= Solution
For $u_0>u_A$, define the <Alfvénic Mach-cone slope>
$$
m=\sqrt{\frac{u_0^2}{u_A^2}-1}
\simeq\frac{u_0}{u_A}.
$$
The equation in part (b) is the <hyperbolic partial differential equation>
$$
\partial_z^2\delta u_x-m^2\partial_x^2\delta u_x=0,
$$
whose <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) gives
$$
\boxed{\delta B_x=-\operatorname{sgn}(z)bW(x+m|z|)}.
$$
The stationary induction and <mass conservation> equations then give
$$
\boxed{\delta u_x=\frac{u_0b}{mB_0}W(x+m|z|)},
$$
$$
\boxed{\delta B_z=\frac b mW(x+m|z|),
\qquad
\delta\rho=\frac{\rho_0b}{mB_0}W(x+m|z|)},
$$
with $\delta u_z=\delta u_y=\delta B_y=0$. In the strongly super-Alfvénic limit, $\delta u_x\simeq u_Ab/B_0$ and $\delta B_z\simeq bu_A/u_0$ inside the disturbed region.
The perturbations have support only where
$$
\boxed{|x+m|z||<\frac L2}.
$$
Because 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.
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