Approximation of overtopping volume and its normalization

The overtopping volume, $Q$, can be defined as the volume of water column above the deck level, shown schematically in Fig.1. The overtopping volume $Q$ only depends on spatial surface profile, if the deck is thin compared to the incident wave amplitude. Therefore, if the spatial wave profile is known, it is possible to calculate the overtopping volume $Q$ accurately. However, there is some difficulty to estimate the spatial profile of random waves. Thus, the overtopping volume $Q$ is calculated by the following assumptions and procedure. First, the profile of the individual wave crest is assumed to be a sinusoidal wave locally. Then, if the deck level $z=z_c$ is given and the crest amplitude $A$ is known, the volume of water column above deck level, $Q$, is simply calculated by

$$Q(A) = \frac{2A}{k}\sqrt{1-\left(\frac{z_c}{A}\right)^2}$$ (11)

where $k$ is the local wave number and $A$ is local amplitude of the wave. Eq.(11) can be roughly simplified for $z_c/A \ll 1$ as

$$Q(A) \simeq \left\{ \begin{array}{l@{\ :\ }l} \display... ..._c)}{k} } & A \ge z_c \\ \ 0 & A < z_c \end{array} \right.$$ (12)

Fig.2 shows an example of Eq.(11) and (12) for the cases of $z_c/A_{max}$ = 0.1 and 0.5. The transformation of the wave amplitude $A$ into overtopping volume $Q$ by the approximate Eq.(12) becomes worse as the height of the deck level $z_c$ increases. The overtopping volume $Q$ decreases as $z_c$ increases as expected. Obviously, the assumptions of the local sinusoidal wave crest profile and local wave number are only valid for narrow banded spectra. The validity of the assumptions will be discussed later in the context of the experimental results.

The overtopping volume $Q$ can be normalized by the rms value of surface elevation $\eta_{rms}$ and wavenumber $k$ as

$$Q' = \frac{kQ}{\eta_{rms}} = \frac{2(A-z_c)}{\eta_{rms}}$$ (13)

From Eq.(13), the inverse relationship between $Q$ and $A$ is given by

$$A' = \frac{1}{2}Q' + z_c'$$ (14)

The upper script $'$ for $A$, $Q$ and $z_c$ will be dropped for simplification hereafter. All equations and variables are normalized by $\eta_{rms}$ following section.

Figure 2: Relationship between wave crest amplitude $A$ and overtopping volume $Q$ for the cases of $z_c=0.1 A_{max}$ and $0.5 A_{max}$.

=12cm