and around active regions dominated by magnetism in the surface layers and above. To set the scene, the quiet Sun’s internal $p$-modes ($p$ for pressure) are essentially acoustic waves travelling in an unmagnetised stratified medium, with dispersion relation $\omega^2-\omega_c^2-c^2k_h^2=c^2k_z^2$, where $\omega$ is the frequency, $c$ is the sound speed (which increases with depth), $k_h$ and $k_z$ are the horizontal and vertical components of the wave vector, and $\omega_c$ is the acoustic cutoff frequency [commonly defined by $\omega_c^2=(c^2/4H^2)(1-2\, dH/dz)$ where $H$ is the density scale height, but see @SchFle98aa for a critique of such formulæ]. Vertical propagation is therefore limited to the cavity where the left hand side is positive. At depth, where $c^2k_h^2\gg\omega_c^2$ typically, the lower turning point is around the Lamb depth $\omega^2=c^2 k_h^2$, whilst at the upper turning point $\omega^2=\omega_c^2+c^2 k_h^2\geq\omega_c^2$. The introduction of strong magnetic field in surface layers has several effects to alter this picture:

1.  Magnetic field alters the density and thermal structure of the plasma by supplying additional magnetic forces, requiring the plasma pressure to adjust accordingly (the ‘indirect’ effect). Changes in the sound speed naturally affect wave travel times.

2.  Magnetohydrodynamic (MHD) wave modes become available: fast, slow, and Alfvén. At depth, where 