New third-order probability density function for the surface elevation in irregular seas

We develop saddle-point approximations of the probability density function (PDF) for the surface elevation in nonlinear, irregular seas, without any assumptions regarding uni-directionality of the wave field or narrow-bandedness of the spectrum. A new, closed-form approximation of the PDF at third order is derived, accounting for both skewness and kurtosis. The approximate PDF matches known asymptotics of the third-order positive tail exactly and is free of unphysical oscillations in the negative tail, which widely plague existing PDFs describing nonlinear seas. A normalisation procedure is introduced to enforce unit integral, zero mean and unit variance. We further show that the input cumulants can be adjusted to likewise yield prescribed skewness and kurtosis. Comparison of the new third-order PDF against challenging datasets stemming from irregular wave theory, a fully nonlinear wave model as well as laboratory experiments establishes accuracy. In the Appendix we additionally develop a novel perturbation method, which is used to derive the full form of the asymptotic positive and negative tails at fourth and fifth order, including (previously unknown) prefactors. Collectively, it is shown that even-order solutions inevitably involve oscillatory negative tails, whereas odd orders admit non-oscillatory negative-tail solutions.