Joint distributions of wave steepness in narrow band sea states

• The joint probability densities of wave steepness and height and of wave steepness and period are derived.
• It starts from Longuet-Higgins׳ joint probability density of wave period and amplitude for narrow band spectra.
• The formulae of the derived bivariate densities are fitted to numerically simulated data with narrow band spectra.
• Comparisons with bivariate steepness-period and steepness-height densities for ocean and tank waves are presented.
• The marginal probability density of wave steepness is obtained by numerical integration.

The joint probability densities of up-crossing wave steepness and height and of up-crossing wave steepness and period are derived from Longuet-Higgins׳ joint probability density of wave period and amplitude for narrow band spectra. The formulae of the derived bivariate densities are fitted to numerically simulated data with narrow band spectra. Comparisons with bivariate steepness-period and steepness-height densities for ocean and tank waves are also presented. Concordance between joint distributions for different data types is verified with bivariate Kolmogorv–Smirnov test. The marginal probability density of wave steepness is obtained by numerical integration of the joint distributions and is compared with different data.