Non-linear Waves. In this problem we will explore the Riemann invariants for a fluid with $P=\lambda^2 \rho^3 / 3$. This is the equation of state of onedimensional non-interacting Fermi gas.
a) From the continuity equation
$$
\partial_t \rho+\partial_x \rho v=0
$$
and Euler's equation of motion
$$
\rho\left(\partial_t v+v \partial_x v\right)=-\partial_x P
$$
deduce that
$$
\begin{aligned}
\left(\frac{\partial}{\partial t}+(\lambda \rho+v) \frac{\partial}{\partial x}\right)(\lambda \rho+v) & =0 \\
\left(\frac{\partial}{\partial t}+(-\lambda \rho+v) \frac{\partial}{\partial x}\right)(-\lambda \rho+v) & =0
\end{aligned}
$$
In what limit do these equations become equivalent to the wave equation for one-dimensional sound? What is the sound speed in this case?
b) Show that the Riemann invariants $v \pm \lambda \rho$ are constant on suitably defined characteristic curves. What is the local speed of propagation of the waves moving to the right or left?
c) The fluid starts from rest, $v=0$, but with a region where the density is higher than elsewhere. Show that that the Riemann equations will inevitably break down at some later time due to the formation of shock waves.