The physical, gain–loss model for the stochasticity of the phase velocity of a wind-driven water wave

Mathematical-model study of the behavior of a ship in an irregular sea (e.g., the roll motion) includes an analysis of the particular case where the stochastic wind-driven wave is a plane wave, i.e. the wave traveling in one spatial direction. The stochasticity of a wind-driven wave is solely due to the stochasticity of wind. By using the basic physical facts on the water-surface/wind interaction, more specifically, the rates of the gain and loss of the wave energy, the present work derives a dynamical model for the phase velocity of a water wave. The model is a linear ordinary differential equation for the cubed phase velocity. This equation which includes the velocity of wind becomes stochastic when the wind is a stochastic process, for example, is described by the first-order Itô’s stochastic differential equation (ISDE). In this case, the water-wave phase velocity appears to be a solution of the second-order ISDE. The derivation of the model implies an analytical relation which explicitly couples the parameters “alpha” and “beta” of the well-known Pierson–Moskowitz spectral function. It is shown that the derived relation agrees well with the experimentally obtained values of the parameters. The derived model can be used not only in models for the ship dynamics in a stochastic sea but also in a number of other ocean problems where the prediction of water waves by mere measurement of the wind is an advantage. These problems in particular include the real-time control of the stochastic water load on the ocean structures by sensing the stochastic wind rather than sensing stochastic waves thereby creating an alternative to more complex and expensive methods.