# Probability Distribution Functions of freak-waves

Presented are the results of numerical experiments on calculation of Probability Distribution Functions for surface elevations of water waves arising during the evolution of wave field which is described by JONSWAP spectrum.

Considering surface gravity waves which propagate in same direction we apply canonical transformation to a water wave equation and drastically simplify the Hamiltonian. This transformation explicitly uses the vanishing exact four-wave interaction for water gravity waves for a 2D potential fluid. The simplification of Hamiltonian allows to derive new compact equation which solves the spatial Cauchy problem for surface gravity wave on the deep water. This new equation can be easily implemented for numerical simulation.

No doubts that estimation of probability of extreme waves, or freak-wave, appearing at the surface of ocean has practical meaning. They are a native part of the surface wave dynamics in the open seas, and amongst different mechanisms of this phenomenon, linear dispersion and modulational instability are generally recognized. In the frame of the new spatial equation we calculate PDF for surface elevations of water waves which is described by JONSWAP spectrum for different values of wind speed.