deletions | additions       diff --git a/k.tex b/k.tex  index 3fd077a..1a0a133 100644  --- a/k.tex  +++ b/k.tex 
...
respectively. This is analogous to compression surfing, where$\ A = A_{p}$,$\ v = v_{z}$, and$\ d = a_{l}$, the altitude lost by a pelican as a function of distance travelled in plain flight,$\ d_{trav}$. We can use the glide angle equation to relate these variables to $\theta$, a variable already written in terms of known quantities. This gives us $\ a_{l} = d_{trav}[(C_{D}/C_{L})]$. Evaluating the Work equation on its functional domain $\ (0 < v_{o} < sqrt(gh/2))$, at $\ (x_{o}, z_{o}, t_{o})$, for a wave of height$\ h$ and wave number$\ k$, ocean depth$\ H$, air density $\rho$, bird wingspan$\ l$, as well as undisturbed windspeed$\ v_{o}$, we obtain the expression we initially set out for!  $\ W_{w}$($\ d_{trav}$)=(($\rho_{air}$)$\ d_{t}A_{wings} / 2)(C_{D} / C_{L}) [[((gh)^2 - 2ghv^2_{o}])$/($\v_{o}^2$)$\]^2$ 2ghv^2_{o}])/(v_{o}^2)]^2$  *$\exp[-2h(k$)($\ *$exp[-2h$($\ k$)($\  v_{o}$/$\(gh - 2v^2_{o}$)$\cos((kl/2)(sqrt(v^2_{pelican} - (v_{ph} + v_{o})^2) / v_{pelican}))]$ Where, for deep water$\ v_{ph}$ = $\sqrt(g/k)$ and shallow water$\ v_{ph}$ = $\sqrt(gH)$. Evaluating the work equation with a complete set of values relevant to pelicans in standard international (SI) units, this equation can be interpretted as the energy in Joules saved by the pelican during compression surfing as a function of distance. despite its foreboding mathematical appearance, from here on out we know all of the remaining variables as predetermined constants, and we know the domain which our expression makes sense in, so now we can make quantitative conclusions. Before doing this it is important to note that often times pelicans will angle themselves such that they are more parallel to the wave's surface; in this case their altitude will be less than the value$\ a$ that I have an equation for. A good way I can make my solution more realistic is to set an upper bound on the amount of work the wind may do, under my work function's domain. To set this upper bound I simply evaluate the equation for$\ v_{z}$ without the exponential decay factor. Therefore the upper bound on the amount of work the wind may do as a function of distance travelled is given as: