Third-order Solution Of Random Waves And Surface Elevation Statistical Distribution

In the context of potential flow theory,surface gravity waves are a conservative energy dispersive system,and the resonance mechanism leads to energy transfer between primary waves,resulting in a series of dynamic physical phenomena,such as modulation instability and frequency down shift.However,for the analytical theory of surface gravity waves,resonance interactions become an obstacle in solving the steady-state solution（periodic solution）of surface gravity waves,resulting in a problem of division by zero.It is difficult to deal with this mathematically singular problem by using the classical perturbation theory.To solve this zero division problem,in the Hamiltonian theory of water waves,we derive a third-order steady-state solution of multidirectional irregular waves by using the canonical transformation and removing the resonance term that causes the dynamic evolution of the system.This derivation method successfully solves the zero division problem caused by four wave resonance,so that the Hamiltonian solution can reasonably describe the third-order part of the resonant steady-state wave.In addition,the nonlinear steady-state solution of multidirectional irregular waves is the main theoretical basis of wave statistical distribution,especially surface elevation statistical distribution.After obtaining the reasonable third-order steady-state solution,surface elevation statistical distribution law that is more consistent with nonlinear wave surface characteristics and higher accuracy can be obtained,thereby enriching the research on surface elevation theoretical distribution.Aiming at the deficiencies of the existing theoretical distribution models,this paper proposes a Gaussian mixture model to estimate the probability density function of surface elevation,and achieves a good fitting effect.This paper can be divided into two parts:One is about the derivation of the third-order steady-state solution of multidirectional irregular waves,and the derivation of integral properties in the random wave field;The other is the research on the theoretical distribution of surface elevation based on the above third-order steady-state solution.The main research work and conclusions are as follows:（1）A third-order approximate solution for surface gravity waves in the finite water depth is studied in the context of potential flow theory.This solution corresponds to the steady-state part of a multidirectional irregular wave field and provides explicit expressions for the surface elevation,free-surface velocity potential,and velocity potential.The amplitude dispersion relation is also provided.Two approaches are used to derive the third-order analytical solution,resulting in two types of approximate solutions:the perturbation solution and the Hamiltonian solution.The perturbation solution is obtained by a classical perturbation technique in which the time variable is expanded in multiscale to eliminate secular terms.The Hamiltonian solution is derived by using the canonical transformation and removing the resonance term that causes the dynamic evolution of the system in the Hamiltonian theory of water waves.（2）By comparing the two types of solutions,it is found that they are completely equivalent for the first-and second-order solutions and the nonlinear dispersion,but for the third-order part only the₉9)+8)8)+)terms are the same.This shows that the zero division problem of the perturbed solution occurs in the remaining third-order part,resulting in the collapse of the solution.However,the Hamiltonian solutions break through the difficulty that the perturbation theory breaks down due to singularities in the transfer functions when the quartet resonance criterion is satisfied,so that it can reasonably describe the third-order part of the resonant steady-state wave.This is attributed to the complete separation of dynamic and bound harmonics,and the removal of resonance term.（3）Based on the Hamiltonian solution,some integral（statistical）properties in the random wave field,such as skewness,kurtosis and volume flux,are derived,and the non-uniqueness of the mean surface and mean induced flow is presented.Through the analysis of the theoretical formulas of skewness and kurtosis,it is found that as the wave steepness increases,the relative water depth decreases,and the spectral width becomes wider,they tend to become larger on the whole,but this change is not monotonic（using the Wallops and JONSWAP spectrum as input spectrum）.（4）Based on the surface elevation solution in the Hamiltonian solution,the approximate moment generation function is obtained.Using the relationship that the probability density function and the moment generation function are Fourier transform,the final theoretical distribution is in line with the Gram-Charlier A series.The theoretical distribution obtained from the cumulative moment generating function conforms to the Edgeworth series.We have clarified the difference between the two derivation methods.Finally,the calculation results show that the fitting effect of Edgeworth series distribution is better than that of Gram-Charlier A series distribution.（5）The Gaussian mixture model GMM is proposed to estimate probability density function of surface elevation,which theoretically overcomes the difficulty of negative probability in Edgeworth series model,and the maximum likelihood estimation based on EM algorithm is used to determine the unknown parameters in GMM.Through Monte Carlo numerical simulation and observation data,it is confirmed that GMM has excellent fitting ability of surface elevation distribution and is not limited by spectral width.