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

The adiabatic sound speed and Alfvén speed areThe entropy is held fixed in the sound-speed derivative.

A displacement perpendicular to the plane spanned by $\mathbf{k}$ and $\mathbf{B}$ has $\mathbf{k}\cdot \tilde{\mathbf{\xi }}={\mathbf{v}}_A\cdot \tilde{\mathbf{\xi }}=0$. The algebraic wave equation then gives the Alfvén waveThe remaining displacement lies in the $\mathbf{k}$-$\mathbf{B}$ plane. Setting the determinant of that two-dimensional system to zero givesIts larger root is the fast magnetosonic wave, in which gas and magnetic pressure act together. Its smaller root is the slow magnetosonic wave, whose motion is guided more strongly along the field. Both are compressive, whereas the Alfvén mode is transverse and incompressible.

With $\mathbf{B}=B{\mathbf{e}}_x$, $k^2=k_x^2+k_z^2$, $\omega =v{k}_x$, and $\mathbf{k}\cdot {\mathbf{v}}_A=k_x{v}_A$. Substitution into the magnetosonic polynomial and collection of the $k_z^2$ terms giveswhere the tube speed isLet $v_{<}^2=\min (v_s^2,v_A^2)$ and $v_{>}^2=\max (v_s^2,v_A^2)$. Since $v_T^2<v_{<}^2$, $k_z$ is real whenand imaginary whenThe endpoints are turning or degenerate cases.

For $\mathbf{B}=B{\mathbf{e}}_x$, the vertical magnetic perturbation isThe $z$ component of the Fourier-transformed equation of motion is thereforeUsing $\omega =v{k}_x$ gives

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 $v$.