Integrating the three conservation laws across a stationary <normal shock wave> gives the <Rankine-Hugoniot conditions for a perfect gas>
$$
\rho_1u_{x1}=\rho_2u_{x2},
$$
$$
p_1+\rho_1u_{x1}^2=p_2+\rho_2u_{x2}^2,
$$
$$
\frac{\gamma p_1}{(\gamma-1)\rho_1}+\frac12u_{x1}^2
=\frac{\gamma p_2}{(\gamma-1)\rho_2}+\frac12u_{x2}^2.
$$
Eliminate $u_{x2}$ with the conserved mass flux and write $v_s^2=\gamma p/\rho$ and $M_x=u_x/v_s$. Solving the remaining two algebraic equations gives
$$
\boxed{\frac{u_{x1}}{u_{x2}}=\frac{\rho_2}{\rho_1}
=\frac{(\gamma+1)M_{x1}^2}
{(\gamma-1)M_{x1}^2+2}},
$$
$$
\boxed{\frac{p_2}{p_1}
=\frac{2\gamma M_{x1}^2-(\gamma-1)}{\gamma+1}}.
$$
For a compressive shock, $M_{x1}>1$, the density and pressure increase, and the downstream normal flow is subsonic in the shock frame.

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 314 / 2 / b / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-314/2/b/solution