A one-dimensional shock wave propagates at a constant speed along a tube aligned with the $x$-axis and containing a perfect gas. In the reference frame where the shock is at rest at $x=0$, the gas has speed $U_{0}$, density $\rho_{0}$ and pressure $p_{0}$ in the region $x<0$ and speed $U_{1}$, density $\rho_{1}$ and pressure $p_{1}$ in the region $x>0$.

Write down equations of conservation of mass, momentum and energy across the shock. Show that

$$\frac{\gamma}{\gamma-1}\left(\frac{p_{1}}{\rho_{1}}-\frac{p_{0}}{\rho_{0}}\right)=\frac{p_{1}-p_{0}}{2}\left(\frac{1}{\rho_{1}}+\frac{1}{\rho_{0}}\right)$$

where $\gamma$ is the ratio of specific heats.

From now on, assume $\gamma=2$ and let $P=p_{1} / p_{0}$. Show that $\frac{1}{3}<\rho_{1} / \rho_{0}<3$.

The increase in entropy from $x<0$ to $x>0$ is given by $\Delta S=C_{V} \log \left(p_{1} \rho_{0}^{2} / p_{0} \rho_{1}^{2}\right)$, where $C_{V}$ is a positive constant. Show that $\Delta S$ is a monotonic function of $P$.

If $\Delta S>0$, deduce that $P>1, \rho_{1} / \rho_{0}>1,\left(U_{0} / c_{0}\right)^{2}>1$ and $\left(U_{1} / c_{1}\right)^{2}<1$, where $c_{0}$ and $c_{1}$ are the sound speeds in $x<0$ and $x>0$, respectively. Given that $\Delta S$ must have the same sign as $U_{0}$ and $U_{1}$, interpret these inequalities physically in terms of the properties of the flow upstream and downstream of the shock.