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Schnell and Hellack, two researchers who, concerned that the values on record for average flight speeds of various birds had not taken wind speed into account when taking measurements, performed observation experiments in order to more accurately determine these flight speeds. The data they collected for the brown pelican shows a low end speed of$\ 6$ meters per second and peak speeds of about$\ 16$ meters per second. (Schnell, Hellack 110) I am going to evaluate for compression surfing at$\ 15m/s$ as I feel it is reasonable to assume a pelican must be going near its top speed in order to surf the wave, just as a surfer must paddle to his or her top speed in order to catch a wave.   In fact, This top speed introduces a new constraint in the system: my equation will only return a real value for the cos($\phi$), assuming wind blowing in the opposite direction of the wave, for$\ v_{pelican}$ $\greq $\geq  (v_{ph} = v_{o})$, and the$\ v_{z}$ no longer holds when there are large waves, because the pelican is unable to catch up with a top speed of $\approx 16m/s$. To avoid imaginary solutions, where the deep water approximation describes $\omega$ I will consider $\hat v_{o} = \hat v_{ph}$. As I plan on evaluating the typical trade wind wave, intuitively it makes sense that the wind would be blowing with the wind, since after all the wind is generating the waves in the deep ocean. Furthermore, winds that blow against the propagation of the waves are called "offshore winds": even the name of this phenomenon leads one to believe it will occur near shore. Therefore where the shallow water approximation describes$\omega$, I will assume $\hat v_{o} = -\hat v_{ph}$ The density of air is a known constant, roughly 1.2 kilograms per meter cubed. g is the gravitational accelleration constant which is exactly 9.8 meters per second squared.$\ k$, the wave number, is defined as$\ 2pi$/$\lambda$. Two meters of swell ($\ h = 1$meter)at a period$\ T$ of eight seconds describes the common tradewind based ocean swell; such a swell would have a wavelength given by $\lambda$ =$\ v_{ph}T$ =$\ gT^2/(2pi)$ $\approx$ 100 meters and speed$\ v_{ph}$ $\approx$ 12.7 m/s, with$\ k = pi/50$. Primarily we'll use a low wind speed, 1 m/s. This will make calculations easier and will give us something to compare to if we want to plug in a higher wind speed (still within the domain of the work function, obviously) to see which gives the pelican more benefit. I'm sure some math could be done to find the optimal wind speed for compression surfing. Regardless, back to our problem: plug in these numbers and it is easy to see that under the deep water dispersion relation$\ v_{z}(x_{o}, z_{o}, t_{o})$ =$\ 2.8(10^-3)$