Distributions of the Maximum Wave Height

This study defines a freak wave as a maximum wave height $H_{max}$ that exceeds 2 times of its significant wave height $H_{1/3}$ in a wave train. However, the wave height distribution described by an exponential function does not have an upper limit. Thus, the maximum wave height in wave trains is only able to derive the maximum value of a finite number of waves (Goda, 2000). If $N$ is sufficiently large as

$$\lim_{N\to \infty}^{N} [1-P(H_{max})]^{N} = \exp\left[ -N P(H_{max}) \right],$$ (10)

the maximum wave height distribution of wave height with total number of waves $N$ can be derived, Substituting Eq.(6) into Eq.(10), gives the distribution of the maximum wave height.

$$p_{max}(H_{max}) %dH_{max} = \frac{d\,\,}{dH_{max}} \exp[- N P(H_{max})]$$

$$= N \, \frac{H_{max}}{4} \exp\Biggl\{ -\frac{H_{max}^2}{8} -N_{0}... ...2}{8} \right) \left[ 1 + \sum_{i,j}\beta_{i,j}E_{i,j}(H_{max}) \right] \Biggr\}$$

$$\hspace{3.0cm} \times \left[ 1 + \sum_{i,j}\beta_{i,j}E_{i,j}(H_{max}) \right] dH_{max}$$ (11)

Furthermore, the exceedance probability of the maximum wave height can be calculated by

$$P_{max}(H_{max}) = 1 - \exp\left\{ - N \frac{H_{max}^2}{... ... 1 + \sum_{i,j}\beta_{i,j}E_{i,j}(H_{max}) \right] \right\}$$ (12)

Note that the Eq.(11) and Eq.(12) are normalized by the root-mean-square value of surface elevation, $\eta _{rms}$. Therefore, the freak wave occurrence probability that is defined by $H_{max}/H_{1/3}\ge2$, is calculated by Eq.(10) and Eq.(12) using the empirical relationship between $H_{1/3}$ and $\eta _{rms}$ by Eq.(9).

Figure 3: Occurrence probability of freak wave, defined by $H_{max}$/$H_{1/3}\ge 2$, as a function of number of waves $N$ with different values of kurtosis $\mu$ ($\mu _3$=0).

Figure 4: Ratio of occurrence probability of freak wave predicted by Eq.(12) to the Rayleigh theory, as a function of kurtosis $\mu _4$ with number of waves $N$ with different number of waves $N=250\sim 5000$ ($\mu _3$=0).

Figure 3 shows the comparison of the occurrence probability of the freak wave as a function of the number of waves $N$ between the Rayleigh theory and Eq.(12) changing $\mu_4=3.0\sim3.5$ with $\mu _3$=0. For the case of $N$=100, the occurrence probability of the freak wave is 3.3% by the Rayleigh distribution, while 13% by Eq.(12) at $\mu _4$=3.5, and for the case of $N$=1055, the occurrence probability of the freak wave is 29.4% by the Rayleigh distribution, while 75.4% by Eq.(12) at $\mu _4$=3.5. 1,000 number of wave $N$=1000 corresponds to a 2 hours and 45 minutes storm for the case of $T_{1/3}=10$s, which is not an unnatural situation in the real sea. If it defines the threshold value of the occurrence probability of a freak wave as 50%, the expected number of waves that include one freak wave as a maximum wave is 2,000 waves predicted by the Rayleigh distribution, and becomes 500 waves predicted by Eq.(12) with $\mu_4=3.5$ and $\mu _3=0$. The freak wave in a strong nonlinear field can occur four times more frequently than in the linear random wave theory.

Figure 4 shows the ratio of freak wave occurrence probability predicted by Eq.(12): $P_{ER}$ to the Rayleigh theory: $P_{R}$, as a function of kurtosis $\mu _4$. For the case of a small number of waves $N=250$, the ratio $P_{ER}/P_{R}$ linearly depends on $\mu_{4}$. The differences between $P_{ER}$ and $P_R$ decreases as the number of waves increases. For the case of $N=3000$ or 5000, because the most of maximum wave exceeds 2 times $H_{1/3}$ in both theory.

Thus, the occurrence probability of the freak wave depends on the value of $\mu _4$, if wave height distribution follow the Edgeworth-Rayleigh distribution, Eq.(4). The differences between the linear and nonlinear theory are significant for a small number of waves and becomes equivalent to large number of waves as $N>10000$.