deletions | additions       diff --git a/k.tex b/k.tex  index bc04070..7b01110 100644  --- a/k.tex  +++ b/k.tex 
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$\ a$ = -($\ h$)$\cos$[($\ kl/2$)$\sqrt( v^2_{pelican} - ( v_{ph} + v_{x})^2)$/$\ v_{pelican}$]  At this point we are almost ready to put all the pieces together: if we evaluate$\ v_{z}$ at time$\ t$ =$\ 0$, the -($\omega$)$\ t$ factor drops out, simplifying the sine function to $\sin$($\ kx$), where$\ x$ = $\pi$/$ 2k$. Plugging this value of$\ x$ in the whole sine function =$\ 1$, making life a little easier. We can do the same trick for$\ v_{x}$ and recognize that $\cos$($\pi$/$\ 2k$) =$\ 0$: this implies that$\ v_{x}$($\pi$/$\ 2k$) =$\ v_{o}$. Now the only difference in z-velocities associated to the deep and shallow water cases lies in the wave velocity dependence of the exponentiall,$\ exponential,$\  v_{ph}$. These can easily be calculated from the relevant dispersion relations. I will continue referring to the pelicans coordinates as $\ x_{o}$ = $\pi$/$\ 2k$,$\ z_{o}$ =$\ a$ (given by the equation above),$\ t_{o}$ =$\ 0$). $\ y$ is an arbitrary and does not effect the work done by the wind and will be ignored. Thus we can express $\ v_{x}$ and$\ v_{z}$ as the following: $\ v_{x}$($\ x_{o}$,$\ z_{o}$,$\ t_{o}) = v_{o}$ (undisturbed wind speed)