 Compression ratio of a perpendicular magnetohydrodynamic shock

Put $c_1^2=\gamma p_1/{\rho }_1$ and $v_{A1}^2=B_1^2/({\mu }_0{\rho }_1)$. The mass, momentum and energy jump conditions for a perpendicular magnetohydrodynamic shock factor into the unchanged state $x=1$ andFor the compressive physical branch $x>1$, positive downstream pressure implies $x<(\gamma +1)/(\gamma -1)$ when $\gamma >1$ and the upstream sound speed or Alfvén speed is nonzero. Indeed the polynomial impliesIt then gives $u_1^2>c_1^2+v_{A1}^2$: the upstream flow exceeds the perpendicular fast magnetosonic wave speed. The exactly cold unmagnetized strong-shock limit reaches the upper compression bound.