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Equation \ref{eqn:dxdt} describes the group velocity of the wave, while equation \ref{eqn:dkdt} describes the evolution of its wave vector, which is related to the momentum of the wave. Note that in the absence of a magnetic field in a spherical star, only the radial component of the wave vector changes, and the horizontal component is conserved. This is not surprising, as in this case the Hamiltonian is spherically symmetric and thus angular momentum (and hence angular wave vector) is conserved.   However, in the presence of a magnetic field, the last two terms of equation \ref{eqn:dkdt} break the spherical symmetry. Except in the unphysical case of a purely radial field or a constant field, these terms are non-zero, and therefore the angular component of the wave vector must change. Additionally, each term in equation \ref{eqn:dkdt} is the same order of magnitude at $r_{MG}$ where $B \sim B_c$, as long as assuming  $|\nabla B|/B \sim 1/r$. Upon wave reflection, we therefore expect a large change in $k_\perp$. $k_\perp$, such that $\Delta k_\perp \sim \Delta k_r \sim k_r$.  Hence, dipole waves will generally obtain high multipole moments when they propagate through strongly magnetized regions of the star.