Theoretically-Based Probability Density Functions for the Surface Elevation in Highly-Nonlinear Irregular Seas

In this work we re-visit the probability density function (PDF) based on the cumulant generating function for the surface elevation in irregular seas. Importantly, this PDF is free of any assumptions involving narrow bandedness of the spectrum or small directionality of the wave field. First, a new ordinary differential equation ( is derived , governing the PDF to any desired order in nonlinearity. Second, a symptotic solutions to this ODE are found analytically in the limit of large surface elevation , newly providing the form of the positive tail beyond second order . These make theoretically clear how high order cumulants (involving e.g. the kurtosis) may affect the positive tail, which gets heavier at each successive order. Third , it is shown that the asymptotic solutions may be utilized to provide necessary boundary conditions, enabling numerical solution of the governing ODE . Hence, the methodology enables novel determination of the PDF to effectively any desired order in nonlinearity. This is a significant advancement, as previous exact solutions for this PDF have been limited to second order, which was only found recently by the present authors. Successful comparison against challenging data sets confirm accuracy of the new PDFs up to fifth order.