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

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.