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

The interface is material in ideal magnetohydrodynamics, so fluid on it remains on it. Its normal displacement ${\xi }_z$ must consequently be the same when approached from either side. Integrating normal momentum balance through an infinitesimal pillbox shows that the Lagrangian total-pressure traction is continuous. Because the equilibrium total pressure is constant on each side and continuous at $z=0$, this reduces to continuity of the Eulerian perturbation $\delta \Pi$. Hence

Write $k_{z+}=+i{q}_{+}$ above the interface and $k_{z-}=-i{q}_{-}$ below it, with $q_{\pm }=|k_{z\pm }|>0$, so both waves decay away from $z=0$. A common interface displacement ${\xi }_0$ givesPressure continuity therefore requires

DefineSolving the matching condition gives the magnetohydrodynamic interface wave speedThe weights ${\Sigma }_{\pm }$ are positive, so $v^2$ is a weighted average of the two squared Alfvén speeds. It therefore lies between them, and so does the positive phase speed $v$.