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

With $\mathbf{B}=B{\mathbf{e}}_x$, $k^2=k_x^2+k_z^2$, $\omega =v{k}_x$, and $\mathbf{k}\cdot {\mathbf{v}}_A=k_x{v}_A$. Substitution into the magnetosonic polynomial and collection of the $k_z^2$ terms giveswhere the tube speed isLet $v_{<}^2=\min (v_s^2,v_A^2)$ and $v_{>}^2=\max (v_s^2,v_A^2)$. Since $v_T^2<v_{<}^2$, $k_z$ is real whenand imaginary whenThe endpoints are turning or degenerate cases.