Surface Elevations and Moments

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 $t$ and $\zeta(t)$ be its Hilbert transform. Assuming both $\eta$ and $\zeta$ are real zero-mean function with variance $\sigma$, we have

$$Z(t) = \eta(t) + i\zeta(t) = A(t)e^{i\phi(t)},$$ (1)

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

$$\phi(t) = \tan^{-1}\left(\frac{\zeta(t)}{\eta(t)}\right).$$ (3)

The characteristic function $\psi_z(t)$ can be described as

$$\psi_z(u,v) = E\left[e^{i(u\eta+v\zeta)}\right],$$ (4)

where $E[\cdot]$ means the expected value operator, and $u$ and $v$ are dummy variables in $(-\infty,\infty)$.

The cumulants $K_{nm}$ is formulated by Eq.(4) as

$$K_{nm} = (-i)^{n+m}\frac{\partial^{n+m}}{\partial^n \partial^m} \ln\psi_z(u,v)\vert _{u=v=0}.$$ (5)

The following relationship is established between the characteristic function $\psi_z$ and cumulant $K_{nm}$. between them, so we have

$$\psi_z(u,v) = \exp\left[ \sum_{n}\sum_{m} \frac{K_{nm}}{n!m!} (iu)^n(iv)^m \right].$$ (6)

Longuet-Higgins (1963) formulated the detailed relationship between Eq.(5) and Eq.(6). Several values of cumulants $K_{nm}$ are derived from Eq.(5) on the basic assumptions, stationariness and orthogonal property of $\eta$ and $\zeta$.

$$K_{10} = E[\eta ] = \bar{\eta } = 0$$ (7)

$$K_{01} = E[\zeta] = \bar{\zeta} = 0$$ (8)

$$K_{20} = E[\eta^2] = \eta_{rms}^2$$ (9)

$$K_{11} = E[\eta\zeta] = 0$$ (10)

$$K_{nm} = E[\eta^n\zeta^m]$$

$$\hspace{0.5cm} \mbox{\rm for } n+m \le 3$$ (11)

$$K_{nm} = E[\eta^n\zeta^m] + (n-1)(m-1)(-1)^{n/2}\sigma^{n+m}$$

$$\hspace{0.5cm} \mbox{\rm for } n+m = 4$$ (12)

Similar results can be extended for the higher order cumulants, but the analysis is not carried out here beyond the fourth order.

Mori and Yasuda (1994) pointed out that the fourth order moment of water surface elevation is a dominative parameter for wave height statistics. In present case, therefore, we should take into consideration up to the fourth order cumulants. The joint probability density function of $\eta$ and $\zeta$ is formulated as

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

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

$$\hspace{0.75cm} + \frac{1}{4!}\sum_{n=0}^{4} \frac{4!}{(4-n)!n!}\kappa_{(4-n)n} H_{4-n}(\eta')H_{n}(\zeta') \biggr]$$ (13)

where

$$\eta' = \frac{\eta }{\eta_{rms}}$$ (14)

$$\zeta' = \frac{\zeta}{\eta_{rms}}$$ (15)

$$\kappa_{nm} = \frac{K_{nm}}{\eta_{rms}^{n+m}}$$ (16)

and $H_n$ is the $n$th order Hermite polynomial expressed as

$$H_n(x) = (-1)^n \exp\left(-\frac{1}{2}x^2\right) \frac{d^n}{dx^n} \exp\left(\frac{1}{2}x^2\right).$$ (17)

It is noticed here that $\kappa_{00}$=$\kappa_{10}$=$\kappa_{01}$=$\kappa_{11}$=0, $\kappa_{20}$ = $\kappa_{02}$ =1 and $\kappa_{21}$=$\kappa_{12}$=$\kappa_{13}$=$\kappa_{31}$=0, which have already been used in Eq.(7). $\kappa_{30}$ is equal to the skewness $\mu_{3}$ of $\eta$ and $\kappa_{40}$ is equal to kurtosis $\mu_{4} - 3$. There is still unknown cumulants such as $\kappa_{03}$, $\kappa_{04}$, $\kappa_{22}$.

Tayfun (1993) formulated $\kappa_{22}$ for weakly nonlinear waves wtih narrow banded spectra using the second order kernel function in deep water as,

$$\kappa_{22} = 2\alpha^2(1-\gamma_1),$$ (18)

$$\gamma_n = \frac{1}{ \bar{\omega} m_0^2} \int\!\!\!\int_{0}^\infty \left( \f... ... \right)^{n} \vert\omega-\omega'\vert S_1(\omega)S_1(\omega')d\omega d\omega',$$ (19)

where $\bar{\omega}$ is the mean angular frequency of spectrum and $m_0$ is sum of spectrum of the first order component.

By way of Schwarz's inequality an equality, $0 \le \vert\gamma_1 - \gamma_0\vert \le \nu^2\sqrt{2}$ is derived. This gives

$$\gamma_1 = \gamma_0 + O(\nu^2)$$ (20)

and

$$\kappa_{22} = 2\alpha^2(1-\gamma_0) + O(\nu^2)$$ (21)

Since, the maximum value of $\kappa_{22}$ is $O(2\mu_3^2)$, it is assumed there that $\kappa_{22}$=0 and also $\kappa_{03}$ = $\kappa_{30}$ and $\kappa_{04}$ = $\kappa_{40}$ under the condition of weak nonlinearlity.