Show that, in the standard notation for a one-dimensional flow of a perfect gas at constant entropy, the quantity $u+2\left(c-c_{0}\right) /(\gamma-1)$ remains constant along characteristics $d x / d t=u+c$.

A perfect gas is initially at rest and occupies a tube in $x>0$. A piston is pushed into the gas so that its position at time $t$ is $x(t)=\frac{1}{2} f t^{2}$, where $f>0$ is a constant. Find the time and position at which a shock first forms in the gas.