There are two types of acoustic waves in a gas: sound waves and shock waves. The latter appear when a body moves in a gas with a supersonic speed (i.e. the relative speed of the body with respect to that of the gas is greater than the sound speed). At supersonic speeds, in front of the body appears a very thin layer of gas with a higher pressure, called compression shock.
For shock waves the pressure/density differences between the two sides of the wavefront can reach very high values. Passing through the wavefront, the thermodynamic parameters vary abruptly, with a sudden jump.
The gas compressed by the shock wave undergoes an irreversible adiabatic process which cannot be described by Poisson's equation. However, an equation for the shock adiabat was deduced towards the end of the XIXth century by the Scottish physicist William Rankine (1820 – 1872) and, independently by him, by the French engineer Pierre Henri Hugoniot (1851 – 1887), using the mass and energy conservation, as well as the momentum equation.
Denoting with p1 and ρ1 the gas pressure and density in front of the compression shock (which are known), and with p2 and ρ2 the same parameters behind the shock (which are unknown), show that the pressure ratio is related with the density ratio by a relation of the form:
p1p2=C+Dρ1ρ2A+Bρ1ρ2
Find the explicit form of the coefficients A, B, C, and D. The adiabatic exponent γ of the gas is known.
Note: For simplicity, use a stream tube with a constant cross section, crossing perpendicularly the wavefront of the normal shock.