Past exam of the astrophysical fluid dynamics course of the University of Cambridge / 2016 / Paper 1 / 2 / b

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.^[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:where $h=\gamma p/[(\gamma -1)\rho ]$ is the specific enthalpy. For upstream normal Mach number $M_1>1$,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. 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.^[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.