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\begin{appendix}  \section{Supplementary Material}  \subsection{Local Analysis}  \subsection{Stellar Models}  \subsection{Local Analysis}  Many of the properties of magnetohydrodynamic waves can be understood from a local analysis for high wavenumbers ${\bf k}$, in which $k r \gg 1$ and $k H \gg 1$. This analysis is technically only valid for large horizontal wavenumbers, $k_\perp r \gg 1$, whereas dipole oscillation modes have $k_\perp r = \sqrt{2}$. We must therefore be careful in extrapolating to global scale waves, especially for arbitrary magnetic field geometries.