Ergodic Analysis Of Shallow-water Waves Based On Displacement Method

The sea waves are the main loads of sea ports,ships and oil production platforms.Waves in the actual ocean are often highly irregular,which is not only reflected in the random fluctuation in time direction,but also in the different evolution of ocean waves at different space points.Therefore,it is necessary to adopt statistical theory to conduct quantitative research on ocean waves.In the current researches on stochastic waves,it is usually assumed that the ocean wave is a stochastic process which are not only stationary but also ergodic.This assumption assumes that the time-averaged value of the random variable is equal to the ensemble-averaged value.Based on this assumption,the statistical law of ocean waves can be described by the time statistical information of a sample observed for a single long time.Although the ergodic hypothesis has been widely accepted in the research and application of statistical waves,it is still not clear whether ocean waves is ergodic or not,especially for the commonly used shallow water wave theory.There seems to be no evidence to support the ergodic assumption.Using the shallow water equation based on displacement and pressure,this paper intends to test numerically the ergodic properties of random shallow water waves.The main content of this thesis are as follows:1.The testing method to check the ergodicity of stochastic process is discussed first.Based on the central limit theorem of multivariable,this thesis developed a hypothesis testing method to test the stationarity of a random process.The probability of misjudgments of this method is discussed in detail,based on which,some techniques are proposed to improve he accuracy and stability of the proposed testing method.On this basis,we introduced the properties of covariance functions of stationary stochastic processes and the variances of the time-averaged values,which provides the basis for the numerical test of the ergodicity.2.The stationarity and ergodicity of random shallow water waves in the case of flat bottom condition are systematically studied.10000 groups of initial random water surface are generated based on Neumann wave spectrum,and the evolution of each random water surface is numerically simulated to obtain 10000 groups of random wave evolutions.On this basis,the stationarity and ergodicity of random shallow water waves in time and space are studied by using the proposed hypothesis testing method.3.The shallow water model with an uneven bottom is further discussed.The stationarity and ergodicity of the water surface height,water bottom pressure and horizontal displacement in time and space are studied.The stationary and ergodic properties of the means and second moments of the water surface height,water bottom pressure and horizontal displacement in time and space are obtained.The stationary properties of their fourth moments are also obtained.The influence of the reflected waves from the boundaries and the uneven bottom on the stationary and ergodicity of are revealed.