,x (1) , given upcrossing, does not depend on the actual crossing level, .…”

“…When crossings are monitored for the sea surface, level crossing may be considered at a point fixed in space, along a line of sight at a given time, or at a moving material point (e.g. a material point with seakeeping motions [3,4] or forward motion, [8,9,10]). 1 Problems of practical interest may relate to the frequency of free-surface crossings, and/or the conditional distribution of wave-related variables, given free-surface crossing.…”