Freeboard dependent reflection and transmission

In case an obstacle becomes flooded, its reflection and transmission properties change as a function of the relative freeboard, defined as the ratio of the difference in dam height and the water level by the (incident) significant wave height, i.e. $F/H_s$. For transmission this dependence can be schematized as described in Section 2.5.1, where transmission increases with increasing freeboard. Another method is to scale the fixed reflection and fixed transmission coefficients as a function of the relative freeboard. This dependence is expressed using a tanh function in which a parameter $\gamma$ is employed to define the range (in terms of the relative freeboard) over which the fixed reflection and/or transmission coefficients varies between their minimum and maximum values. This dependency is given by

  $$R = \left [ 1 +\tanh \left ( \frac{2}{\gamma_{\tiny\mbox{R}}} \frac{F}{H_s} \right ) \right ] \frac{R_0}{2}$$ (2.176)

in case of reflection with a fixed value $R_0$, and

  $$T = \left [ 1 +\tanh \left ( -\frac{2}{\gamma_{\tiny\mbox{T}}} \frac{F}{H_s} \right ) \right ] \frac{T_0}{2}$$ (2.177)

in case of transmission with a fixed value $T_0$.


This parameterization has two gamma parameters, one for the reflection and one for the transmission. The parameterization is shown schematically in Figure 2.4. In this figure it is assumed that the fixed reflection coefficient $R_0$ has a value of 0.6, and that the maximum or fixed transmission coefficient is equal to $T_0$ = 0.9. In this example the significant wave height has a value of 2 m. The structure of above equations is such that the parameters $\gamma_{\tiny\mbox{R}}$ and $\gamma_{\tiny\mbox{T}}$ coincide with the positions where the reflection and transmission coefficient have their minimum or maximum value.

Figure 2.4: Parameterization of the reflection coefficient $R$ and transmission coefficient $T$ as a function of the relative freeboard $F/H_s$.

Quay option
The above parameterization works for an obstacle in water of a certain depth, irrespective from which side the waves are propagating. For handling this method for a quay, it is assumed that the water depth on each side of the obstacle line are different, where the quay area is less deep than the deeper area in front of the quay. This requires special attention of the model setup to ensure that the depths are different on the grid points at each side of the obstacle representing the edge of a quay. When waves propagate from the deeper side to the shallower (quay) side of the obstacle, the same method is used as above. When waves travel from the shallower part to the deeper part, it is assumed that the reflection coefficient $R=0$ and the transmission coefficient $T=1$.