(a) Show that the equations for one-dimensional unsteady flow of an inviscid compressible fluid at constant entropy can be put in the form

$$\left(\frac{\partial}{\partial t}+(u \pm c) \frac{\partial}{\partial x}\right) R_{\pm}=0$$

where $u$ and $c$ are the fluid velocity and the local sound speed, respectively, and the Riemann invariants $R_{\pm}$are to be defined.

Such a fluid occupies a long narrow tube along the $x$-axis. For times $t<0$ it is at rest with uniform pressure $p_{0}$, density $\rho_{0}$ and sound speed $c_{0}$. At $t=0$ a finite segment, $0 \leqslant x \leqslant L$, is disturbed so that $u=U(x)$ and $c=c_{0}+C(x)$, with $U=C=0$ for $x \leqslant 0$ and $x \geqslant L$. Explain, with the aid of a carefully labelled sketch, how two independent simple waves emerge after some time. You may assume that no shock waves form.

(b) A fluid has the adiabatic equation of state

$$p(\rho)=A-\frac{B^{2}}{\rho}$$

where $A$ and $B$ are positive constants and $\rho>B^{2} / A$.

(i) Calculate the Riemann invariants for this fluid, and express $u \pm c$ in terms of $R_{\pm}$ and $c_{0}$. Deduce that in a simple wave with $R_{-}=0$ the velocity field translates, without any nonlinear distortion, at the equilibrium sound speed $c_{0}$.

(ii) At $t=0$ this fluid occupies $x>0$ and is at rest with uniform pressure, density and sound speed. For $t>0$ a piston initially at $x=0$ executes simple harmonic motion with position $x(t)=a \sin \omega t$, where $a \omega<c_{0}$. Show that $u(x, t)=U(\phi)$, where $\phi=\omega\left(t-x / c_{0}\right)$, for some function $U$ that is zero for $\phi<0$ and is $2 \pi$-periodic, but not simple harmonic, for $\phi>0$. By approximately inverting the relationship between $\phi$ and the time $\tau$ that a characteristic leaves the piston for the case $\epsilon=a \omega / c_{0} \ll 1$, show that

$$U(\phi)=a \omega\left(\cos \phi-\epsilon \sin ^{2} \phi-\frac{3}{2} \epsilon^{2} \sin ^{2} \phi \cos \phi+O\left(\epsilon^{3}\right)\right) \quad \text { for } \quad \phi>0$$