The equation of state relating pressure $p$ to density $\rho$ for a perfect gas is given by

$$\frac{p}{p_{0}}=\left(\frac{\rho}{\rho_{0}}\right)^{\gamma}$$

where $p_{0}$ and $\rho_{0}$ are constants, and $\gamma>1$ is the specific heat ratio.

(a) Starting from the equations for one-dimensional unsteady flow of a perfect gas of uniform entropy, show that the Riemann invariants,

$$R_{\pm}=u \pm \frac{2}{\gamma-1}\left(c-c_{0}\right)$$

are constant on characteristics $C_{\pm}$given by

$$\frac{d x}{d t}=u \pm c$$

where $u(x, t)$ is the velocity of the gas, $c(x, t)$ is the local speed of sound, and $c_{0}$ is a constant.

(b) Such an ideal gas initially occupies the region $x>0$ to the right of a piston in an infinitely long tube. The gas and the piston are initially at rest. At time $t=0$ the piston starts moving to the left with path given by

$$x=X_{p}(t), \quad \text { with } X_{p}(0)=0 .$$

(i) Solve for $u(x, t)$ and $\rho(x, t)$ in the region $x>X_{p}(t)$ under the assumptions that $-\frac{2 c_{0}}{\gamma-1}<\dot{X}_{p}<0$ and that $\left|\dot{X}_{p}\right|$ is monotonically increasing, where dot indicates a time derivative.

[It is sufficient to leave the solution in implicit form, i.e. for given $x, t$ you should not attempt to solve the $C_{+}$characteristic equation explicitly.]

(ii) Briefly outline the behaviour of $u$ and $\rho$ for times $t>t_{c}$, where $t_{c}$ is the solution to $\dot{X}_{p}\left(t_{c}\right)=-\frac{2 c_{0}}{\gamma-1}$.

(iii) Now suppose,

$$X_{p}(t)=-\frac{t^{1+\alpha}}{1+\alpha}$$

where $\alpha \geqslant 0$. For $0<\alpha \ll 1$, find a leading-order approximation to the solution of the $C_{+}$characteristic equation when $x=c_{0} t-a t, 0<a<\frac{1}{2}(\gamma+1)$ and $t=O(1)$.

[Hint: You may find it useful to consider the structure of the characteristics in the limiting case when $\alpha=0$.]