Approximation of maximum value of overtopping rate

To derive the overtopping rate, there are two way. First approach is to formulate the overtopping rate $q$ by the overtopping volume $Q$ and wave period $T$. Another approach is to formulate the overtopping rate $q$ by and wave amplitude $A$. First approach is reasonable to formulate the mean wave overtopping rate $\bar{q}$ and second one is good to formulate the maximum wave overtopping rate $q_{max}$. In this section, we try to formulate the maximum overtopping rate $q_{max}$ by using and $A$.

It is assumed that the profile of the individual wave crest is assumed to be a sinusoidal wave locally as same as previous section. Assuming a linear sinusoidal wave for velocity field, the maximum overtopping rate $q_{max}$ on the deck is occurred at the crest and it can be described by

$$q_{max} = \int_{z_c}^{A} u(z) dz$$ (31)

Expanding $\mbox{\rm e}^{kA}$ up to $O(ka)$ in Eq.(21) and substituting Eq.(21) into (31) gives the maximum overtopping rate $q_{max}$ as a function of wave crest amplitude $A$.

$$q_{max} = (A-z_c)\omega A\left[ 1+ \frac{k}{2}(A+z_c) \right]$$ (32)

Eq.(32) is cubic function for $A$. To make simplify the problem, assuming wave steepness $kA$ and characteristic deck steepness $kz_c$ is enough smaller than 1, Eq.(32) can be reduced into quadric function for $A$.

$$q_{max} = (A-z_c)\omega A$$ (33)

This assumption is equivalent to the assumption of the vertically uniform horizontal velocity assumption. The estimated error of this assumption for linear regular waves with out deck case ($z_c$=0) is 5% and 15% for $ka$=0.1 and 0.3, respectively. Therefore, this assumption is not cruel for linear regular waves for small wave steepness. This will be verified by the experimental data later. The maximum overtopping rate $q_{max}$ by Eq.(33) is independent of wave steepness, because $kA$ term was neglected through the formulation. Normalize $q_{max}$ by $g$, $k$ and $\eta_{rms}$, Eq.(33) becomes

$$q_{max}' = \frac{q_{max}}{\eta_{rms}^2\sqrt{gk}} = (A'-z_c')A'$$ (34)

Figure 18: Model/data comparison of relationship between normalized wave crest amplitude $A'$ and overtopping rate with experimental results (solid circle), theory 1; Eq.(32) (dash) and theory 2; Eq.(33) (solid line) for (a) Case 1 and (b) Case 2.

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(a) Case 1.

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(b) Case 2.

Before formulate the PDF of the maximum overtopping rate model, it is necessary to check the assumptions used in the derivations again. A local sinusoidal velocity field and small steepness were assumed for relatively large amplitude waves. To check these assumptions, Fig.18 compares the measured relationship between normalized wave crest amplitudes $A'$ and maximum overtopping rate with the approximated solution by Eq.(32) and simplified solution by Eq.(33). There is no significant difference between the approximated solution, Eq.(32), and simplified solution, Eq.(32). The both of the solutions give nice agreement with the measured data. The correlation coefficients between the simplified solution, Eq.(33), and the experimental results are 0.81 for Case 1 and 0.90 for Case 2, respectively. This result is inconsistent with ugly relationship between and $A$ because the correlation coefficients between and $A$ are less than 0.3. The main reason of this agreement is that the measured is estimated by $\eta$ and that gives one point data with vertically uniform profile assumption. Thus, the assumptions of local sinusoidal velocity filed and small steepness are valid for large amplitude waves with small steepness and relatively high deck level condition.

Solving Eq.(34) for $A'$, we have

$$A' = \frac{1}{2}\left(z_c' \pm \sqrt{4q_{max}' + z_c'^2}\right)$$ (35)

Substituting Eq.(35) into Eq.(9) gives the PDF of maximum overtopping rate.

$$p_{\ensuremath{q_{max}}\,}(\ensuremath{q_{max}^\prime}\,, z_c')d\ensuremath{q_{max}}\, = \frac{\alpha\beta}{2^{\alpha-1}\sqrt{4\ensuremath{q_{max}^\prime}... ...( \sqrt{4\ensuremath{q_{max}^\prime}\,+ z_c'^2}+ z_c' \right)^{\alpha-1} \!\!\!$$

$$\hspace{-2.0cm}\times \exp\left\{ - \beta \left\{ \left[ \frac{1}... ...\right]^{\alpha} - z_c'^\alpha \right\} \right\} d\ensuremath{q_{max}^\prime}\,$$ (36)

Integrating Eq.(36) between $[0, \infty)$, we have the exceedance probability of on the deck given by

$$P_{\ensuremath{q_{max}}\,}(\ensuremath{q_{max}^\prime}\,, z... ...' \right) \right]^{\alpha} - z_c'^\alpha \right\} \right\}$$ (37)

Eq.(36) and (37) represent the overtopping rate distribution based on a modified form of the Weibull wave amplitude distribution and is simply referred to as the ``modified Weibull overtopping rate distribution" hereafter for clarity. For the special case of $\alpha$=2 and $\beta$=1/2 which is the Rayleigh condition for the wave amplitude distribution, Eq.(36) and (37) are simplified as

$$p^R_{\ensuremath{q_{max}}\,}(\ensuremath{q_{max}^\prime}\,, z_c')d\ensuremath{q_{max}^\prime}\, = \frac{1}{2} \left( \frac{ z_c' }{ \sqrt{4\ensuremath{q_{max}^\prime}\,+ z_c'^2}} + 1 \right) \!\!\!$$

$$\hspace{-2.0cm}\times \exp\left\{ -\frac{1}{4} \left[ z_c'\left( ... ... 2\ensuremath{q_{max}^\prime}\, \right] \right\} d\ensuremath{q_{max}^\prime}\,$$ (38)

$$P^R_{\ensuremath{q_{max}}\,}(\ensuremath{q_{max}^\prime}\,,... ...' \right) + 2\ensuremath{q_{max}^\prime}\, \right] \right\}$$ (39)

Following the same manner, Eq.(38) and (39) represent the $q_{max}$ distribution based on a modified form of the Rayleigh wave amplitude distribution and is simply referred to as the ``modified Rayleigh overtopping rate distribution" hereafter for clarity. loped a theoretical framework for the

Figure 19: Dependence of the exceedance probability of overtopping rate on $\alpha$ with $z_c'$=1.0 and $\kappa$=0.1.

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Figure 20: Dependence of the exceedance probability of overtopping rate on deck level $z_c'$ with $\alpha$=2.0 $\kappa$=0.1.

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Figure 21: Model/data comparison of PDF of wave overtopping rate for (a) Case 1 and (b) Case 2 with experimental data (solid), modified Rayleigh overtopping distribution (dash), and modified Weibull overtopping distribution (chain).

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(a) Case 1.

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(b) Case 2.

Figure 22: Model/data comparison of exceedance probability of wave overtopping rate for (a) Case 1 and (b) Case 2 with experimental data (solid), modified Rayleigh overtopping distribution (dash), and modified Weibull overtopping distribution (chain).

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(a) Case 1.

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(b) Case 2.

Fig.19 shows the sensitivity of the exceedance probability of the modified Weibull overtopping rate distribution on $\alpha$ for the case of $z_c'$=1.0 and $\kappa$=0.1. With a change in $\alpha$ from $\alpha$=1.5 to $\alpha$=2, the exceedance the derivations. probability of decreases one order of magnitude at . The decrease is slower than overtopping volume $Q$ or . This is because the function in the exponential in Eq.(37) is proportional to , although the exceedance probability of $Q$ is proportional to . Thus, decreases slower than $Q$ and has different power law in comparison with $Q$ and . Fig.20 shows the sensitivity of the exceedance probability of to the deck level $z_c$ for the case of $\alpha$=2.0. With a change of $z_c'$ from $z_c'$=0 to $z_c'$=2, the exceedance probability seems almost constant. The deck level doesn't effect significantly on the shape of the exceedance probability of than the value of $\alpha$

Fig.21 compares the PDF of the normalized maximum wave overtopping rate for the experimental results with the modified Rayleigh overtopping rate distribution, Eq.(38), and the modified Weibull overtopping rate distribution, Eq.(36). The modified Weibull overtopping rate distribution shows qualitatively better agreement with the experimental data than the modified Rayleigh overtopping rate distribution, particularly for larger values of . Fig.22 compares the exceedance probability of maximum wave overtopping rate for the experimental results with the modified Rayleigh overtopping rate distribution, Eq.(39), and the modified Weibull overtopping rate distribution, Eq.(37). The modified Weibull overtopping rate distribution shows fair agreement with the experimental data for whole range of in Case 1, However, both of the modified Weibull overtopping rate distribution and the modified Rayleigh overtopping rate distribution are underestimated for Case 2. This is because both of the modified Weibull exceedance probability for $Q$ and are underestimated in comparison with the experimental data. The quantitative comparison with the theories and experimental data will be required for large number of waves with different conditions. Nevertheless, the qualitative trends of can be estimated by Eq.(36) and (37).