With $\mathbf B=B\mathbf e_x$, $k^2=k_x^2+k_z^2$, $\omega=vk_x$, and $\mathbf k\mathbin\cdot\mathbf v_A=k_xv_A$. Substitution into the magnetosonic polynomial and collection of the $k_z^2$ terms gives
$$
\boxed{k_z^2=
\frac{(v^2-v_s^2)(v^2-v_A^2)}
{(v^2-v_T^2)(v_s^2+v_A^2)}\,k_x^2},
$$
where the <tube speed> is
$$
\boxed{v_T^2=\frac{v_s^2v_A^2}{v_s^2+v_A^2}}.
$$
Let $v_<^2=\min(v_s^2,v_A^2)$ and $v_>^2=\max(v_s^2,v_A^2)$. Since $v_T^2<v_<^2$, $k_z$ is real when
$$
\boxed{v_T^2<v^2<v_<^2\quad\hbox{or}\quad v^2>v_>^2},
$$
and imaginary when
$$
\boxed{0\leq v^2<v_T^2\quad\hbox{or}\quad v_<^2<v^2<v_>^2}.
$$
The endpoints are turning or degenerate cases.

Solution

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

{title2=$v_T$}
{wiki}

The tube speed combines sound and Alfvén speeds according to
$$
v_T^2=\frac{v_s^2v_A^2}{v_s^2+v_A^2}.
$$
It is smaller than both constituent speeds.


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