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

For $u_0>u_A$, define the Alfvénic Mach-cone slopeThe equation in part (b) is the hyperbolic partial differential equationwhose characteristic curves are $x\pm mz=\text{constant}$. Let $W(s)$ be one for $|s|<L/2$ and zero otherwise. Selecting the characteristics that trail downstream, toward negative $x$, and imposing part (c) givesThe stationary induction and mass conservation equations then givewith $\delta u_z=\delta u_y=\delta B_y=0$. In the strongly super-Alfvénic limit, $\delta u_x\simeq u_{A}b/B_0$ and $\delta B_z\simeq b{u}_A/u_0$ inside the disturbed region.

The perturbations have support only whereBecause the wing is infinite in $y$, this is the union of two inclined slabs bounded by the characteristic planes $x+m|z|=\pm L/2$. These two trailing slabs are the Alfvén wings generated by the conductor.