Relationship between $H_{1/3}$ and $\eta _{rms}$

The exceedance probability of wave heights is necessary to estimate the maximum wave height. Therefore, Eq.(6) will be expaned to the maximum wave height distribution. However, wave height $H$ in Eq.(6) is normalized by $\eta _{rms}$. Thus, the relationship between $H_{1/3}$ and $\eta _{rms}$ have to obtain to predict the occurrence probability of the freak wave, which is defined $H_{max}/H_{1/3}\ge2$ in this study.

Following Goda (2000), $1/N$ wave height is given by

$$H_{1/N} = \frac{ \displaystyle \int_{H_N}^{\infty}H\,p(H)d... ...^{\infty}p(H)dH } = H_N + \frac{N}{a}\mbox{\rm Erf}(a\,H_N)$$ (7)

where $a$ is constant to normalize the wave height and Erf is the error function as Erf $(x)=\int_{x}^\infty \ensuremath{\mbox{\rm e}}^{-t^2}dt$. In this case, $H$ is normalized by $\eta _{rms}$, and $a=1/(2\sqrt{2})$. If the wave height distribution is described by Eq.(4), weak nonlinearity influences on $1/N$ wave height. The formulation of $1/N$ wave height is also formulated by Eq.(4).

$$H_{1/N} = N\biggr[ (\mu_4-3)E_{41}(H_N) + \mu_3^2 E_{42}(H_N) + (\mu_4-3)E_{61}(H_N)$$

$$\hspace{2.3cm} + \mu_3^2\mu_4 E_{62}(H_N) + \mu_3^4 E_{63}(H_N)\biggl] \times \exp\left(\frac{H_N^2}{ a^2 }\right)$$

$$\hspace{1.1cm} + \frac{N}{a}\biggl[ 1 + (\mu_4-3)F_{41}(H_N) + \mu_3^2 F_{42}(H_N) +(\mu_4-3)F_{61}(H_N)$$

$$\hspace{2.4cm} + \mu_3^2\mu_4 F_{62}(H_N) + \mu_3^4 F_{63}(H_N)\biggr] \times \mbox{\rm Erf}\left(\frac{H_N}{a}\right)$$ (8)

where $E_{ij}$ and $F_{ij}$ are polynomials for $H_N$ (see Appendix).

Figure 1: Dependence of skewness $\mu _3$ and kurtosis $\mu _4$ on $H_{1/3}/\eta _{rms}$ by Eq.(8).

Figure 2: Comparison of $H_{1/3}/\eta _{rms}$ between theoretical result; Eq.(8) with $\mu _3=0$ and experimental data (Mori, 2003) (solid line; Eq.(8), dash; Rayleigh theory, $\circ$; experimental data, chain; empirical curve by Eq.(9)).

Eq.(8) is a transcendental equation and is difficult to solve analytically. The second term on the right hand side in Eq.(8) is exceedance probability and is therefore monotonically decreasing function. Thus, the numerical solution of Eq.(8) can be calculated. The relationship between and $\ensuremath{\eta_{rms}}\,$ was calculated by Newton-Rapson method. Figure 1 shows the numerical solution of $H_{1/3}/\eta _{rms}$ by Eq.(8) as a function of $\mu _3$ and $\mu _4$. The profile of $H_{1/3}/\eta _{rms}$ is symmetric to $\mu _3$ and $H_{1/3}/\eta _{rms}$ increases rapidly for $\vert\mu_3\vert\gg 0.5$. The reason why $H_{1/3}/\eta _{rms}$ is symmetric to $\mu _3$ is that the wave height $H$ in Eq.(4) is assumed two times of its wave amplitude $A$. Therefore, there is no difference between positive and negative value of $\mu _3$ in Eq.(8) Dependence of $\mu _4$ on $H_{1/3}/\eta _{rms}$ is more significant than $\mu _3$ because $\mu _3$ is smaller than 0.5 for deep-water waves. $H_{1/3}/\eta _{rms}$ monotonically increases to $\mu _4$. Figure 2 shows the comparison of $H_{1/3}/\eta _{rms}$ with the experimental data by Mori (2003) and the numerical solution with $\mu _3=0$. The experiment was conducted under a deep-water condition with JONSWAP spectrum in 2D wave channel (see Mori, 2003 in detail). The dependence of $\mu _4$ on $H_{1/3}/\eta _{rms}$ has similar tendency both of the numerical solution of Eq.(8) and the experimental data, qualitatively. However, there is quantitative difference between them. The chained line in the figure indicates the empirical curve calculated by the least-squares curve fitting for the quadric function:

$$\frac{ H_{1/3} }{ \eta_{rms} } = -0.109 \mu_4^2 + 1.023 \mu_4 + 1.727$$ (9)

For the case of $\mu_4=3$, a linear random wave, $H_{1/3}/\eta _{rms}$ by Eq.(8) is equal to 4.004 and is equivalent to the Rayleigh theory $H_{1/3}/\eta _{rms}$ is typically about 3.8 in the field under the deep-water condition (Goda, 2000). The experimental data gives $H_{1/3}/\eta_{rms}=3.82$ at $\mu_4=3$ by Eq.(9). Therefore, the experimental data is consistent with field observation. Remarkably, / is not constant and it depends on $\mu _4$. It fluctuates about 10% with the value of kurtosis $\mu _4$. Longuet-Higgins (1980) also reported the wave steepness dependency on $H_{1/3}/\eta _{rms}$ considering regular wave nonlinearity to random wave field. However, we conclude that the major nonlinear contributions to $H_{1/3}/\eta _{rms}$ is kurtosis than the wave steepness.