Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 4 / 39C (Waves)

For one-dimensional homentropic flow,and $dp=c^{2}d\rho$. For a perfect gas, $dc/c=(\gamma -1)d\rho /(2\rho )$. Combining the two equations therefore shows that the Riemann invariantssatisfyTheir characteristic curves are parametrized byand $R_{\pm }$ is constant along the corresponding curve until characteristic intersection creates a shock.

Before shock formation, the piston launches a right-moving simple wave into the undisturbed gas. The opposite invariant is fixed at its ambient value:Therefore the simple-wave relation is

The motion is isentropic before a shock forms, and the perfect-gas relations giveWriting $\delta =\delta c/c_0$ and applying the generalized binomial expansion,

The fluid velocity at the piston equals its velocity,Using $c-c_0=(\gamma -1)u/2$ in part (ii) givesOver $0\le t\le 1$,ThusThe linear pressure oscillation has zero mean, while the quadratic compressibility produces a positive mean pressure, the finite-amplitude acoustic analogue of radiation pressure.