Wave and crest height distribution

First, it is assumed unidirectional, stationarity and ergodicity for wave field. The most widely used statistical distribution describing wave height and crest/trough amplitude is the Weibull distribution. As a starting point of this study, it is assumed that the PDF of wave/crest height can be described by the Weibull distribution (KendallStuart) given by

$$p_{w}(x)dx = \alpha\beta x^{\alpha-1}\exp(-\beta x^\alpha)dx$$ (1)

where $\alpha$ and $\beta$ are coefficients of the Weibull distribution, and $x$ is a random variable such as crest or wave height normalized, for example, by the root-mean-square of the free surface fluctuations, $\eta_{rms}$. For the cases of $\alpha$=2 with $\beta$=1/8 or $\beta$=1/2, Eq.(1) is equivalent to the Rayleigh distribution for wave height or crest/trough amplitude distributions, respectively. In this study, the one parameter Weibull distribution is adopted to simplify the problem. Therefore, the value of $\beta$ is fixed as 1/8 for wave height and 1/2 for crest height distributions, and $\alpha$ is assumed larger than 1. If vertical wave symmetry is assumed, then the wave height $H$ is defined by twice the wave amplitude $A$. For the Rayleigh distribution of wave height, Eq.(1) gives

$$\frac{ \bar{H} }{ \eta_{rms} } = \int_{0}^{\infty} x p_w(x)dx = \sqrt{2\pi}$$ (2)

$$\frac{ H_{rms}^2 }{ \eta_{rms}^2 } = \int_{0}^{\infty} x^2 p_w(x)dx = 8$$ (3)

and for the Rayleigh distribution of wave amplitude, Eq.(1) gives

$$\frac{ \bar{A} }{ \eta_{rms} } = \int_{0}^{\infty} x p_w(x)dx = \sqrt{\frac{\pi}{2}}$$ (4)

$$\frac{ A_{rms}^2 }{ \eta_{rms}^2 } = \int_{0}^{\infty} x^2 p_w(x)dx = 2$$ (5)

where the over-bar indicates the mean value.

For the deck overtopping problem, the wave crest amplitude is the dominant variable compared with either the wave trough amplitude or wave height. Moreover, the wave nonlinearities enhance vertical asymmetry of waves, and, as a result, the wave crest amplitude is generally larger than its trough amplitude. Therefore, only Eq.(1) for the PDF of wave crest amplitude is used hereafter. This implies that for fixed $\beta$=1/2, $\alpha$ is the only empirical parameter controlling profile of the distribution. The mean and rms value of Eq.(1) with $\beta=1/2$ are given by

$$\frac{ \bar{A} }{ \eta_{rms} } = \frac{ 2^{ \frac{1}{\alpha} } }{ \alpha } \,\Gamma\left( \frac{ 1 }{ \alpha },0 \right)$$ (6)

$$\frac{ A_{rms}^2 }{ \eta_{rms}^2 } = \frac{ 2^{ 1 + \frac{2}{\alpha} } }{ \alpha } \,\Gamma\left( \frac{ 2 }{ \alpha },0 \right)$$ (7)

where $\Gamma$ is the incomplete gamma function defined by

$$\Gamma\left( a,z \right) = \int_{z}^{\infty} t^{a-1} e^{-t} dt$$ (8)

Eq.(6) and (7) with $\alpha$=2 are equivalent to the Rayleigh distribution defined by Eq.(4) and (5).

If the deck is thin compared to the incident wave amplitude, then the influence of the deck on the incident wave profile is negligible. In this case, the PDF of the wave amplitude distribution on the deck can be described by a truncated form of Eq.(1):

$$p_w(A')dA' = \left\{ \begin{array}{l@{\ \ \ }l} \alpha\... ...]dA' & A' \ge z_c' \\ 0 & A' < z_c' \\ \end{array} \right.$$ (9)

where the $'$ indicates a normalized value, and $A'$ and $z'_c$ are the wave amplitude and deck level normalized by the rms value of the surface elevation $\eta_{rms}$ in the absence, or well seaward, of the deck. Eq.(9) gives the exceedance probability of the wave amplitudes on the deck:

$$P_w(A') = \left\{ \begin{array}{l@{\ \ \ }l} \exp\left[... ...ght] & A' \ge z_c' \\ 1 & A' < z_c' \\ \end{array} \right.$$ (10)

Obviously, the equivalence assumption of $\alpha$ between Eq.(1) and Eq.(9) depends on the thickness of the deck in comparison with the incident wave height. The validity of equivalency of $\alpha$ will be checked later through comparisons with laboratory observations.

Figure 1: Illustration of geometrical location of deck and incident wave.

=10cm