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

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

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

Now suppose instead that, owing to dissipative effects,

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

where $\alpha$ is a positive constant. Suppose also that $u$ is prescribed at $t=0$ for all $x$, say $u(x, 0)=v(x)$. Demonstrate that, unless a shock forms,

$$u(x, t)=v\left(x_{0}\right) e^{-\alpha t}$$

where, for each $x$ and $t, x_{0}$ is determined implicitly as the solution of the equation

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

Deduce that a shock will not form at any $(x, t)$ if

$$\alpha>\frac{\gamma+1}{2} \max _{v^{\prime}<0}\left|v^{\prime}\left(x_{0}\right)\right|$$