Pressure in a perfect-gas simple wave
= Pressure in a perfect-gas simple wave
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.