= Compression ratio of a perpendicular magnetohydrodynamic shock
{title2=$x=\rho_2/\rho_1$}
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 $x=1$ and
$$
(2-\gamma)v_{A1}^2x^2+[(\gamma-1)u_1^2+2c_1^2+\gamma v_{A1}^2]x-(\gamma+1)u_1^2=0.
$$
For 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 implies
$$
\frac{p_2}{\rho_1}=
\frac{c_1^2[(\gamma+1)x-(\gamma-1)]/\gamma
+(\gamma-1)v_{A1}^2(x-1)^3/2}
{(\gamma+1)-(\gamma-1)x}.
$$
It 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.
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