# Measurements of High-Froude Number Boat Wakes near a Seawall
## 2 Materials and Methods
### Data Processing
#### 2.3.3 Wavelet Analysis  
Continuous wavelet analysis [58; 59; 60] of an infinite time series is defined as the convolution of a function with a collection of translations and dilations of a zero-mean \"mother\" wavelet function \(\psi\). Wavelets were used to decompose the \(h^{\prime}(t)\) signals into the time-frequency domain [61; 62]. The decomposition expanded the one-dimensional, real data into a two-dimensional, complex domain  
\[Wh^{\prime}\big{(}t^{\prime},a\big{)}=a^{-1}\int_{-\infty}^{\infty}h^{\prime }(t)\Big{[}\psi^{*}\left(a^{-1}\big{(}t-t^{\prime}\big{)}\right)\Big{]}dt \tag{3}\]  
This allowed for separation of amplitude and phase signal elements. Here, \(\psi\) is a continuous function with compact support, and * indicates complex conjugate. The parameter \(a\) is the'scale' or dilation parameter whose inverse can often be directly related to a Fourier frequency [63], so the transform can be equivalently examined in time-frequency space \(Wh^{\prime}(t^{\prime},f)\). The Morse wavelet [64; 65] was used in this analysis because it is a continuous, oscillatory complex function whose Fourier transform [65] exists only on the positive real axis. Morse wavelets provide good correlation with the type of signals generated by both boat wakes and wind-driven waves. Normalization by \(a^{-1}\) in (2) scales the magnitude of \(Wh^{\prime}\) such that spectral components are more easily compared [66]. This is in contrast to the more common scaling of \(a^{-1/2}\)[67] that suppresses the amplitude of high-frequency signals relative to lower-frequency signals. The continuous transform is not orthonormal, but it is invertible. In this case,  
\[h^{\prime}(t)=\frac{1}{C_{\psi}}\!\!\int_{0}^{\infty}\int_{-\infty}^{\infty} \Big{[}a^{-1}\psi^{*}\Big{(}a^{-1}\big{(}t-t^{\prime}\big{)}\Big{)}\Big{]} \mathcal{H}\big{(}t^{\prime},a\big{)}Wh^{\prime}\big{(}t^{\prime},a\big{)} dt^{\prime}da \tag{4}\]  
where  
\[0<C_{\psi}=\int_{-\infty}^{\infty}\frac{\big{|}\hat{\psi}(\omega)\big{|}^{2}} {|\omega|}d\omega<\infty \tag{5}\]  
and \(\omega=2\pi f\) is the angular frequency. The limits on Equation (4) are known as the admissibility condition. The window \(\mathcal{H}\) is a localized time-frequency filter suitable for highlighting specific signals but can also be configured as a spectral bandpass by removing dependence on \(t^{\prime}\), or as a full inverse by setting \(\mathcal{H}=1\). Equation (4) was used below with \(\mathcal{H}(t^{\prime},a)\
eq 1\) to isolate the wake signal from the wind-generated wave field and calculate wake amplitude and duration.  
Boat wakes are often distinguishable from wind driven waves under time-frequency decomposition [33; 35; 41]. Wakes from small boats, such as those examined in this study, are coherent wave packets with nominal periods of several seconds, and duration of several minutes. When measured by a stationary instrument, their waveform shifts from low frequency to high frequency over time [41], which is reflected in their amplitude \(\|Wh^{\prime}(t,f)\|\) as a distributed, arching maxima. Such a changing frequency structure is commonly referred to as a \"chirp\" [68]. In contrast, wind-driven coastal waves (below) more closely resemble a random process at high frequencies.