# Measurements of High-Froude Number Boat Wakes near a Seawall
## 3 Results
### Wave Forms  
Examination of \(\|Wh^{\prime}(t,f)\|\) presented two primary types of signals in \(h^{\prime}\) that occurred both individually and in combination. The first type was found in a narrow-band (nominally 0.5-1 Hz) with modulating amplitude that produced a 'keyboard'-like structure in \(\|Wh^{\prime}(t,f)\|\) (Figure 4a,c). Directional analysis indicated multiple sources from which these waves were propagating (Figure 5a). This type of signal was associated with wind-driven waves and persisted for minutes to many hours. The second type (Figure 4b,d) was a smooth wake packet that progressed from lower to higher frequencies (roughly 0.15 to 0.8 Hz) and displayed a temporally narrow chirp structure in wavelet space. These were associated with boat wakes and generally persisted for 1-10 min. Many wake signals were found to be isolated, with one direction of origin (Figure 5b), indicating direct propagation from the boat's travel line to the location of the gauge. The general lack of spectral overlap of wind waves and boat wakes at frequencies below \(\sim\)0.5 Hz was important for the development of the AWD in Section 2.3.4. Figure 5b also provided a test of the angular resolution. A single wake packet can be presumed to have originated from a unique direction. The angular resolution of the DIWASP 1.4 software is an adjustable parameter and, in this case, was set to 30\({}^{\circ}\). This resolution was reasonable for the purposes of this study (Figure 5b). The practical resolution will depend on the array configuration, sampling rate, and signal frequency. A third type of signal was found composed of two overlapping wakes (Figure 4e,g). The \(h^{\prime}\) in these cases generally showed a beating pattern found when two signals of similar frequency are summed. In this case, the frequency of both wakes (\(f_{1}(t)\) and \(f_{2}(t)\)) increased with time, yielding an increasingly rapid carrier wave of period \((f_{1}+f_{2})^{-1}\) and slowing envelope with period \((f_{1}-f_{2})^{-1}\). The difference in arrival times at the gauge of these wakes were finite and typically between 30 and 90 s. The delay was found to be reasonably estimated for a boat traveling parallel to a seawall over a flat bottom (Figure 6), the travel time for the reflected wave to reach the gauge would be \((L_{1}+L_{2})/c_{g}=(d_{1}+d_{\beta})/\big{(}c_{g}\sin\theta\big{)}\), where \(\theta=\phi\) was the angle of propagation relative to the shipline, and \(d_{\beta}\) is the distance from the seawall to the gauge. Similarly, the travel time directly from the boat to the gauge would be \((d_{1}-d_{\beta})/\big{(}c_{g}\sin\theta\big{)}\). The difference in travel time was then \(d_{\beta}/(c\sin\theta)\), where \(c=2c_{g}\) is the celerity of the wave. This yielded a delay time of 30 s between arrival of the primary wake and the reflected wake, assuming 1.3 m of water, \(d_{\beta}=62\) m, and an angle of \(\theta=35^{\circ}\) (from the Kelvin solution), comparable to the measured values. Direction of boat travel, angle of propagation, and dependence of \(c_{g}\) on depth are likely the most significant sources of uncertainty in this estimate. Figure 4: Common wave types (**a,c**) wind waves; (**b,d**) boat wake; (**e,g**) double (reflected) wake; (**f,h**) wind wave and wake. Shown are (**a,b,e,f**) wavelet amplitude and (**c,d,g,h**) corresponding wave signal \(h^{\prime}\). The assumption that the secondary wake was a reflection was further supported by the directional spectrum of this signal, which showed two directions from which the signals propagated (e.g., Figure 5c). One direction was from the deeper region of Safety Harbor, where most boat activity occurs; the second direction was from the seawall. The DIWASP package could not be used to separately verify the different directions of the two wave packets as directional spectra for short (\(\sim\)1 min) time series were sensitive to small changes in length of the input data. Reflected wakes were further documented in the supervised boat passes (below).