Solution

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

U = - u_{0} e_{x}, B_{0} = B_{0} e_{z},

(1)

and take every perturbation to be independent of

y

. The cold-plasma linearized ideal magnetohydrodynamic equations are

- ρ_{0} u_{0} \partial_{x} δ u = \frac{1}{4 π} (\nabla \times δ B) \times B_{0},

(2)

- u_{0} \partial_{x} δ B = (B_{0} \cdot \nabla) δ u - B_{0} \nabla \cdot δ u .

(3)

Their relevant components are

- ρ_{0} u_{0} \partial_{x} δ u_{x} = \frac{B _{0}}{4 π} (\partial_{z} δ B_{x} - \partial_{x} δ B_{z}),

(4)

- u_{0} \partial_{x} δ B_{x} = B_{0} \partial_{z} δ u_{x}, - u_{0} \partial_{x} δ B_{z} = - B_{0} \partial_{x} δ u_{x} .

(5)

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 π ρ _{0}},

(6)

the result is

(u_{A}^{2} - u_{0}^{2}) \frac{\partial ^{2} δ u _{x}}{\partial x ^{2}} + u_{A}^{2} \frac{\partial ^{2} δ u _{x}}{\partial z ^{2}} = 0 .

(7)

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

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