Rankine-Hugoniot conditions

The Rankine-Hugoniot conditions are the jump relations that constrain the state variables across a Shock wave in a compressible fluid. They arise not from the differential form of the Euler equations, which is undefined at a discontinuity, but from the integral form of conservation laws: mass, momentum, and energy must balance across the shock front. The conditions determine the shock speed and the post-shock state uniquely from the pre-shock state and the equation of state, effectively turning the discontinuity into a boundary value problem that is solvable without knowing the microscopic physics of the transition layer. This decoupling — global conservation determining the jump, local dissipation determining the internal structure — is one of the most elegant examples of scale separation in applied mathematics, and it raises a persistent question: if the shock structure is invisible to the conservation laws, what exactly are we conserving when we say a quantity is conserved?

See also: Riemann problem