Magnetosonic critical speed ($c_{\mathrm{f}},c_{\mathrm{s}}$)

For propagation in a direction with field component $v_{an}$ of the Alfvén velocity, the squared fast magnetosonic wave and slow magnetosonic wave speeds are the roots of $c^4-(c_s^2+v_a^2)c^2+c_s^2{v}_{an}^2=0$. They are real and nonnegative, and satisfy $c_{\mathrm{s}}^2\le v_{an}^2\le c_{\mathrm{f}}^2$. A steady acceleration equation can become singular when the flow equals either of these speeds.

If a steady wind equation has the form $(w^2-c_{\mathrm{f}}^2)(w^2-c_{\mathrm{s}}^2)w^{\prime }/w=\mathcal{N}$, a smooth nondegenerate crossing requires $\mathcal{N}=0$ at the corresponding magnetosonic critical speed. Subsequent differentiation or the full system determines the admissible slope. The numerator condition alone does not prove that a global smooth wind exists.