Question 7.11: A new sea wall is to be designed to limit wave overtopping u......
A new sea wall is to be designed to limit wave overtopping under severe conditions to below a critical value Q_{c}. Write down a failure function and make appropriate simplifying assumptions to derive an approximate failure function that is dependent on water level and wave height only.
We use the formula due to Owen (1980) for overtopping (see Section 9.4). Substituting Equations 9.6 and 9.7 into Equation 9.5
\begin{aligned} & R_{u 2 \%} / H_{m o} = 1.75 \gamma_b \gamma_f \gamma_\beta \xi_{m-1,0} \quad \text { with a maximum of } (9.6) \\ & R_{u 2 \%} / H_{m o} = 1.07 \gamma_f \gamma_\beta\left[4.0-1.5 /\left(\gamma_b \xi_{m-1,0}\right)\right]\end{aligned} (9.7)
R/H_{i} = \left(\pi /2 \alpha _{Wall} \right) ^{1/2} for \pi /4\lt \alpha \lt \pi /2. (9.5)
gives
G=Q_c-Q=Q_c-g T_m H_s A e^{-B\left\lgroup\frac{C L-W L}{r T_m \sqrt{g H_s}}\right\rgroup } (7.52)
where A and B are constants depending on sea wall geometry, C L is the crest level, WL is the still water level and r is wall roughness.
One current approach is to take T_{z} as being directly related to H_{s} through the assumption that storm waves have a similar (i.e., constant) wave steepness. That is,
T_z=\left\lgroup\frac{2 \pi H_s}{g S}\right\rgroup ^{1 / 2} (7.53)
where S denotes wave steepness. Assuming a JONSWAP wave spectrum we have
T_{\text {m }}=1.073 T_{z} (see Section 3.4.3), so
T_m=1.073\left\lgroup{\frac{2 \pi H_s}{gS} }\right\rgroup ^{1 / 2} \equiv a H_s^{1 / 2} (7.54)
The distribution of T_m is thus completely determined by the distribution of H_s, and we may substitute Equations \mathrm{A}, \mathrm{B} and \mathrm{C} to eliminate one variable in the reliability function. There is likely to be dependence between H_s and water level because of wave breaking due to depth limitation. However, there is unlikely to be much, if any, physical cause for T_m and WL to have strong dependence. Due to wave generation and propagation processes we might expect some dependence of both H_s and T_m on wave direction. A common way of accounting for wave direction is to undertake a series of ‘conditional’ calculations, one for each direction sector of interest. The results for each sector can be considered in turn and the worst case(s) used for design purposes. As construction of a new sea defence is being considered, we will take C L, r, A and B as being known values, although they could also be taken as random variables with known probability distributions. The failure function thus becomes
which is a function of two dependent variables H_{s} and WL. In nearshore locations there can be a degree of correlation between wave heights and water level due to the water depth modulation of waves by the tides. The level of correlation would have to be determined from observations.