deletions | additions       diff --git a/Magnetic Trapping.tex b/Magnetic Trapping.tex  index 1b6451c..a46e4ad 100644  --- a/Magnetic Trapping.tex  +++ b/Magnetic Trapping.tex 
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In our case, the location of $r_{\rm MG}$ is similar to such an interface. Just above $r_{\rm MG}$, incoming magneto-gravity waves have $k = \omega/(\sqrt{2} v_A \mu)$, whereas just below the interface an Alfven wave has $k = \omega/(v_A \mu)$. Across the location of $r_{\rm MG}$ a transmitted wave has a sudden jump in wave number and group velocity by a factor of $\sqrt{2}$. Thus, the interface has an effective index of refraction of $n = \sqrt{2}$, similar to that of many common liquids (water has $n=1.33$).   Despite a relatively low effective index of refraction, it is not clear how well the waves will be transmitted across $r_{\rm MG}$. Above Just above  $r_{\rm MG}$, the group velocity of the incoming magneto-gravity waves is primarily horizontal and is approximately \begin{align}  {\bf v}_g & \sim \bigg[ \frac{\omega}{k_\perp} {\bf \hat{n}}_\perp + \frac{\omega^2}{N k_\perp} {\bf \hat{r}} \bigg] \nonumber \\  & \sim \bigg[ \frac{\omega}{k_\perp} {\bf \hat {n}}_\perp + v_A {\bf \hat{r}} \bigg] \, .