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

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