Petya G. Petrova

The Effect of Third-Order Nonlinearities on the Statistical Distributions of Wave Heights, Crests and Troughs in Bimodal Crossing Seas

This paper investigates the effect of third-order nonlinearities on the statistical distributions of wave heights, crests, and troughs of waves mechanically generated in a deep-water basin and simulating two crossing systems characterized by bimodal spectra. The observed statistics exhibits various effects of third-order nonlinearities, in a manner dependent on both the distance from the wave-maker and the angle between the mean directions of the component wave systems. In order to isolate and demonstrate the effects of third-order nonlinearities by themselves, the vertically asymmetric distortions induced by second-order bound waves are removed from the observed time series. It appears then that the distributions of wave crests, troughs and heights extracted from the nonskewed records clearly deviate from the Rayleigh distribution, suggesting that the waves are characterized by nonlinear corrections of higher-order than the typical of second-order waves. Nonetheless, some models developed for weakly nonlinear second-order waves can still be used in describing wave heights, crests and troughs in mixed seas, provided that the relevant distribution parameters are modified, so as to reflect the effects of third-order corrections and some basic characteristics of the mixed seas.

Distribution of Wave Height Maxima in Storm Sea States

The effect of the coefficient of kurtosis, as a measure of third order nonlinearity, on the distribution of wave height maxima has been investigated. Measurements of the surface elevation during a storm at the North Alwyn platform in the North Sea have been used. The mean number of waves in the series is around 100. The maximum wave statistics have been compared with nonlinear theoretical distributions. It was found that the empirical probability densities of the maximum wave heights describe qualitatively the shift of the distribution modes toward higher values. The tendency for the peak of distribution to diminish with an increase in the coefficient of kurtosis up to 0.6 is also clearly seen. However, the empirical peak remains higher than the theoretically predicted one. The exceedance probability of the maximum wave heights was also estimated from the data and was compared with the theory. For the highest coefficients of kurtosis, estimated at nearly 0.6, the theoretical distribution approximates very well the empirical data. For lower coefficients of kurtosis, the theory tends to overestimate the exceedance probability of the maximum wave heights.

Influence of the Third Order Nonlinearity on the Distribution of Wave Height Maxima in an Offshore Basin

The effect of the coefficient of kurtosis, λ40 , on the distribution of wave height maxima has been investigated. The data set used consists of water surface displacements in irregular deep water unidirectional wave fields generated in an offshore basin and defined by the JONSWAP spectrum. The full-scale records are of almost 3h17min duration. The measurements have been performed at ten equidistant gauges along the basin, which permits to follow the changes in wave statistics away from the wave-generator. Subsequently, the records have been split into series of different length, corresponding to N = 100, 200 and 300 waves, and the probability density functions and the exceedance probabilities of the maximum wave heights have been constructed conditional on λ40 . They have been compared with the modified Edgeworth-Rayleigh model of Mori and Janssen [1] applied to the maximum wave heights. The theoretical expressions are formulated as a simple function of the coefficient of kurtosis and the number of waves in the sample. The coefficient of kurtosis, reflecting the third order nonlinearity, is found to increase with the distance from the wave-maker. The considered theoretical density curves describe only qualitatively the shift of the empirical mode towards higher values. The tendency of the peak of the distribution to diminish with increase of λ40 has been observed. However, the most probable wave height remains underestimated by the theory for all classes of λ40 , regardless of the length of the time series. Finally, the probability that a certain normalized height level Hmax /Hs will be exceeded increases with the increase of λ40 , as being theoretically predicted, although it is overestimated by the theory in the lower range of values of λ40 and underestimated over the higher range of values of λ40 .Copyright

Theoretical Analysis of Average Wave Steepness Related to Peak Period or to Mean Period

The average wave steepness may be defined as the ratio between the significant wave height Hs and the wavelength associated to either the zero-up-crossing mean period Lm or the peak period Lp (Bitner Gregersen et al., 1998). This parameter may be calculated from wave data at fixed locations, as well as by starting from theoretical spectra. In this paper the average wave steepness is firstly analyzed by considering a JONSWAP spectrum. It is shown that for this spectrum the ratio Sm = Hs /Lm , as well as Sp = Hs /Lp , depends upon the values of the spectrum parameters. The theoretical values are then compared with wave data in the Mediterranean Sea, Pacific Ocean, Atlantic Ocean and North Sea. The values of Sm and Sp are deeply investigated for severe sea states, with large values of Hs . It is obtained that in severe sea state the observed values of wave steepness, defined as either Sm or Sp , are always in the range defined by theoretical spectra; therefore, the extreme values of Sm and Sp , which are of interest for naval architecture, may be obtained from theoretical analysis, as a function of extreme values of Hs .Copyright

Wave Height Distributions of Laboratory Generated Bimodal Seas with Abnormal Waves

The study analyses bimodal data recorded in an offshore basin. The measured sea states represent two wave systems propagating at an angle of 60°, 90° and 120°, respectively, from each other. The wave spectra are separated into swell and wind sea components and the energy ratio between the areas under the associated curves (SSER) is estimated. The sea states are grouped into three classes with respect to SSER. Abnormal waves are identified in the three classes of SSER, although the majority of the cases pertain to wind-sea dominated wave fields. The probabilities of exceedance of the individual wave heights for the sea states with abnormal waves are estimated and approximated with probabilistic models, which take into account either the spectral bandwidth or the wave nonlinearities. Strong correlation is found between SSER and the fourth order normalized cumulant of the surface elevation, Λ40. The distributions for the maximum wave heights are constructed and compared with theoretical predictions depending only on the number of waves in the sample and Λ40. Furthermore, the probability of a wave to exceed a given threshold level conditional on Λ40 is compared with the theoretical estimates.