deletions | additions       diff --git a/k.tex b/k.tex  index 52be21e..b6a2f1f 100644  --- a/k.tex  +++ b/k.tex 
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Anyways, the velocity equations that correspond to these conditions are given by$\ v_{x}$($\ x$,$\ z$,$\ h$) = $\gamma$($\ h$)cos($\ kx$)[exponential] +$\ v_{o}$ and$\ v_{z}$($\ x$,$\ z$,$\ h$) = $\beta$($\ h$)sin($\ kx$)[exponential]. In each of these$\ k$ is the wave number. $\gamma$ and $\beta$ represent how much the x and z velocities actually change due to the wave. After solving the whole problem under these conditions, a serious contradiction presented itself. After I set $\alpha$, $\gamma$, and $\beta$ such that they solved Bernoulli's relation and the continuity equation everywhere, I plugged the conditions that I initially imposed on$\ v_{x}$ and$\ v_{z}$ back into my equations to see if they made sense. The conditions on$\ v_{z}$ were not met; in fact my equation stated that the minimum z-velocity would occur at $\ x$ = $\pi$/$\ 2k$. This was exactly where I required that the maximum velocity to occur. In formulating the contradictory scenario, I had $\gamma$ come out to the positive value ($\ gh$)/($\ v_{o}$), and the product ($\beta$)($\alpha$) equal the negative quantity, (-$\ kgh$)/($\ v_{o}$). Clearly as the magnitude of the disturbances due to the wave die off with additional altitude, $\alpha$ must be set positive such that the exponential term decays in this manner. This implies $\beta$ would be negative. Reviewing my calculations, it became apparent that both the continuity equation and the condition that$\ v_{z}$(max) happens at$\ x$ = $\pi$/$\ 2k$ could not be satisfied simultaneously with a negative $\beta$ value. Clearly it would be unreasonable to change the continuity equation. However, if we change the requirements on velocity so that now the minimum value of$\ v_{x}$ occurs at the trough, where$\ x$ =$\ 0$, In can easily be verified that $\xi$,$\ v_{z}$,$\ v_{x}$, bernoulli's equation, the continuity equation, can all be solved in harmony under the revised conditions imposed on the wind velocity components.  Using the same ensatz, and the revised wind velocity conditions, First I invoked Bernoulli's equation using$\ v_{x}$ values for the trough on one side and valuse for$\ v_{x}$ at the crest on the other. This analysis gave me $\gamma$($\ h$) = (-$\ gh$)/$\ v_{o}$. By approximating the pressure differences in the air at the crest and trough to be negligible, the continuity equation for air becomes $\grad$($\ v$) = 0. Plugging in $\ v_{x}$,$\ v_{z}$, and$\ v_{y}$ (which is a constant and therefore does not effect $\grad$($\ v$)) we obtain a relationship between the product ($\alpha$)($\beta$) = ($\ kgh$)/$\ v_{o}$. In order to determine $\alpha$ and $\beta$, I substituted $\beta$ = $\ kgh$/$\ v_{o}$  into the Bernoulli relation between the trough ($\ x$ =$\ 0$,$\ z$ = -$\ h$)  and equilibrium ($\ x$ = $\pi$/$\ 2k$,$\  z$ = $\ 0$) using the full wind velocity, given by$\ v^2$ = ($\ v^2_{x}$ + $\ v^2_{z}$)