Shock wave acceleration in a gravitational field. A special case

A one-dimensional problem of shock wave acceleration in a uniform gravitational field is exactly solved. In front of the shock wave, the medium state is initially in equilibrium and its density decreases according to a power law. The shock wave is generated using a piston moving freely in the gravitational field. The adiabatic index is assumed to be equal to 3. The obtained solution is represented in terms of elementary functions.