The Rankine-Hugoniot equation governs the behaviour of shock waves. It is named after physicists William John Macquorn Rankine and Pierre Henri Hugoniot, French engineer, 1851-1887.
The idea is to consider one-dimensional, steady flow of a fluid subject to the Euler equations and require that mass, momentum, and energy are conserved. This gives three equations from which the two speeds, and , are eliminated.
It is usual to denote downstream conditions with subscript 1 and upstream conditions with subscript 2. Here, is density, speed, pressure. The symbol means internal energy per unit mass; thus if ideal gases are considered, the equation of state is .
The following equations
Eliminating the speeds gives the following relationship: