Mathematical formulation

In the previous section we have obtained for homogeneous random waves a clear relationship between spectral shape and kurtosis through the resonant and non-resonant four-wave interactions of the Zakharov equation. Following a central limit theorem, linear, dispersive random waves have a Gaussian PDF for the surface elevation. Finite amplitude effects result, however, in deviations from the Normal distribution, as measured by a finite skewness and kurtosis. () investigated the influence of second-order nonlinearity on wave crest distributions. However,for narrow band wave trains it will be shown that the wave height distribution only depends on the kurtosis. Therefore, we shall formulate the relationship between wave height distribution and kurtosis to examine analytically the effects of kurtosis on freak wave occurrence.

We assume that waves to be analyzed here are unidirectional with narrow banded spectra and satisfy the stationary and ergodic hypothesis. Let $eta(t)$ be the sea surface elevation as a function of time $t$ and $zeta(t)$ an auxiliary variable such that $eta(t)$ and $zeta(t)$ are not correlated. Assuming both $eta(t)$ and $zeta(t)$ are real zero-mean function with variance $sigma$, we have

$$Z(t) = \eta(t) + i\zeta(t) = A(t)e^{i\phi(t)},$$ (35)

$$A(t) = \sqrt{\eta^2(t) + \zeta^2(t)},$$ (36)

$$\phi(t) = \tan^{-1}\left(\frac{\zeta(t)}{\eta(t)}\right),$$ (37)

where $A$ is the envelope of the wave train and $phi$ the phase. () investigated the wave height distribution as a function of kurtosis and skewness using the joint probability density function of $eta(t)$ and $zeta(t)$ for a narrow banded weakly nonlinear wave train. We will follow this approach closely. For weakly nonlinear waves deviations from the Normal distribution are small. In those circumstances the PDF of the surface elevation can be described by the Edgeworth distribution. As there is no-correlation between $eta(t)$ and $zeta(t)$, the joint probability density function of $eta(t)$ and $zeta(t)$ becomes

$$p(\eta,\zeta) = \frac{1}{2\pi}\exp\left[-\frac{1}{2}(\eta^{2}+\zeta^{2})\right]$$

$$\hspace{0,5cm} \times \Biggl[ 1 + \frac{1}{3!}\sum_{n=0}^{3} \frac{3!}{(3-n)!n!}\kappa_{(3-n)n} H_{3-n}(\eta)H_{n}(\zeta)$$

$$\hspace{0.75cm} + \frac{1}{4!}\sum_{n=0}^{4} \frac{4!}{(4-n)!n!}\kappa_{(4-n)n} H_{4-n}(\eta)H_{n}(\zeta) \Biggr]$$ (38)

where $H_n$ is the $n$th order Hermite polynomial

$$H_n(x) = (-1)^n \exp\left(+\frac{1}{2}x^2\right) \frac{d^n}{dx^n} \exp\left(-\frac{1}{2}x^2\right).$$ (39)

All variables will be normalized by the variance of the surface elevation $sigma=m_{0}^{1/2}$ (where $m_{0}$ is the zero moment of the wave spectrum) and have zero-mean.

Keeping () and () in mind, the PDF of the envelope $A$ follows now immediately from an integration of the joint probability distribution

$$p(A,\phi)=A\;p(\eta,\zeta)$$ (40)

over $phi$, hence

$$p(A)= \int_0^{2\pi}{\rm d}\phi\; p(A,\phi).$$ (41)

Performing the integration over $phi$ it is found that the first term of () gives the usual Rayleigh distribution $Aexp(-A^2/2)$, while the terms involving the skewness $kappa_{30}$, etc, all integrate to zero because they are odd functions of $phi$. The third term does give contributions and as a result we find

$$p(A)=A e^{-\frac{1}{2}A^2} \left[1+\frac{1}{4} \left(\kappa_{40} + \kappa_{22}\right) \left(1-A^2+\frac{1}{8}A^4\right) \right],$$ (42)

where we have used $kappa_{40}=kappa_{04}$, a relation that can easily be verified. Finally, using $kappa_{22}=kappa_{40}/3$ the final result for the narrow-band approximation of the PDF of the envelope becomes

$$p(A)=A e^{-\frac{1}{2}A^2} \left[ 1 + \frac{1}{3}\kappa_{40}\left(1-A^2+\frac{1}{8}A^4\right) \right].$$ (43)

It is emphasized that, as expected, the PDF for the envelope does not contain contributions that are linear in the skewness. However, as follows from Eqns.() and (), the skewness is, compared to the kurtosis, a large quantity. Quadratic terms in skewness could give an equally important contribution to the PDF of $A$ as the kurtosis terms. This was pointed out by () and it required to extend the Edgeworth distribution to sixth order. Nevertheless, this additional term, proportional to $mu_3^2$, gives for narrow-band wave trains only a small contribution to the PDF for the same reason as the bound wave contribution to the kurtosis may be neglected.

From the result () interesting consequences for the distribution of maximum wave heights may be derived. In the narrow band approximation wave height $H$ equals $2A$ and hence the wave height PDF becomes

$$p(H)=\frac{1}{4}H e^{-\frac{1}{8}H^2}\left[1+\kappa_{40}A_H(H)\right],$$ (44)

where

$$A_H(H)=\frac{1}{384}\left(H^4-32H^2+128\right).$$ (45)

The exceedance probability $P_H(H)$ for wave height then follows from an integration of () from $H$ to $infty$:

$$P_H(H)=e^{-\frac{1}{8}H^2}\left[1+\kappa_{40}B_H(H)\right],$$ (46)

where

$$B_H(H)=\frac{1}{384}H^2\left(H^2-16\right).$$ (47)