deletions | additions       diff --git a/subsection_Local_Analysis_The_properties__.tex b/subsection_Local_Analysis_The_properties__.tex  index 9f4d97c..019aba2 100644  --- a/subsection_Local_Analysis_The_properties__.tex  +++ b/subsection_Local_Analysis_The_properties__.tex 
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\subsection{Local Analysis}  The properties of magnetohydrodynamic waves in red giant cores can be understood using a local (WKB) analysis for high wavenumbers ${\bf k}$, in which $k r \gg 1$ and $k H \gg 1$. 1$ (where $H$ is a pressure scale height).  We show below the WKB limit is always a good approximation in our stellar models. In the adiabatic, anelastic, and ideal MHD approximations, the local dispersion relation for MHD waves is \citep{unno:89}  \begin{equation}