Piston-driven normal shock
= Piston-driven normal shock
If a piston moves at speed $V$ into stationary perfect gas and drives a shock at speed $U$, then the shock-frame upstream and downstream speeds are $U$ and $U-V$. If $p_2-p_1=\beta p_1$, the jump conditions give
$$
V^2=
\frac{2\beta^2}{2\gamma+(\gamma+1)\beta}
\frac{p_1}{\rho_1},
\qquad
\frac{U^2}{c_1^2}
=1+\frac{\gamma+1}{2\gamma}\beta.
$$