= Solution
Take $p_1\ll\rho_1u_1^2$. Let $q=u_2/u_1$. <Mass conservation> gives $\rho_2=\rho_1/q$, and normal <momentum conservation> gives
$$
p_2\simeq\rho_1u_1^2(1-q),\qquad
\frac{p_2}{\rho_2}=q(1-q)u_1^2.
$$
Insert this into the energy condition from part (a):
$$
\frac\gamma{\gamma-1}q(1-q)+\frac12q^2=\frac12.
$$
Rearranging and factoring,
$$
(\gamma+1)q^2-2\gamma q+(\gamma-1)
=(q-1)[(\gamma+1)q-(\gamma-1)]=0.
$$
The root $q=1$ is the unshocked state. The compressive <Strong-shock Rankine-Hugoniot conditions> instead give
$$
\boxed{u_2=\frac{\gamma-1}{\gamma+1}u_1.}
$$
Correspondingly, $\rho_2/\rho_1=(\gamma+1)/(\gamma-1)$ and $p_2=2\rho_1u_1^2/(\gamma+1)$. These relations require $\gamma>1$ and negligible upstream thermal <pressure>.
Back to article page
