Comparison of the theory with laboratory data

The validity of the theory is examined by means of a comparison with experimental data. The laboratory experiment was conducted in a glass channel which is 65m long, 1m wide, 2m high and was filled to a depth of about 1.0m. Water surface displacements were measured with twelve capacitance type wave gauges. Measurements with a sampling frequency of 32Hz were performed for over a period of 330s long. The number of waves per wave train was about 350-450. The initial spectra were given by Wallops type spectra with band widths of $m$=5, 10, 30, 60 and 100, and peak frequency of $f_p$=1Hz and a dimensionless water depth of $k_ph$=3.99 (with $h$ the water depth), so that the waves were deep-water waves. Initial phases of the waves were given by uniformly distributed random numbers. The details of the experimental setup and conditions are given in ().

The comparison of exceedance probability of wave heights is shown in Figure . The filled circles $bullet$ denote experimental data, the Rayleigh distribution is represented by the dotted line, Eq.() corresponds to the solid line, and the wave height distribution including skewness effects proposed by () (denoted as ER, Edgeworth-Rayleigh, in the figure) corresponds to the dashed line. For simplicity we refer to Eq.() as Modified ER (MER) hereafter. Due to the nonlinear effects, the exceedance probability obtained from the laboratory data departs for large wave height from the Rayleigh distribution. Both the MER and ER distributions for the exceedance probability of wave heights follow this separation at large amplitude region. Surprisingly, the MER distribution shows better agreement with the laboratory data than the ER distribution, although the corrections to the Rayleigh distribution only stem from effects of finite kurtosis. This same conclusion holds for larger values of the kurtosis (Figure (b)). Thus, Eq.() can be regarded as a theory to predict the height distribution for large amplitude waves in a narrow-band, weakly-nonlinear wave field.

It is noted that the MER distribution has a much simpler form than the ER distribution (, ), as only effects of finite kurtosis are retained. As argued before, for a narrow-band wave train there is no need to include effects of bound waves on the wave height distribution, hence effects of skewness can be discarded. Also, () thought that the cross correlation term $kappa_{22}$ was small, while it, in fact, gives a considerable contribution. This last point explains why the MER gives a better agreement with data than the ER.