= Solution
Interstellar shocks arise when relative flow is supersonic: stellar winds, <supernova> ejecta, colliding clouds and gas moving through strong disturbances can drive them. Neighboring compressive <characteristics> steepen or a fast flow encounters material it cannot decelerate through upstream sound communication. A thin dissipative front then converts directed <kinetic energy> into thermal <energy> and increases <entropy>.
Integrate <mass>, normal <momentum> and total <energy> conservation through a planar steady shock in its rest frame. With no <mass> source, negligible external impulse across the thin layer and no radiative <energy> loss, their normal fluxes are respectively
$$
j=\rho u,\qquad p+\rho u^2,\qquad
j\left(h+\frac{u^2}{2}\right).
$$
For a <perfect gas> of constant <adiabatic index>, $h=\gamma p/[(\gamma-1)\rho]$. Equality of the fluxes on the two sides yields the three <Rankine-Hugoniot conditions> in the question. Adiabatic means no <energy> escapes this control <volume>; it does not make the shock isentropic.
In the strong-shock limit, let $r=\rho_2/\rho_1=u_1/u_2$ and neglect $p_1$. <Momentum> conservation gives $p_2=\rho_1u_1^2(1-1/r)$. Substitution into <energy> conservation gives
$$
\frac{2\gamma}{\gamma-1}(r-1)+1=r^2,
\qquad
(r-1)\left(r-\frac{\gamma+1}{\gamma-1}\right)=0.
$$
The nontrivial compressive root is
$$
\boxed{\frac{\rho_2}{\rho_1}=\frac{u_1}{u_2}=\frac{\gamma+1}{\gamma-1}.}
$$
The $r=1$ root describes no shock. A <monatomic gas> has the familiar compression four.
For identical colliding clouds in their center-of-momentum frame, <reflection symmetry> fixes the contact plane and the uniform compressed material on either side at rest. The unshocked clouds still approach at speeds $\pm V$; their supersonic upstream flow has not received a smooth <pressure> adjustment ahead of the shock. Let the right shock move to the right at speed $S$. In its own frame the upstream and downstream normal speed magnitudes are $V+S$ and $S$. The same <shock compression ratio> therefore gives
$$
\frac{V+S}{S}=\frac{\gamma+1}{\gamma-1},\qquad
\boxed{S=\frac{\gamma-1}{2}V.}
$$
This is the <shock speed in a symmetric cloud collision>. For $\gamma=5/3$, the shocks move outward at $V/3$, not at the upstream speed or the relative cloud speed $2V$. As a check, the post-shock <pressure> is $p_2=(\gamma+1)\rho_1V^2/2$; in the monatomic case $p_2/\rho_2=V^2/3$, giving specific internal <energy> $V^2/2$, exactly the initial bulk <kinetic energy> per unit cloud <mass> when brought to rest.
\Image[/past-exam-of-the-astrophysical-fluid-dynamics-course-of-the-university-of-cambridge/2012/paper-1-symmetric-cloud-shocks.png]
{title=Symmetric cloud collision: stationary compressed gas and two outward-moving shocks}
When a shock reaches a cloud's rear boundary, its supply of cold upstream gas ends. If the surrounding <pressure> is low, the hot compressed gas expands outward and a rarefaction travels back into it, converting thermal <energy> into motion and cooling the gas. A sustained uniform stationary shocked region is no longer the solution. Interaction with an appreciable exterior medium can instead drive another outward disturbance; the boundary <pressure> determines that part of the evolution. The stated adiabatic model provides no radiative mechanism that would keep a permanently cold dense sheet at the contact.
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