Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 4 / 39C / b / i / Solution

In a right-moving simple wave entering gas initially at rest with sound speed $c_0$, the other Riemann invariant is constant, so
$$
c=c_0+\frac{\gamma-1}{2}u.
$$
The isentropic relation
$$
\frac p{p_0}=\left(\frac c{c_0}\right)^{2\gamma/(\gamma-1)}
$$
converts the velocity disturbance into its nonlinear pressure disturbance.


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 Incoming links: Pressure in a perfect-gas simple wave