Transmission

There are several mechanisms for transmission of waves. In SWAN, the user may compute transmission of waves passing over a dam with a closed surface or may choose a constant transmission coefficient.


If the crest of the breakwater is at a level where (at least part of the) waves can pass over, the transmission coefficient $K_{t}$ (defined as the ratio of the (significant) wave height at the downwave side of the dam over the (significant) wave height at the upwave side) is a function of wave height and the difference in crest level and water level. It must be noted that the transmission coefficient can never be smaller than 0 or larger than 1. In SWAN, two expressions can be employed. The first is taken from Goda et al. (1967):

  $$K_t = \left\{ \begin{array}{ll} 1 \, , & \frac{F}{H_i} < -\bet... ...lpha - \beta \\ 0 \, , & \frac{F}{H_i} > \alpha - \beta \end{array} \right.$$ (2.171)

where $F=h-d$ is the freeboard of the dam and where $H_i$ is the incident (significant) wave height at the upwave side of the obstacle (dam), $h$ is the crest level of the dam above the reference level (same as reference level of the bottom), $d$ the mean water level relative to the reference level, and the coefficients $\alpha$, $\beta$ depend on the shape of the dam (Seelig, 1979) as given in Table 2.1. It should be noted that this formula is only valid for slopes more gentle than 1:0.7 (1.4:1 or 55 degrees).

Table 2.1: Parameters for transmission according to Goda et al. (1967).

case	$\alpha$	$\beta$
vertical thin wall	1.8	0.1
caisson	2.2	0.4
dam with slope 1:3/2	2.6	0.15

Expression (2.172) is based on experiments in a wave flume, so strictly speaking it is only valid for normal incidence waves. Since there are no data available on oblique waves, it is assumed that the transmission coefficient does not depend on direction. Furthermore, it is assumed that the frequencies remain unchanged over an obstacle (only the energy scale of the spectrum is affected and not the spectral shape).


For an impermeable rough low-crested dam, the following expression of d'Angremond et al. (1996) is chosen:

  $$K_t = -0.4 \frac{F}{H_i} + 0.64 (\frac{B_k}{H_i})^{-0.31} (1 - e^{-0.5{\xi}_p})$$ (2.172)

with $B_k$ the crest width and ${\xi}_p \equiv \tan \alpha/\sqrt{H_i/L_{0p}}$ the breaker parameter. For this, the slope of the breakwater $\alpha$ must be given and $L_{0p} = g T_p^2/ 2\pi$ is the deep water wave length. The restriction to eq. (2.173) is as follows

  $$0.075 \leq K_t \leq 0.9$$ (2.173)

In most cases, the crest width is such that $B_k < 10 H_i$. However, if this is not the case, the following expression should be used instead of (2.173):

  $$K_t = -0.35 \frac{F}{H_i} + 0.51 (\frac{B_k}{H_i})^{-0.65} (1 - e^{-0.41{\xi}_p})$$ (2.174)

with the restriction:

  $$0.05 \leq K_t \leq -0.006\frac{B_k}{H_i} + 0.93$$ (2.175)

The formula's (2.173) and (2.175) give a discontinuity at $B_k = 10 H_i$. Following Van der Meer et al. (2005), for practical application, Eq. (2.173) is applied if $B_k < 8 H_i$, Eq. (2.175) if $B_k > 12 H_i$ and in between ( $8 H_i \leq B_k \leq 12 H_i$), a linear interpolation is carried out.