# Extreme Wave Loading on a Vertical Circular Cylinder
## 2 Experimental Modeling and Data Analysis
### Incident Wave Field  
A range of wave conditions with varying wave steepness and spectral peak wave frequencies were tested, as listed in Table 1. Here, \(k_{p}\) and \(\lambda_{p}\) are the wave number and the wavelength corresponding to the spectral peak wave period \(T_{p}\). \(A_{p}\) is the peak wave amplitude, and _Ur_ is the Ursell number (\(=(A_{p}/h)/(k_{p}h)^{2}\)). It can be seen that the ratio of the peak wave amplitude (\(A_{p}\)) to the cylinder radius (\(R\)) was up to 1.2 (i.e., \(A_{p}/R\leq 1.2\)), and the cylinder size (\(k_{p}R\)) ranged from 0.149 to 0.364. These correspond to the field scale conditions in which the dynamic responses of a structure are expected to occur.  
A focused wave group consistent with a JONSWAP spectrum with \(\gamma=3.3\) was used, providing a model for transient events and thus ensuring that the experimental results over the time of interest are not contaminated by reflected waves [33].  
In this context, the NewWave-type focused wave group is applied [26],  
\[\eta_{NW}=\sum a_{NW}(f_{i})\cos[k_{i}(x-x_{0})-\omega_{i}(t-t_{0})+\varphi_{g}] \tag{1}\]  
where \(x_{0}\) and \(t_{0}\) are the focus point and time, respectively. In this work, these are pre-determined as 9.3 m and 32 s, respectively. \(\varphi_{g}\) is the global phase difference, and the amplitude of each wave component is calculated as  
\[a_{NW}(f_{i})=A_{p}\frac{S(\omega_{i})\times\Delta\omega_{i}}{\sum S(\omega_{ i})\times\Delta\omega_{i}} \tag{2}\]  
In which \(\Delta\omega\) is the increment of the angular frequency, and \(S(\omega)\) is the wave energy spectrum (JONSWAP spectrum in this work). Here, \(\omega\) and \(k\) are the wave angular frequency and the wave number, respectively, satisfying the dispersive equation  
\[\omega^{2}=gk\text{tanh}kh \tag{3}\]  
where \(g\) is the acceleration due to gravity. Recall that \(h\) is the water depth and \(A_{p}\) is the peak wave amplitude.  
An example time history of a NewWave-type focused wave group is shown in Figure 2 (top right), where all Fourier wave components are forced to come into phase with each other to produce a very violent wave-structure interaction, see, e.g., in Figure 2 (top left). It is clear from the photo that a mass of water is projected vertically upwards during the violent impact event, forming a thin sheet of water that wrapped around the upstream half of the cylinder. A similar phenomenon has been reported by [5; 14; 34], among others, in their experiments on wave interactions with a vertical bottom-mounted cylinder, and by [22] in their wave-in-deck experiments with focused waves into a solid deck.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \(T_{p}\) (s) & \(h\) (m) & \(k_{p}\) (m\({}^{-1}\)) & \(A_{p}\) (m) & \(R\) (m) & \(k_{p}R\) & \(k_{p}A_{p}\) & \(h/\lambda_{p}\) & _Ur_ \\ \hline \hline \(1\) & & 4.41 & 0.07 & & 0.364 & 0.309 & 0.246 & 0.084 \\ \hline \(1.5\) & & 2.53 & 0.09 & & 0.208 & 0.227 & 0.141 & 0.329 \\ \hline \(1.5\) & & 2.53 & 0.10 & & 0.208 & 0.253 & 0.141 & 0.366 \\ \hline \(2\) & & 1.80 & 0.08 & & 0.149 & 0.144 & 0.100 & 0.575 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Incoming wave fields considered in this work.