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\subsection{Local Analysis}  Many of the properties of magnetohydrodynamic waves can be understood from a local (WKB)  analysis for high wavenumbers ${\bf k}$, in which $k r \gg 1$ and $k H \gg 1$. This %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. In the adiabatic, anelastic, and ideal MHD approximations, the local dispersion relation for MHD waves is \citep{unno:89}  \begin{equation} 
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\label{eqn:alven}  \omega_A^2 = \frac{ \big( {\bf B} \cdot {\bf k} \big)^2}{4 \pi \rho},  \end{equation}  where $B$ ${\bf B}$  is the magnetic field and $\rho$ is the density. The Alfven frequency can also be expressed as \begin{equation}  \label{eqn:alven2}  \omega_A^2 = v_A^2 k^2 \mu^2,  \end{equation}  where $v_A$ is the Alfven velocity, speed,  \begin{equation}  \label{eqn:valfven}  v_A^2 = \frac{B^2}{4 \pi \rho} \, .