Show that, for a one-dimensional flow of a perfect gas (with $\gamma>1$ ) at constant entropy, the Riemann invariants $R_{\pm}=u \pm 2\left(c-c_{0}\right) /(\gamma-1)$ are constant along characteristics $d x / d t=u \pm c .$

Define a simple wave. Show that in a right-propagating simple wave

$$\frac{\partial u}{\partial t}+\left(c_{0}+\frac{1}{2}(\gamma+1) u\right) \frac{\partial u}{\partial x}=0$$

In some circumstances, dissipative effects may be modelled by

$$\frac{\partial u}{\partial t}+\left(c_{0}+\frac{1}{2}(\gamma+1) u\right) \frac{\partial u}{\partial x}=-\alpha u$$

where $\alpha$ is a positive constant. Suppose also that $u$ is prescribed at $t=0$ for all $x$, say $u(x, 0)=u_{0}(x)$. Demonstrate that, unless a shock develops, a solution of the form

$$u(x, t)=u_{0}(\xi) e^{-\alpha t}$$

can be found, where, for each $x$ and $t, \xi$ is determined implicitly as the solution of the equation

$$x-c_{0} t=\xi+\frac{\gamma+1}{2 \alpha}\left(1-e^{-\alpha t}\right) u_{0}(\xi)$$

Deduce that, despite the presence of dissipative effects, a shock will still form at some $(x, t)$ unless $\alpha>\alpha_{c}$, where

$$\alpha_{c}=\frac{1}{2}(\gamma+1) \max _{u_{0}^{\prime}<0}\left|u_{0}^{\prime}(\xi)\right|$$