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The first situation is the simplest and thus I will hash out this scenario before the others. The magnitude of the lift force on an object in flight is given by the density of the fluid the object is traveling through times the velocity of the object times the circulation, where circulation is the closed path integral of the fluid velocity around a wing's cross section. For a bird in flight, the circulation ( $\Gamma$ ) can be approximated as $\Gamma$ = $\ (C_{L}/2)(v_{o}A_{p})$ where$\ A_{p}$ is the planiform area of the wing: that is, the area of the wing projected onto the relevant xy plane. Therefore the magnitude of the lift force is given by (1/2)($\rho_{o}$)($\ A_{p})(v^2)$. The drag force follows the same relation, but$\ C_{L}$ is replaced by$\ C_{D}$. Choosing the xy plane to be parallel with the plane of the bird's flight in a standard, right handed cartesian coordinate system, we can set $\Sigma(F_{x})$ and $\Sigma(F_{z})$ both equal to zero as we are solving for the angle where the bird flies at a constant altitude. Solving for this angle, $\theta$, we find that   $\tan$( $\theta$ ) =$\ C_{D}$/$\ C_{L}$. I will refer to this as the glide angle equation. From this expression we can easily formulate an equation that gives the work done by a pelican flying at a constant altitude$\ W_{p}$ as a function of distance using the definition of potential energy,$\ U = mga$. If we label the altitude lost while gliding under constant velocity descent as$\ a_{l}$ and the distance travelled in the horizontal xy plane as$\ d_{trav}$, we obtain the work equation for a pelican (mass M_{p})flying on its own. This will be good for comparison once we derive the equation that gives the work done by the wind. Anyways, for a pelican flying at constant altitude (no wave yet) Work, which can be interpretted as the energy expended by the bird, in joules:  $\ W_{p} = M_{p}a_{l}g = M_{p}g(d_{trav})[arctan(C_{D}/C_{L})]$  The solution to the second problem is beyond the scope of simple geometric arguments. To solve for the air flow over ocean swells, a good method is to introduce an ensatz, or in other words, a guess. As a first step, I approximated an ocean swell as a sinusoidal wave of amplitude$\ h$. From there I required that at a large height above the wave the wind would feel no effects from the wave, and would travel with$\ v_{x}$ = constant =$\ v_{o}$, the undisturbed wind speed. Both$\ v_{x}$ and$\ v_{z}$ decay at equal rates with additional altitude$\ z$ and therefore obey the same exponential decay factor, exp[-($\alpha$)($\ z$)] Since at this point I do not have any intuition for what this factor is, I introduced the undetermined constant $\alpha$. Initially I thought that due to radial accelleration along the trough, combined with the force of gravity pushing air down the backside of an arbitrary swell, that the maximum wind velocity would occur at the deepest point of the trough. On the flipside the wind would be least on top of the wave as it has to fight gravity in traversing up the face. One may think the wave would block the wind, but I am considering waves under the constraint that wavelength $\lambda$ >>$\ h$ to negate that factor. For purposes of analysis I picked a segment of wave to work with such that the maximum$\ v_{z}$ value occurs at$\ x$ = $\pi$/$\ 2k$, water displacements from equilibrium given by $\xi$($\ x$) = -$\ h$cos($\ kx$) I picked this zone because when I put the pelican in the picture it will center its body directly over the line in the$\ y$ axis where$\ z$ = $\ 0$. This will make the third independent problem of solving for $\phi$ much easier, let alone the solution for$\ W_{w}$, the work done by the wind. Following from this condition, the minimum $\ v_{z}$ occurs at $\ x$ =$\ 3pi$/$\ 2k$, maximum$\ v_{x}$ occurs at $\ x$ = $\ 0$ and minimum $\ v_{x}$ occurs at $\ x$ = $\pi$/$\ k$. 
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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 $\ a_{l} = d_{trav}[arctan(C_{D}/C_{L})]$. Evaluating the Work equation at $\ (x_{o}, z_{o}, t_{o})$ for a wave of height$\ h$ and wave number$\ k$, ocean depth$\ H$, air density $\rho$, as well as undisturbed windspeed$\ v_{o}$, we obtain the expression we initially set out for!  $\ W_{w}$($\ d_{trav}$)=(($\rho_{air}$)$\ d_{t}A_{wings}/2)arctan(C_{D}/C_{L})[ghk^2/(gh - 2v^2_{o})]^4exp[(2h/k)(gh/v_{o} 2v^2_{o})]^2exp[(2h/k)(gh/v_{o}  - 2v_{o})cos((kl/2)(sqrt(v^2_{pelican} - (v_{ph} + v_{o})^2)/v_{pelican}))]$Where, for deep water$\ v_{ph}$ = $\sqrt(g/k)$ and shallow water$\ v_{ph}$ = $\sqrt(gH)$.  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. This expression makes it simple to gain intuition for, despite its foreboding mathematical appearance.    An ancient creature, estimates have it that this majestic bird has been cruising the skies for at least thirty million years. Historical geologists would call this period of time the "Oligocene epoch" of the "Paleogene period," but to the rest of us these implications can be put in much simpler terms--Pelicans are dinosaurs! Perhaps it is to this that the Brown Pelican owes its aerial expertise. Thirty million years has allowed evolution to take its course, and over the generations the pelican has been able to develop the artform of "compression surfing."