依据波动的射线理论及随机波的线性模型，提出了一种三维海浪要素统计分布的完整理论。依此理论，只要获得方向谱的10个参数就能确定各种波浪要素的统计分布。结果表明，波高仍然服从Rayleigh分布；波高与周期的联合分布仅取决于一维谱而与给定的弥散关系无关；与波长有关的分布不仅依赖于谱的高频行为并取决于谱的方向特征；波陡分布由方向谱的6个参数确定；周期与波长的联合分布是存在的。当然，详尽的观测或实验检验是完全必要的。

本文是随机海浪二阶非线性特征量研究的第一部分。在精确导出任意均匀水域上三维随机波动一阶解、二阶解的基础上，更严密、更合理地重新推导了波面高度分布的偏度与二阶功率谱的理论表达式，并首次给出了有限水深波面二阶谱的理论形式。

本文首先对文献[1]中导出的理论结果给予明确、合理的物理解释；然后，将这些理论结果分别应用于三维深水波动、二维有限深水及二维深水波动，系统地给出这些主要波动模式中二阶非线性特征量的理论表达式；最后，论述了在大型风浪槽内设计并进行的风浪实验，用获得的实验资料验证了本文导出的理论结果，发现两者相当符合。

依据作者提出的海浪能量外观分布，即外频谱的概念，重新推导了Bretschne Lder风浪谱，极大地简化了Bret schneider原来的推演过程，并对谱的系数予以订正，将订正后的Bretschneider谱与Pie rSOll和Mos cowitz谱进行了比较，结果表明，两种风浪频谱在数学表述上是完全等价的。

Based upon the nonlinear model of random sea waves, the statistical distribution of wave surface elevation exact to the fourth order is derived as the truncated Gram-Charlier series, by calculating directly each order moment. The phenomenon found by Huang et al. that the agreement between observed data and investigated series deteriorates much more when the series is kept to λs is explained, The effect of the appmximation order on the truncation of series and the determination of coefficients is investigated. For the mth order approximation, the derived series is truncated at H3m-3 with the absence of H3m-4, and the coefficients of H3m-3 and H3m-6 are connected by a simple algebraic relation.

Based upon the nonlinear model of Languet-Higgins the joint distribution of wave surface slopes is theoretically investigated. It is shown that in the fourth order approximation, the distribution is given by truncated Gram-Charlier series. The types of wave-wave coupling interactions are related to the order of approximation to nonlinearity of sea surface, which eventually defines the truncated term of the Gram-Charlier series. For each order approximation, the coefficients in the series are modified comparatively to the corresponding ones for the previous order approximation. If the nonlinear effect of the kurtosis is considered, the wave surface must be as accurate at least as to the third order approximation, and with regard to skewness, the wave surface must be as accurate at least as to the fourth order approximation.

A concept of apparent energy distribution of sea waves is suggested and named "the outer Dequency spectrum" to distinguish it from the frequency spectrum in common sense In line with the concept, a kind or outer spectrum with three parameters in Cauchy's form is derived theoretically from the joint distribution of wave periods and heights reached previously by the present authors and the Bretschnelder spectrum is rededuced simply in a distinct manner. The foundation of medlanics of the snnilarity between the frequency, spectrum m common sense and the outer one is briefly discussed, in view of the fact that Bretschneider spectrum has a wide application in engineering, the advantages and probabihty of asmg the outer spectrum as a common one are also considered.

Based upon the nonlinear model of random sea waves, a commonly applicable statistical distribution of wave surface elevation exact to the third order is derived through the" direct calculations of each order moment. The distribution arrived reduces, in the sense of being exact to He, to the Gram-Charlier series due to Longuet-Higgins for deep water, provided that only the two simplest kinds of wave-wave interactions are Taken into account. The reason why the agreement of Oram-Charlier series with experimental data becomes worse and worse as the terms of series are increased is explicited for the first time.