Derive Riemann's equations for finite amplitude, one-dimensional sound waves in a perfect gas with ratio of specific heats $\gamma$.

At time $t=0$ the gas is at rest and has uniform density $\rho_{0}$, pressure $p_{0}$ and sound speed $c_{0}$. A piston initially at $x=0$ starts moving backwards at time $t=0$ with displacement $x=-a \sin \omega t$, where $a$ and $\omega$ are positive constants. Explain briefly how to find the resulting disturbance using a graphical construction in the $x t$-plane, and show that prior to any shock forming $c=c_{0}+\frac{1}{2}(\gamma-1) u$.

For small amplitude $a$, show that the excess pressure $\Delta p=p-p_{0}$ and the excess sound speed $\Delta c=c-c_{0}$ are related by

$$\frac{\Delta p}{p_{0}}=\frac{2 \gamma}{\gamma-1} \frac{\Delta c}{c_{0}}+\frac{\gamma(\gamma+1)}{(\gamma-1)^{2}}\left(\frac{\Delta c}{c_{0}}\right)^{2}+O\left(\left(\frac{\Delta c}{c_{0}}\right)^{3}\right)$$

Deduce that the time-averaged pressure on the face of the piston exceeds $p_{0}$ by

$$\frac{1}{8} \rho_{0} a^{2} \omega^{2}(\gamma+1)+O\left(a^{3}\right)$$