Past exam of the astrophysical fluid dynamics course of the University of Cambridge / 2021 / Section B / 4 / b

Take $p_1\ll {\rho }_1{u}_1^2$. Let $q=u_2/u_1$. Mass conservation gives ${\rho }_2={\rho }_1/q$, and normal momentum conservation givesInsert this into the energy condition from part (a):Rearranging and factoring,The root $q=1$ is the unshocked state. The compressive Strong-shock Rankine-Hugoniot conditions instead giveCorrespondingly, ${\rho }_2/{\rho }_1=(\gamma +1)/(\gamma -1)$ and $p_2=2{\rho }_1{u}_1^2/(\gamma +1)$. These relations require $\gamma >1$ and negligible upstream thermal pressure.