# Extreme Wave Loading on a Vertical Circular Cylinder
## 3 Higher-Order Wave Components Driving Nonlinear Loading on a Structure  
### Nonlinear Wave Evolutions  
Figures 6 and 7 show time histories (left) and the corresponding spectra (right) of the free surface elevation measured at various spatial locations in the absence of the model cylinder to investigate nonlinear wave evolutions. Typically, the peak wave period \(T_{p}=1\) s and the peak wave amplitude \(A_{p}=0.07\) m in Figure 6. And these are 2 s and 0.08 m, respectively, for results in Figure 7. According to Table 1, the former corresponds to the case with stronger nonlinear (i.e., larger wave steepness) but smaller shallow-water effects (i.e., the ratio of water depth and the wavelength is smaller), while the latter is the opposite.  
It can be seen from measurements at the location of WG1 (\(x=2\) m) that a transient wave group is generated due to frequency dispersion, as expected. The wave group becomes more compact at the location of WG2 (\(x=6.7\) m), and at the focus point (WG3; \(x=9.3\) m), all wave components come in phase, producing a large energetic event in the form of NewWave. The crests/troughs preceding and succeeding the peak are symmetric, suggesting that linear components dominate even for the wave of large steepness up to 0.31. This is confirmed by the amplitude spectra shown on the right of Figure 6.  
In contrast, the shape, and thus the kinematic of the wave group, shown in Figure 7, differ substantially from the predictions of linear wave theory. Even at the location of WG1, a significant amount of energy is transferred to the higher harmonic wave components at frequencies \(2f_{p}\) and \(3f_{p}\). They are about 40% and 10% of the linear wave components, respectively. Recall that the wave steepness for this case is 0.144, and the non-dimensional water depth (\(h/\lambda_{p}\)) is 0.1. Thus, it is reasonable to infer that for this case, the shallow-water effect, i.e., wave-bottom interaction, plays a more important role in facilitating this significant energy transfer to high-wavenumber components and occurs shortly after releasing/propagating from the wave paddle.  
Figure 5: Time histories of the free surface elevation and the wave impact force on the vertical truncated cylinder. From top to bottom are the free surface elevation measured at \(x=2\) m, 8.9 m, and the horizontal and vertical forces. \(T_{p}=2\) s, \(A_{p}=0.08\) m, and the air gap \(h_{\rm ag}=0\) m.  
Figure 6: Time histories of the free surface elevation at various locations (**left**) and the corresponding wave spectra (**right**). Here, \(T_{p}\) = 1 s, \(A_{p}\) = 0.07 m. From top to bottom are results at WG1, WG2 and WG3, respectively.  
Figure 7: Time histories of the free surface elevation at various locations (**left**) and the corresponding wave spectra (**right**). Here, \(T_{p}\) = 2 s, \(A_{p}\) = 0.08 m. From top to bottom are results at WG1, WG2 and WG3, respectively.  
In general, there is significant energy transfer among wave components, and the wave steepness is not the best measure of wave nonlinearity for the cases considered, as expected. This will be further discussed later.