Wave Statistics in Nonlinear Sea States

2011 ◽  
Author(s):  
Mohamed Latheef ◽  
Chris Swan
Keyword(s):  
Empirical Distribution ◽  
Nonlinear Effects ◽  
Rayleigh Distribution ◽  
Second Order ◽  
Wave Crest ◽  
Statistical Distributions ◽  
Sea State ◽  
Wave Statistics ◽  
Wave Basin ◽  
Wave Heights

This paper concerns the statistical distribution of both wave crest elevations and wave heights in deep water. A new set of laboratory observations undertaken in a directional wave basin located in the Hydrodynamics laboratory in the Department of Civil and Environmental Engineering at Imperial College London is presented. The resulting data were analysed and compared to a number of commonly applied statistical distributions. In respect of the wave crest elevations the measured data is compared to both linear and second-order order distributions, whilst the wave heights were compared to the Rayleigh distribution, the Forristall (1978) [1] empirical distribution and the modified Glukhovskiy distribution ([2] and [3]). Taken as a whole, the data confirms that the directionality of the sea state is critically important in determining the statistical distributions. For example, in terms of the wave crest statistics effects beyond second-order are most pronounced in uni-directional seas. However, if the sea state is sufficiently steep, nonlinear effects arising at third order and above can also be significant in directionally spread seas. Important departures from Forristall’s empirical distribution for the wave heights are also identified. In particular, the data highlights the limiting effect of wave breaking in the most severe seas suggesting that many of the commonly applied design solutions may be conservative in terms of crest height and wave height predictions corresponding to a small (10−4) probability of exceedance.

Physics-Based Approach to Wave Statistics and Probability

2013 ◽  
Author(s):  
Alexander V. Babanin
Keyword(s):  
Extreme Events ◽  
Modulational Instability ◽  
Probability Distributions ◽  
Rayleigh Distribution ◽  
Order Theory ◽  
Second Order ◽  
Ocean Engineering ◽  
Actual Distribution ◽  
Wave Statistics ◽  
Wave Heights

Design criteria in ocean engineering, whether this is one in 50 years or one in 5000 years event, are hardly ever based on measurements, and rather on statistical distributions of relevant metocean properties. Of utmost interest is the tail of these distributions, that is rare events such as the highest waves with low probability. Engineers have long since realised that the superposition of linear waves with narrow-banded spectrum as depicted by the Rayleigh distribution underestimates the probability of extreme wave crests, and is not adequate for wave heights either, which is a critical shortcoming as far as the engineering design is concerned. Ongoing theoretical and experimental efforts have been under way for decades to address this issue. Here, we will concentrate on short-term statistics, i.e. probability of crests/heights of individual waves. Typical approach is to treat all possible waves in the ocean or at a particular location as a single ensemble for which some comprehensive solution can be found. The oceanographic knowledge, however, now indicates that no single and united comprehensive solution is possible. Probability distributions in different physical circumstances should be different, and by combining them together the inevitable scatter is introduced. The scatter and the accuracy will not improve by increasing the bulk data quality and quantity, and it hides the actual distribution of extreme events. The groups have to be separated and their probability distributions treated individually. The paper offers a review of physical conditions, from simple one-dimensional trains of free waves to realistic two-dimensional wind-forced wave fields, in order to understand where different probability distributions can be expected. If the wave trains/fields in the wave records are stable, distributions for the second-order waves should serve well. If modulational instability is active, rare extreme events not predicted by the second-order theory should become possible. This depends on wave steepness, bandwidth and directionality. Mean steepness also defines the wave breaking and therefore the upper limit for wave heights in this group of conditions. Under hurricane-like circumstances, the instability gives way to direct wind forcing, and yet another statistics is to be expected.

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

2011 ◽  
Author(s):  
Petya G. Petrova ◽  
M. Aziz Tayfun ◽  
C. Guedes Soares
Keyword(s):  
Rayleigh Distribution ◽  
Second Order ◽  
Statistical Distributions ◽  
Third Order ◽  
Weakly Nonlinear ◽  
Distribution Parameters ◽  
Wave Heights ◽  
Component Wave ◽  
The Mean ◽  
Basic Characteristics

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, simulating two crossing systems characterized by bimodal spectra. Observed statistics exhibit 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, 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 non-skewed series clearly deviate from the Rayleigh distribution, suggesting that 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 relevant distribution parameters are modified so as to reflect the effects of third-order corrections and some basic characteristics of mixed seas.

Analysis of Short-Term and Long-Term Wave Statistics by Time Domain Simulations Statistics

2017 ◽  
Author(s):  
Øistein Hagen ◽  
Ida Håøy Grue ◽  
Jørn Birknes-Berg ◽  
Gunnar Lian ◽  
Kjersti Bruserud
Keyword(s):  
Wave Height ◽  
Time Domain ◽  
Statistical Distributions ◽  
Short Term ◽  
Sea State ◽  
Wave Statistics ◽  
Individual Wave ◽  
Wave Heights ◽  
Crest Height

In the design of new structures and assessment of existing structures, short- and long term statistical distributions of wave height, crest height and wave periods, as well as joint distributions, are important for structural integrity assessment. It is important to model the statistical distributions accurately to calculate wave design criteria and to assess fatigue life. A detailed study of the wave statistics for an offshore location at the Norwegian Continental Shelf field is carried out. Extensive time domain simulations for the complete scatter diagram of possible sea states are carried out by a second order wave model. Time series of the surface elevation are generated for JONSWAP and Torsethaugen wave spectra, and for several wave spreading models. Statistics for individual wave heights, crest heights and wave periods are established. The simulated results for the short-term statistics are compared with existing short term models that are commonly used, viz. the Forristal, Næss and Rayleigh wave height distributions, and the Forristall 2nd order crest height distribution. Also, parameterized distributions for wave height and for crest height are fitted to the simulated data. The long-term distributions F(H) and F(C) of all simulated individual wave heights H and crest heights C are determined by weighting the simulations with the long-term probability of occurrence of the sea state. Likewise, the long-term distributions F(Hmax) and F(Cmax) of the maximum simulated individual wave heights Hmax and crest heights Cmax in the sea states are determined. The design criteria for return periods R = 1, 10, 100 and 10 000 years are determined from the appropriate quantile levels. The effect of statistical uncertainty is investigated by comparing the confidence intervals for the estimated extreme values results as function of the number N of 3-hour time domain simulations per sea state for 10<N<500.

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

2013 ◽  
Vol 135 (2) ◽  
Author(s):  
Petya G. Petrova ◽  
M. Aziz Tayfun ◽  
C. Guedes Soares
Keyword(s):  
Rayleigh Distribution ◽  
Second Order ◽  
Statistical Distributions ◽  
Third Order ◽  
Weakly Nonlinear ◽  
Distribution Parameters ◽  
Wave Heights ◽  
Component Wave ◽  
Basic Characteristics ◽  
The Waves

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.

Second Order Model for Wave Crests Used in Prediction of Green Water Load and Volume on Ships in Random Waves

2004 ◽  
Author(s):  
Hanne Therese Wist ◽  
Dag Myrhaug ◽  
Ha˚vard Rue
Keyword(s):  
Wave Theory ◽  
Rayleigh Distribution ◽  
Second Order ◽  
Green Water ◽  
Wave Crest ◽  
Random Waves ◽  
Sea State ◽  
Water Load ◽  
Height Model ◽  
Crest Height

The probability that a wave crest in a random sea will exceed a specified height has long been recognized as important statistics in practical work, e.g., in predicting green water load and volume on a ship. Nonlinear probability density functions for predicting green water load and volume are presented. The models are based on the linear model of [1] in combination with transformation of a second order wave crest height model. The wave crest height model is obtained from second order wave theory for a narrow-banded sea state in combination with transformation of the Rayleigh distribution. Results from the models are compared with model tests of a cargo ship presented in [1].

An Experimental Study of Shallow Water Wave Statistics on Mild Bed Slopes

2011 ◽  
Author(s):  
Vasiliki Katsardi ◽  
Chris Swan
Keyword(s):  
Water Depth ◽  
Measured Data ◽  
Intermediate Water ◽  
Statistical Distributions ◽  
Spectral Bandwidth ◽  
Peak Period ◽  
Wave Statistics ◽  
Wave Heights ◽  
Bed Slope ◽  
Water Depths

This paper describes a new series of laboratory observations, undertaken in a purpose built wave flume, in which a number of scaled simulations of realistic ocean spectra were allowed to evolve over a range of mild bed slopes. The purpose of the study was to examine the distribution of wave heights and its dependence on the local water depth, d, the local bed slope, m, and the nature of the input spectrum; the latter considering variations in the spectral peak period, Tp, the spectral bandwidth and the wave steepness. The results of the study show that for mild bed slopes the statistical distributions of wave heights are effectively independent of both the bed slope and the spectral bandwidth. However, the peak period plays a very significant role in the sense that it alters the effective water depth. Following detailed comparisons with the measured data, the statistical distributions for wave heights in relatively deep water are found to be in reasonable agreement with the Forristall [1] and Glukhovskii [2] distributions. For intermediate water depths, the Battjes & Groenendijk [3] distribution works very well. However, for the shallowest water depths none of the existing distributions provides good agreement with the measured data; all leading to an over-estimate of the largest wave heights.

Statistical Analysis of a Set of Basin Waves

2011 ◽  
Author(s):  
Jule Scharnke ◽  
Janou Hennig
Keyword(s):  
Statistical Analysis ◽  
Wave Height ◽  
Water Depth ◽  
Water Waves ◽  
Rayleigh Distribution ◽  
Second Order ◽  
Distribution Functions ◽  
Wave Crest ◽  
Intermediate Water ◽  
Wave Measurements

In a recent paper the effect of variations in calibrated wave parameters on wave crest and height distributions was analyzed (OMAE2010-20304, [1]). Theoretical distribution functions were compared to wave measurements with a variation in water depth, wave seed (group spectrum) and location of measurement for the same initial power spectrum. The wave crest distribution of the shallow water waves exceeded both second-order and Rayleigh distribution. Whereas, in intermediate water depth the measured crests followed the second order distribution. The distributions of the measured waves showed that different wave seeds result in the same wave height and crest distributions. Measured wave heights were lower closer to the wave maker. In this paper the results of the continued statistical analysis of basin waves are presented with focus on wave steepness and their influence on wave height and wave crest distributions. Furthermore, the sampling variability of the presented cases is assessed.

Wave Distributions and Sampling Variability

2007 ◽  
Author(s):  
O̸istein Hagen
Keyword(s):  
Wave Height ◽  
Extreme Values ◽  
Height Distribution ◽  
Short Term ◽  
Sea State ◽  
Sampling Variability ◽  
Wave Statistics ◽  
Individual Wave ◽  
Wave Heights ◽  
Crest Height

The paper describes the effect of sampling variability on the predicted extreme individual wave height and the predicted extreme individual crests height for long return periods, such as for the 100-year maximum wave height and 100-year maximum crest height. We show that the effect of sampling variability is different for individual crest or wave height as compared to for significant wave height. The short term wave statistics is modeled by the Forristall crest height distribution and the Forristall wave height distribution [3,4]. Samples from the 3-hour Weibull distribution are simulated for 100.000 years period, and the 100-year extreme values for wave heights and crest heights determined for respectively 20 minute and 3 hour sea states. The simulations are compared to results obtained by probabilistic analysis. The paper shows that state of the art analysis approaches using the Forristall distributions give about unbiased estimates for extreme individual crest or wave height if implemented appropriately. Direct application of the Forristall distributions for 3-hour sea state parameters give long term extremes that are biased low, and it is shown how the short term distributions can be modified such that consistent results for 20 minute and 3 hour sea states are obtained. These modified distributions are expected applicable for predictions based on hindcast sea state statistics and for the environmental contour approach.

The Second Order Statistics of High Waves in Wind Sea and Swell

2011 ◽  
Vol 133 (3) ◽  
Author(s):  
Peter Tromans ◽  
Luc Vanderschuren ◽  
Kevin Ewans
Keyword(s):  
Wave Height ◽  
Probabilistic Method ◽  
Nonlinear Effects ◽  
Second Order ◽  
Wave Crest ◽  
Extreme Wave ◽  
Type Method ◽  
Second Order Statistics ◽  
Wind Sea ◽  
Steep Waves

The statistics of extreme wave crest elevation and wave height have been calculated for realistic, directionally spread sea and swell using a probabilistic method tested and described previously. The nonlinearity of steep waves is modeled to the second order using Sharma and Dean kinematics, and a response surface (reliability type) method is used to deduce the crest elevation or wave height corresponding to a given probability of exceedance. The effects of various combinations of sea and swell are evaluated. As expected, in all cases, nonlinearity makes extreme crests higher than the corresponding linear ones. The nonlinear effects on the wave height are relatively small.

Experimental Investigation of the Nonlinear Effects on the Statistics of Vertical Motions and Loads of a Containership in Irregular Waves

2004 ◽  
Vol 48 (02) ◽  
pp. 148-167 ◽  
Author(s):  
N. Fonseca ◽  
C. Guedes Soares
Keyword(s):  
Experimental Data ◽  
Experimental Investigation ◽  
Probability Distributions ◽  
Nonlinear Behavior ◽  
Nonlinear Effects ◽  
Rayleigh Distribution ◽  
Irregular Waves ◽  
Cross Sectional ◽  
Wave Heights ◽  
Vertical Accelerations

The paper presents the results of an experimental investigation of the nonlinear effects on the vertical motions and loads on a containership model advancing in irregular waves. The experimental data are compared with numerical results from a nonlinear time domain strip method. The tests were carried out in a seakeeping tank using three sea states with significant wave heights of 4.2 m, 6.1 m, and 9.9 m, thus including very severe conditions. The measured responses include the absolute and relative motions, vertical accelerations, and cross-sectional loads at midship and ¼ Lpp from the forward perpendicular. The statistics of the experimental records demonstrate partly the nonlinear behavior of the responses, especially of the structural loads. The probability distributions of the positive and negative peaks show that the heave and pitch motions are only slightly asymmetric and their distributions compare well with the Rayleigh distribution. The vertical loads present distributions of peaks that are highly asymmetric and deviate from the Rayleigh distribution. Comparisons between simulated results and experimental data show that the numerical model is able to represent the nonlinear characteristics of the responses.

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