Self-similar solutions for the emergence of energy varying shock waves from plane-parallel atmospheres

We present a self-similar solution to describe the propagation of a shock wave whose energy is deposited or lost at the front. Both of the propagation of the shock wave in a medium having a power-law density profile and the expansion of the medium to a vacuum after the shock breakout are described with a Lagrangian coordinate. The Chapman-Jouguet detonation is found to accelerate the medium most effectively. The results are compared with some numerical simulations in the literature. We derive the fractions of the deposited/lost energy at the shock front in some specific cases, which will be useful when applying this solution to actual phenomena.