= Solution
Work in the locally planar <shock frame>, taking $v_u,v_d>0$ towards the surface. For a stationary <normal shock wave> with negligible radiative loss inside the discontinuity, the <Rankine-Hugoniot conditions for a perfect gas> express <conservation of mass>, <conservation of momentum>, and <conservation of energy> per unit area as
$$
\boxed{\rho_uv_u=\rho_dv_d\equiv J,}
$$
$$
\boxed{P_u+\rho_uv_u^2=P_d+\rho_dv_d^2,}
$$
$$
\boxed{J\left(\frac{v_u^2}{2}+h_u\right)
=J\left(\frac{v_d^2}{2}+h_d\right),\qquad
h=\frac{\gamma}{\gamma-1}\frac P\rho.}
$$
Here $h$ is <specific enthalpy>, including the <pressure> work; using only <internal energy> would omit that work. The <gravitational potential> is continuous across an infinitesimally thin front. For cold upstream gas, $P_u$ and $h_u$ are negligible, while a <monatomic gas> has $\gamma=5/3$ and $h=5P/(2\rho)$.
The hydrodynamic equations apply to flow parallel to the guiding <magnetic field> when its stress contributions cancel across the front. An oblique <magnetohydrodynamic shock> requires magnetic stresses and <Poynting flux>; a radiation-mediated front or appreciable escaping radiation requires the corresponding radiative <momentum> and <energy flux> terms. These are additional physics, not terms hidden in $h$ here.
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