Mathematical formulation

There are several possibilities to categorize a freak wave. We use the most simple freak wave definition which defines a freak wave as one having a maximum wave height $H_{max}$ exceeding twice the significant wave height $H_{1/3}$ of the wave train. Hence, in the context of the above freak wave definition, the PDF of maximum wave height is necessary.

The PDF of maximum wave height $p_m$ in wave trains can be obtained once the PDF of wave height $p(H)$ and exceedance probability of wave height $P(H)$ is known (, ), thus

$$p_m(H_{max})dH_{max} = N [1-P(H_{max})]^{N-1} p(H_{max})dH_{max}$$ (48)

with $N$ the number of waves. For sufficiently large $N$ one may use the approximation

$$\lim_{N\to \infty}^{N} [1-P(H_{max})]^{N} \simeq \lim_{N\to \infty}^{N} \exp\left[ -N P(H_{max}) \right],$$ (49)

Substituting Eq.() into Eq.(), gives the PDF of the maximum wave height, $p_m$,

$$p_{m}(H_{max}) dH_{max} = \frac{N}{4} H_{max} \mbox{\rm e}^{-\frac{H_{max}^2}{8}} \left[ 1 + \kappa_{40} A_H(H_{max}) \right]$$

$$\hspace{3.5cm} \times \exp\left\{ - N \mbox{\rm e}^{-\frac{H_{max}^2}{8}} \left[ 1 + \kappa_{40} B_H(H_{max}) \right] \right\} dH_{max}$$ (50)

and the exceedance probability of maximum wave height $P_m$,

$$P_{m}(H_{max}) = 1 - \exp\left\{ - N \mbox{\rm e}^{-\fra... ...{8}} \left[ 1 + \kappa_{40} B_H(H_{max}) \right] \right\}.$$ (51)

Equations Eq.() and () are evaluated as a function of $N$ and $kappa_{40}$ (or $mu_4$). For $kappa_{40}=0$ results are identical to the ones following from the Rayleigh distribution. For simplicity it will be assumed that $H_{1/3}=4 m_0^{1/2}$, although it is $H_{1/3}=4.004 m_0^{1/2}$ in an exact linear random wave theory. The freak wave condition in this study therefore becomes $H_{max}/m_0^{1/2} > 8$, and we obtain from Eq.() the following simple formula to predict the occurrence probability of a freak wave as function of $N$ and $kappa_{40}$,

$$P_{freak} = 1 - \exp\left[ - \beta N ( 1 + 8\kappa_{40} ) \right]$$ (52)

where $\beta=\mbox{\rm e}^{-8}$ is constant.

Using Eq.() it is seen that the effect of kurtosis becomes already of the same order as linear theory for $kappa_{40}=1/8$. This corresponds to $mu_4=3.125$, and is not a strong nonlinear condition. Hence, both the effects of finite kurtosis and the number of waves $N$ are important for determining the probability of maximum wave height in the nonlinear wave train.

Figure shows for increasing $mu_4$ from 3.0 to 3.5 the comparison between linear (Rayleigh) theory and present theory of the occurrence probability of a freak wave, $P_{freak}$, as a function of the number of waves $N$. For the case of $N=100$, the occurrence probability of a freak wave predicted by linear theory is 3.3%, while it is 15.4% according to Eq.() with $mu_4=3.5$, and for the case of $N=1000$, the occurrence probability of the freak wave is 28.5% according to linear theory, while it is 81.3% according to Eq.() with $mu_4=3.5$. The number of waves $N=1000$ corresponds to a duration of about 3 hours for the case of $T_{1/3}=10$s, which is not an unusual situation in the stormy conditions. Alternatively, defining the threshold value of the occurrence probability of a freak wave as 50%, the expected number of waves that include at least one freak wave as a maximum wave is 2,000 waves predicted by linear theory, and becomes 500 waves predicted by Eq.() with $mu_4=3.5$. Thus, in a strong nonlinear field freak waves can occur several times more frequently than in a linear wave field.

Figure shows the ratio $R_{freak}$ of freak wave occurrence probability predicted by the present approach and Rayleigh theory as a function of kurtosis $mu_4$,

$$R_{nonlinear} = \frac{ P_{freak} }{P_{freak}\vert _{\kappa_{40}=0}} - 1$$ (53)

For the case of a small number of waves $Nle250$, the ratio $R_{freak}$ depends linearly on $mu_{4}$. If $mu_4$ is 3.1 and $Nle500$, the occurrence probability of freak waves is 50% more than according to linear theory. On the other hand, the increment of $R_{freak}$ decreases as the number of waves increases. This is because for very large number of waves even in linear theory the maximum wave height almost always exceeds 2 times $H_{1/3}$.