in type and magnitude to those found here, though it is difficult to compare precisely since line-of-sight selects fast and slow waves differently, depending on their velocity polarizations with respect to magnetic field. Their result that phase change is minimal along the line of sight is consistent with our understanding that the atmospheric slow (acoustic) wave that will be preferentially selected by that viewing angle has the same phase as the fast wave incident from below on $a=c$ [there is no phase jump in the transmitted wave; see @TraKauBri03aa Eqn. (22)].

Comparison with BVP Results {#sec:BVP}
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![image](\figdir/BVP_1_5_1){width=".8\hsize"}

For the case of a uniform magnetic field and a horizontally invariant atmosphere, the wave propagation equations may be reduced to a $4^\mathrm{th}$ (2D) or $6^\mathrm{th}$ (3D) order system of ordinary differential equations, assuming an $\exp[{\rmn{i}}(k_x x+k_y y-\omega t)]$ dependence on horizontal coordinates and time. Boundary conditions may be applied so as to allow the ingress of only a pure fast wave at the bottom of the computational region (a few Mm below the solar surface), with outgoing radiation or evanescent (as appropriate) conditions on all other waves at top and bottom. This defines an ordinary differential Boundary Value Problem (BVP). Details 