= Solution
Integrate each steady one-dimensional conservation law across a thin region enclosing the <shock>. <Mass conservation> gives constant <mass flux> $j=\rho u$. <Momentum conservation> includes both advected momentum and <pressure> force, so $p+\rho u^2$ is constant. The total energy flux is $u(\rho e+p)=j(e+p/\rho)$, where $e=c_vT+u^2/2$. For a genuine <shock> $j\ne0$, division by the <mass flux> and the equation of state $c_vT=p/[(\gamma-1)\rho]$ give the conserved stagnation enthalpy
$$
e+p/\rho=\frac{u^2}2+\frac{\gamma p}{(\gamma-1)\rho}.
$$
Thus the <Rankine-Hugoniot conditions for a perfect gas> are
$$
\boxed{[\rho u]=0,\quad[p+\rho u^2]=0,\quad\left[\frac{u^2}2+\frac{\gamma p}{(\gamma-1)\rho}\right]=0.}
$$
The velocities here are measured in the stationary <shock frame>; a moving-<shock> problem must first be transformed to that frame.
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