2. Consider plane shocks.

(a) Starting from the jump conditions (Rankine-Hugoniot relations) in mass flux for a plane shock, show that if the upstream and downstream velocities in any convenient frame of reference are u1 and u2, respectively and the densities in those regions are 1 and 2, that the velocity of the shock itself in that frame, us is given by

(b) Using that result show directly that if a piston is observed moving to the right at speed up into gas with adiabatic index at rest, and the condition that in the limit for a strong shock, , the shock speed in this frame is given by

(c) Assume that the piston is actually a supernova explosion of up = 104 km s1 expanding into the interstellar medium with a number density of n = 1 cm3 and T = 104 K. Assuming an adiabatic index of = 5/3, estimate the temperature behind the shock.