deletions | additions       diff --git a/Theory.tex b/Theory.tex  index 5e67d0a..2058b50 100644  --- a/Theory.tex  +++ b/Theory.tex 
...
\section{Theory}  Appleton's equation is given by\[\eta^2=1-\frac{\omega_{pe}^2}{\omega^2\left((1+\frac{i\nu}{\omega})-\frac{\omega_{ce}^2\sin^2\theta}{2\omega^2\left(1-\frac{\omega_{pe}^2}{\omega^2}\right)}\pm\sqrt{\frac{(\omega_{ce}^2\sin^2\theta)^2}{4(1-\frac{\omega_{pe}^2}{\omega^2})}+\frac{\omega_{ce}^2cos^2\theta}{\omega^2}}\right)}\]for an infinite plasma[2]. This describes the index of refraction for a whistler wave where $\eta^2=\left(\frac{kc}{\omega}\right)^2$. $\omega_{pe}$ is the plasma frequency,$\omega_{ce}$ is the electron cyclotron frequency, and $\nu$ is the rate of collisions in the plasma.The angle $\theta$ refers to the angle the waves make with respect to the background magnetic field $B_0$.The first assumption made is that the first experiment's waves are made with $\theta=0$ for waves parallel to $B_0$ and that damping is slight so that $\nu\approx0$.Making these assumptions it is found that Appleton's equation reduces to \[n^2=1-\frac{\omega_{pe}^2}{(\omega+\omega_{ce})(\omega-\omega_{ce})}\].Further assumptions can be made to reduce this equation by noting that $\omega_{ci}>>\omega_{ce}$,$\omega>\omega_{ce}$, and that the wave frequency $\omega$ is less than $\omega_{ce}$.This results in an equation for index of refraction that is is simple provided the above assumptions \[n^2\approx\frac{\omega_{pe}^2}{\omega\omega_{ce}}\]. Appleton's equation can be derived from the plasma force equation,maxwells equations, and using fourier analysis to show pertubations of the magnetic field are in the form $\vec{B_1}(\vec{r},t)=Be^{i(\vec{k} \cdot \vec{r}-\omega t)}$.The plasma force equation is written assuming that the particles are cold so that there are no pressure gradients and that the quantity $\vec{v}\cdot\nabla\approx0$ in the convective derivative then the force equation is $m\frac{\partial\vec v}{\partial t}=$