deletions | additions       diff --git a/subsection_Local_Analysis_Many_of__.tex b/subsection_Local_Analysis_Many_of__.tex  index 273447d..55a9a54 100644  --- a/subsection_Local_Analysis_Many_of__.tex  +++ b/subsection_Local_Analysis_Many_of__.tex 
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%\label{eqn:integral}  T = \exp{\int^{r_2}_{r_1} i k_r dr} \simeq \exp{\int^{r_2}_{r_1} - \frac{\sqrt{\ell (\ell +1)}}{r} dr } \, ,  \end{equation}  where $r_1$ and $r_2$ are the boundaries of the evanescent region (in this case, the upper boundary occurs where $\omega$T$ evaluates to  \begin{equation}  %\label{eqn:integral}  T \simeq \bigg( \frac{r_1}{r_2} \bigg)^{\sqrt{l(l+1)}} ,  \end{equation}  The fraction of transmitted energy flux through the evanescent region is $T^2$, while the fraction of reflected energy is $R^2=1-T^2$. Waves that tunnel through the evanescent region into the radiative core continue to propagate inward as gravity waves, since magnetic effects are likely negligible near the top of the radiative zone. As the waves propagate inward, however, the background magnetic field strength likely increases ($B \propto r^{-3}$ in the simple case of a dipole field). Moreover, the Brunt-Vaisala frequency also increases sharply toward the H-burning shell (see Figure \ref{Fig:Struc}). Consequently, the magneto-gravity frequency $\omega_{\rm MG}$ (equation \ref{eqn:maggrav}) increases inward, such that magnetic restoring forces begin to dominate the wave dynamics.