The wing and its forcing are time independent in the co-moving frame, so after transients have propagated away the perturbation is stationary there. Write the background velocity and field as
$$
\mathbf U=-u_0\mathbf e_x,
\qquad
\mathbf B_0=B_0\mathbf e_z,
$$
and take every perturbation to be independent of $y$. The cold-plasma <linearized ideal magnetohydrodynamic equations> are
$$
-\rho_0u_0\partial_x\delta\mathbf u
=\frac1{4\pi}(\nabla\times\delta\mathbf B)\times\mathbf B_0,
$$
$$
-u_0\partial_x\delta\mathbf B
=(\mathbf B_0\mathbin\cdot\nabla)\delta\mathbf u
-\mathbf B_0\nabla\mathbin\cdot\delta\mathbf u.
$$
Their relevant components are
$$
-\rho_0u_0\partial_x\delta u_x
=\frac{B_0}{4\pi}
(\partial_z\delta B_x-\partial_x\delta B_z),
$$
$$
-u_0\partial_x\delta B_x=B_0\partial_z\delta u_x,
\qquad
-u_0\partial_x\delta B_z=-B_0\partial_x\delta u_x.
$$
Differentiate the momentum equation with respect to $x$ and use the two induction relations. With the <Alfvén speed>
$$
u_A^2=\frac{B_0^2}{4\pi\rho_0},
$$
the result is
$$
\boxed{(u_A^2-u_0^2)\frac{\partial^2\delta u_x}{\partial x^2}
+u_A^2\frac{\partial^2\delta u_x}{\partial z^2}=0}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 314 / 3 / b / Solution

{title2=linearized ideal MHD equations}

The linearized ideal magnetohydrodynamic equations retain terms first order in perturbations about a chosen ideal-MHD background. Around a uniform state they describe <Alfvén waves> and <magnetosonic waves>.

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