# Measurements of High-Froude Number Boat Wakes near a Seawall
## 2 Materials and Methods
### Data Processing
#### 2.3.4 Wake Identification  
An objective method for identifying wakes was developed based on the chirp signal of the wakes as represented in wavelet space [69]. A family of parabolic curves \(K(n,t,f)\) was generated such that time \((t_{Ki})\) coordinate for curve \(n\) at frequency \(f_{i}\) was  
\[t_{Ki}=\frac{n\Delta t-t_{0}}{(f_{1}-f_{0})^{2}}(f_{i}-f_{0})^{2}+t_{0} \tag{6}\]  
These were segments of horizontal parabolas with vertices at \((t_{0},f_{0})\) and terminations at \((n\Delta t,f_{1})\), where \(n\Delta t\) was the wake duration and \(\Delta t\) = 5 s. The latter being equivalent to the resolution of the duration measure and was much less than the typical wake duration. Quadratics were used to approximate the dispersion of wakes emanating from shiplines of varying orientation and distance from the gauge and poorly known propagation paths over varying bathymetry. For each time, these K-curves were convolved with the masked wavelet amplitude \(\|W_{m}h^{\prime}(t,f_{i})\|=\|Wh^{\prime}(t,f_{i})*m(t,f_{i})\|\). The convolution took the form of a summation over each K-line  
\[C(n,t)=\sum_{i=0}^{N}\big{[}\big{\|}W_{m}h^{\prime}(t_{Ki}-t,f_{i})\big{\|} \big{]} \tag{7}\]  
\(N+1\) was the number of frequencies included in the integration, from \(f_{0}\) = 0.1 Hz to \(f_{1}\) = 0.6 Hz, the nominal range of the observed wakes. This choice reduced overlap with the typically higher-frequency wind waves. Mask values \(m\) were equal to one everywhere but were set to zero for \(\ln(\|Wh^{\prime}(t,f)\|)<-8\), an empirical value corresponding to the nominal amplitude found by examining \(\|W_{m}h^{\prime}\|\) during time periods of low wind and no boat traffic. This value is likely dependent on \(\psi\) and the local wave environment. Applying the mask \(m\) decreased noise in \(C\). The K-lines were designed to detect divergent wakes typically generated when \(F_{L}>0.4\) and \(F_{h}>1\). The parameter \(n\) ranged from 1 to 120, yielding K-curves from 5 sec to 10 min long. A range of \(n\) values was used since the duration of any particular wake signal was unknown. Wake duration varies with distance traveled by the wake due to wave dispersion--the further the wake travels the longer the signal and the lower its amplitude [70]. The \(n\) that maximized \(C(n,t)\) for each time was selected to create a time series \(M(t)=C(n_{max}(t),t)\). Long-term background values \(\overline{M}(t)\) were calculated using a 20-min Hanning window and removed \(M^{\prime}(t)=M-\overline{M}\). This rectification reduced the likelihood of a small peak being falsely identified due to high background values, as was often found during periods of high wind. Peaks of sufficient \(M^{\prime}(t)\) indicated high correlation between the curves and the wavelet amplitude, and were identified as wakes with duration \(\tau_{1}=n_{max}\Delta t\). This convolutional approach was less computationally intensive then other methods of wake identification when considering unknown shape and duration, such as application of machine learning [71], ridge regression [72], and manual confirmation [35]. For brevity this method will be referred to as AWD (Algorithm for Wake Detection) through the rest of this article. To summarize, the sequential steps of the AWD are:  
1. Set-up 1. Compute masked \(\|Wh^{\prime}\|\);* Determine wake frequency range defined by \([f_{0},f_{1}]\);
* Find the nominal minimum frequency (\(f_{w}\)) of wind waves;
* Select the wake duration resolution \(\Delta t\);
* Select range of \(n\) such that maximum \(n\Delta t\) matches or exceeds longest expected wake duration;
* Define generic set of K-curve array positions for \(\|Wh^{\prime}(t,f)\|\). * Calculate \(M(t)\)
* Integrate \(\|Wh^{\prime}(t,f)\|\) over each K-curve starting at the initial \(t\), yielding \(C(n,t_{0})\);
* Determine the \(n\) that maximizes \(C\);
* Define \(M(t_{0})\) and \(n_{max}(t_{0})\) at this maximal value of \(C\);
* Repeat for all times of interest.