# Extreme Wave Loading on a Vertical Circular Cylinder
## 2 Experimental Modeling and Data Analysis
### Iterative Correction for Generating Controlled and Accurate Focused Wave Group  
A downshift in the actual focus point/time is often observed due to the nonlinear dispersion of the wave groups between the paddle and the structure; see, e.g., [35; 36]. As a consequence, some of the recorded wave groups may be focused better than others (depending on the wave nonlinearity), and there may be a discrepancy between the target and the measured focused wave groups at the desired focus point (measured by WG3 at \(x_{0}\) in this work).  
Figure 3: Time histories for all four full signals that have the same linear envelope but different global phases. Top: measurements at WG1. Bottom: measurements at WG3. Here, \(T_{p}=1\) s and \(A_{p}=0.07\) m.  
This critical challenge of generating controlled and accurate wave groups was addressed through the use of the iterative correction technique by [37]. In this method, the signal driving the wave paddle motion is iteratively corrected by comparing the measured to the desired target amplitude spectra [37],  
\[\begin{split} a_{in}^{n}(f_{i})&=a_{in}^{n-1}(f_{i}) \frac{a_{tgt}(f_{i})}{a_{out}^{n-1}(f_{i})}\\ \varphi_{in}^{n}(f_{i})&=\varphi_{in}^{n-1}(f_{i})+ \left[\varphi_{tgt}(f_{i})-\varphi_{out}^{n-1}(f_{i})\right]\end{split} \tag{5}\]  
where \(a(f_{i})\) and \(\varphi(f_{i})\) are the amplitude and phase of a spectral component at frequency \(f_{i}\), respectively. The superscript \(n\) is the number of iterations, and the subscripts _'in'_ and _'out'_ indicate that these are associated with the input and output signals, respectively. Note that for \(n=1\), \(a_{in}^{1}(f_{i})=a_{tgt}(f_{i})=a_{\text{NW}}(f_{i})\) and \(\varphi_{in}^{1}(f_{i})=\varphi_{tgt}(f_{i})=(\omega_{i}t_{0}-k_{i}x_{0}+ \varphi_{8})\), i.e., the pre-selected linear spectrum (\(a_{tgt}\), \(\varphi_{tgt}\)) is used for driving the paddle signal initially.  
More specially, this work uses the linearized part of the measured total wave signal, and thus, the measured output spectrum (\(a_{out}\), \(\varphi_{out}\)) is obtained by performing an FFT (fast Fourier transform) calculation on the linear component time history, i.e., \(\eta^{(1)}\) in Equation (4) isolated by the four-phase decomposition method [24]. And the output amplitudes and phases (\(a_{out}\), \(\varphi_{out}\)) are calculated based on the free surface elevations measured at WG3 (i.e., \(x=9.3\) m, the predefined focus point).  
The final linearized undisturbed wave shape (dashed red line) is compared to the target (black solid line) in Figure 4. Here, the peak wave period \(T_{p}=1\) s and \(A_{p}=0.07\) m. It can be seen that a satisfactory agreement is achieved, indicating that the applied methodology, i.e., iterative correction scheme in tandem with the four-phase-based decomposition method, works well for generating the desired focused wave in a physical flume with a piston-type wavemaker. The aforementioned issues associated with spatial and temporal downshift of the focal location are overcome.  
Comparisons have also been made for other wave conditions considered in this work (details are to follow later in this section and see Table 1), with comparable agreement, but are not shown here for brevity.  
In addition, the good agreement also suggests that the nonlinear process in water waves of high steepness (\(k_{p}A_{p}\) up to 0.3) that propagate in intermediate, nearly shallow waters (\(h/\lambda_{p}\) down to 0.1 in which \(\lambda_{p}\) is the wavelength) may still be described by a Stokes-type perturbation expansion in both frequency and wave steepness. This assumption is the key for the four-phase decomposition method discussed above to be applicable.