# Extreme Wave Loading on a Vertical Circular Cylinder
## 3 Higher-Order Wave Components Driving Nonlinear Loading on a Structure
### Harmonic Structure of Nonlinear Waves  
It is clear from Section 3.1 that nonlinear wave-wave and wave-bottom interactions may introduce a significant harmonic generation, redistributing the wave energy from the fundamental frequency into higher harmonics of this linear component. And given that each harmonic for a wave group spreads over a range of frequencies, the four-phase decomposition method discussed above (Equation (4)) is adopted to extract the harmonic structure of the nonlinear wave fields. Typical results are shown in Figure 8; from top to bottom are long-wave difference component, linear, second, third, and fourth sum harmonics, respectively. The fourth sum harmonic is separated from the second-order difference term by digital filtering. The envelopes of the harmonics in time are also calculated and plotted on top of the corresponding wave harmonics, displaying the structure of the wave signal in time. The envelope of the linear wave component is obtained using the formula  
\[\mathrm{E}(\eta^{(1)})=\sqrt{\left(\eta^{(1)}\right)^{2}+\left(\left(\eta^{(1) }\right)^{H}\right)^{2}} \tag{6}\]  
Recall that \((\eta^{(1)})^{H}\) is the Hilbert transform of the linear wave time history. Figure 8: Harmonic structure of the free surface elevation measured at WG3, the focus point. (**Left**): \(T_{p}=1\) s, \(A_{p}=0.07\) m. (**Right**): \(T_{p}=2\) s, \(A_{p}=0.08\) m. From top to bottom are results for 2 nd order difference term, linear wave, 2 nd order sum term, 3 rd harmonic and 4 th harmonic waves, respectively. Black lines: time histories, and red lines are the corresponding envelopes. If the nonlinear process follows a Stokes-type expansion, the envelopes of the \(n\)th harmonics can be approximated by raising the fundamental envelope to the power \(n\), which is then scaled according to the magnitude of the \(n\)th harmonic component [37]. That is,  
\[\mathrm{E}(\eta^{(n)})=s[\mathrm{E}(\eta^{(1)})]^{n} \tag{7}\]  
where \(s\) is the scaling factor obtained by the least-square method to match the size of envelopes to the maximum of the measured \(n\)th-order harmonics. This assumption is further assessed in Figure 8 by plotting the approximated envelopes on top of the wave time series at each other. It can be seen that the extracted harmonics fit the estimated envelopes fairly well, although some discrepancies are found. The mismatch is slightly more obvious in the case shown on the right. The second-order harmonic wave curve shows a small secondary pulse at approximately \(t\) = (34-38) s, which may be attributed to the error waves from the wave paddle. The error wave results from the fact that the linear wave signal was used for driving the paddle, while the waves generated were inherently nonlinear. In addition, the wiggles of the third-order harmonic wave spread slightly outside their approximated envelopes. This is not surprising as the methodology is well known to work less well for the third harmonics [14]. Distinct/extra nonlinear mechanisms, including Morison drag from ringing forces [5; 38] and the secondary load cycle [6], may be responsible for this. It is clear that the Stokes-like approximation and, thus, the applied four-phase decomposition method works well, even for the waves of stronger nonlinear and shallow-water effects. It also can be seen from Figure 8 that contributions from higher-order harmonics (ratio between the linear and higher harmonics) are larger for the case with a smaller (non-dimensional) water depth (right plot), consistent with the observations in Figures 6 and 7. The peaks of each harmonic (non-dimensionalized by dividing by the total peak wave amplitude \(\eta_{t}\)) for the waves studied in this work (see Table 1) are then summarized in Figure 9. As suggested by the analysis in Section 3.1 and in Figure 8, we plot the variation in the peaks against the so-called Ursell number (\(Ur=(A_{p}/h)/(k_{p}h)^{2}\)), which indicates the nonlinearity of long surface water waves. The values of the Ursell number are also listed in Table 1 for reference.