Define
$$
\Sigma_\pm=\frac{\rho_\pm}{|k_{z\pm}|}.
$$
Solving the matching condition gives the <magnetohydrodynamic interface wave> speed
$$
\boxed{v^2=
\frac{\Sigma_+v_{A+}^2+\Sigma_-v_{A-}^2}
{\Sigma_++\Sigma_-}}.
$$
The weights $\Sigma_\pm$ are positive, so $v^2$ is a weighted average of the two squared <Alfvén speed>[Alfvén speeds]. It therefore lies between them, and so does the positive phase speed $v$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 314 / 3 / e / iii / Solution

{title2=MHD interface wave}

An MHD interface wave propagates along a discontinuity and decays away from it. Continuity of normal displacement and total pressure determines its dispersion relation.


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