A perfect gas occupies the region $x>0$ of a tube that lies parallel to the $x$-axis. The gas is initially at rest, with density $\rho_{1}$, pressure $p_{1}$, speed of sound $c_{1}$ and specific heat ratio $\gamma$. For times $t>0$ a piston, initially at $x=0$, is pushed into the gas at a constant speed $V$. A shock wave propagates at constant speed $U$ into the undisturbed gas ahead of the piston. Show that the excess pressure in the gas next to the piston, $p_{2}-p_{1} \equiv \beta p_{1}$, is given implicitly by the expression

$$V^{2}=\frac{2 \beta^{2}}{2 \gamma+(\gamma+1) \beta} \frac{p_{1}}{\rho_{1}}$$

Show also that

$$\frac{U^{2}}{c_{1}^{2}}=1+\frac{\gamma+1}{2 \gamma} \beta$$

and interpret this result.

[Hint: You may assume for a perfect gas that the speed of sound is given by

$$c^{2}=\frac{\gamma p}{\rho}$$

and that the internal energy per unit mass is given by

$$e=\frac{1}{\gamma-1} \frac{p}{\rho}$$