= Solution
A finite-amplitude compression can steepen because different parts of the wave travel at different characteristic speeds. For a right-moving <simple wave> in a <perfect gas>, one may write $dx/dt=c_0+(\gamma+1)u/2$ in terms of the local <velocity> perturbation $u$. Faster compressive characteristics catch slower ones. When the smooth inviscid description would become multivalued, a physical compression shock replaces it. A rarefaction normally spreads rather than forming an entropy-violating expansion shock.<Lecture notes: Astrophysical fluid dynamics by Gordon I. Ogilvie>[§§6.1.3–6.2, simple-wave reduction and characteristic crossing]{ref}
Across a locally planar shock, use <velocities> relative to its moving front. The <Rankine-Hugoniot conditions for a perfect gas> conserve <mass>, momentum and total <energy> flux:
$$
\rho_1u_1=\rho_2u_2=j,\qquad p_1+\rho_1u_1^2=p_2+\rho_2u_2^2,\qquad h_1+\frac12u_1^2=h_2+\frac12u_2^2,
$$
where $h=\gamma p/[(\gamma-1)\rho]$ is the <specific enthalpy>. For upstream normal <Mach number> $M_1>1$,
$$
\frac{\rho_2}{\rho_1}=\frac{(\gamma+1)M_1^2}{2+(\gamma-1)M_1^2},\quad\frac{p_2}{p_1}=\frac{2\gamma M_1^2-(\gamma-1)}{\gamma+1},\quad M_2^2=\frac{1+(\gamma-1)M_1^2/2}{\gamma M_1^2-(\gamma-1)/2}.
$$
Thus a normal compression shock increases <pressure>, density and <temperature>, reduces the normal <velocity>, and changes a supersonic upstream flow to subsonic downstream flow in the <shock frame>. In the strong-shock limit, compression approaches $(\gamma+1)/(\gamma-1)$, which is four for a <monatomic gas>. \b[A shock conserves total <energy> but generates <entropy>; an adiabatic shock is not an isentropic compression.] For a calorically <perfect gas>, <stagnation temperature> is conserved while <stagnation pressure> falls.<Normal Shock Wave Equations by NASA Glenn Research Center>[normal-shock property ratios, total-temperature conservation and total-pressure loss]{ref}
Real dissipative processes resolve the narrow front; the jump conditions do not require its detailed internal profile. In a continuum-gas experiment a <shock tube> can produce a moving shock by releasing a <pressure> discontinuity. <pressure> probes separated along the tube measure arrival speed and the <pressure> jump; distinguishing the shock from the following contact surface is essential. A contact surface can have a density/<entropy> jump with continuous <pressure> and normal <velocity>, whereas a shock carries nonzero <mass> flux through its front.
\Image[/past-exam-of-the-astrophysical-fluid-dynamics-course-of-the-university-of-cambridge/2016/paper-1-shock-and-sound.png]
{title=Original characteristic-crossing and longitudinal <acoustic wave> diagrams}
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