nevertheless has its own upper turning point substantially higher in the atmosphere. For simplicity, assume that this is high enough that $a\gg c$ and therefore ignore sound speed altogether. The dispersion relation is then $\omega^2=a^2(k_h^2+k_z^2)$, indicating reflection where $\omega^2=a^2k_h^2$. This is an important point: the fast wave reflects where its horizontal phase speed $\omega/k_h$ coincides with the local Alfvén speed. After it reflects, the fast wave re-enters the solar interior wave field, and therefore is part of the seismology of the Sun. Its journey through the atmosphere must therefore have some effect on wave timings.

However, that is not the full story. The fast wave itself may partially mode-convert to upward or downward propagating Alfvén waves, depending on magnetic field orientation relative to the wave vector [@CalGoo08aa; @CalHan11aa; @HanCal12aa; @KhoCal11aa; @KhoCal12aa; @Fel12aa], thereby removing energy from the seismic field and potentially altering its phase (see the schematic diagram Figure \[fig:Alfschem\]). This conversion typically occurs in a broad region near the upper turning point of the fast wave, but only if the wave is propagating across the vertical plane containing the magnetic field lines; we let $\phi$ denote the angle between the vertical magnetic flux and wave propagation planes. Such a 