On the probability and averaged shape of extreme water waves

Extreme waves are reported more frequently than the probability predicted from standard statistical models in the open ocean. This departure from the classical statistical distributions is primarily due to nonlinear physics. Wave nonlinearity also modifies the most probable shape of extreme water waves as these waves propagate relative to that expected in a linear model. Better insights into both aspects of extreme water waves are crucial for the safety design of carriers and marine structures. The horizontal asymmetry and contraction of the averaged shape of extreme events can be observed in naturally occurring water waves. Under experimental conditions, horizontal asymmetry is a somewhat transient behaviour whereas contraction seems to be permanent. We also observe limited finite water effect on these nonlinear modifications, particularly for extremely steep ones. The probability of extreme events can be measured by the kurtosis of the free surface as a convenient proxy. The analytical derivations for how this evolves in wave flumes are found to be accurate for low steepness. For high steepness cases, the analytical models miss the transient maxima and over-predict the kurtosis at the steady state. This is primarily because the kurtosis is dependent on the spectral bandwidth in such situations. The analytical models underestimate the steady state kurtosis at finite water depth, particularly for the critical water depth. Data driven methods are applied to predict the space-time probability distribution of extreme events. Two practical examples with second order wave fields indicate the substantial potential of data driven methods in predicting the probability of extreme events. These benefits are particularly clear as the complexity of the problem increases. In this study, we closely examine the averaged shape and the probability of extreme water waves. Nonlinear modifications on both aspects lead to significant departure from the linear theory. These nonlinear modifications challenge the safety of the engineering design.