Perpendicular magnetohydrodynamic shock ($B_2/B_1={\rho }_2/{\rho }_1$)

For a planar magnetohydrodynamic shock with normal velocity and tangential magnetic field, mass conservation gives $\rho u=m$ and the ideal magnetohydrodynamic induction equation gives $uB$ constant across the front. Hence $B/\rho$ is unchanged, expressing magnetic flux freezing. Normal momentum flux is $\rho u^2+p+B^2/(2{\mu }_0)$, and energy flux per unit mass flux is $u^2/2+\gamma p/[(\gamma -1)\rho ]+B^2/({\mu }_0\rho )$, apart from a smooth common gravitational potential. The amplified tangential field contributes both magnetic pressure and magnetic energy transport.

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.