Demos
Wave statistics and wave distributions in WAFO
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Empirical and theoretical wave statistics  from Chapter 3 in the Tutorial
This section contains WAFO algorithms for empirical analysis of wave data and for computation of exact wave characteristic distributions from wave models.
Statistical analysis of empirical wave data
One of the main objects of the WAFO toolbox is computation of wave
characteristic distributions from different wave models. To help with the
evaluation of the accuracy of the theoretical results WAFO also contains tools
for empirical analysis of observed wave records 
onedimensional analysis of time and space distributions, as well as
twodimensional analysis of
joint distributions of, for example, wave period and wave amplitude, are
possible.
Several wave period and wavelength statistics can be handled by WAFO,
for example
The figure shows the observed distribution of crest period in a wave data set. The solid curve shows a kernel estimate of the probability density function. The smoothness of estimate can be regulated  in this example there is very little smoothing for short waves. 

Joint wave period and wave height distributions are found and smootheed by kernel density estimation.  
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Theoretical wave distributions
Wave characteristic distributions in standard wave models and exact distributions in transformed Gaussian models
By WAFO one calculate theoretical wave characteristic distributions in two ways. Simplest is to use any of the standard approximations suggested by LonguetHiggins (1975, 1983) or by Cavanié et al. (1976), or the common Rayleigh approximation based on the narrow band approximation. More exact (and just as simple by WAFO) is to calculate the exact distribution based on the full wave energy spectrum (or the equivalent correlation function). The only assumption needed is that the wave process is Gaussian, or that it can be transformedto a Gaussian wave by a memoryless transformation.
Both time domain and space domain variables can be calculated. We give an example with a twopeaked spectrum and compare the distribution of crest period and crest length.
The LonguetHiggins and Cavanié approximations
The LonguetHiggins approximation for joint crest period (Tc) and height (Ac) distribution is based on the joint distribution of the envelope amplitude and the time derivative of the envelope phase. The figure shows the result for a Gaussian sea with JONSWAP spectrum with peak period 10 s and significant wave height 5 m.  
In the Cavanié approximation of the joint crest period (Tc) and height (Ac) distribution every local maximum above the mean level is considered as a crest of a wave; the wave is then approximated by a sine wave with the same curvature and height as the maximum. The figure shows the result for a Gaussian sea with JONSWAP spectrum with peak period 10 s and significant wave height 5 m. 
Exact distributions in transformed Gaussian models
Some of the most useful routines in WAFO calculate the exact joint
distribution of wave period/length quantities and the corresponding amplitude or
height variables in a Gaussian sea. It can also calculate the joint distribution
for the height of successive local maximum and minimum values for use in fatigue
analysis.
Exact distribution of crest period (Tc) and height (Ac) for a Gaussian sea with JONSWAP spectrum with peak period 10 s and significant wave height 5 m.  
A Torsethaugen spectrum with significant wave height 6 m and primary peak period 8 sec with two peaks, corresponding to wind driven waves and swell.  
The density of crest period for the Torsethaugen spectrum computed by WAFO. Note the tendency to a mixed distribution, corresponding to the two spectral peaks.  
WAFO contains an option to swithch between time and space variables.
Here is the density (red curve) of crest length Lc in meters, i.e. the
distance between a mean level upcrossing and the next downcrossing. Note the
tendency to many very short waves.
The very short waves can be considered as mere noise, and a more interesting quantity is the crest length for those wave which have a crest height exceeding some moderately high level. The black curve shows the calculated distribution of crest length Lc for waves with crest height Ac>1.5 m (=Hs/4). 

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