the Rankine-Hugoniot condition
/ RANG-kin OO-go-nyoh /
Once a shock has formed, the solution jumps from one value to another across a moving front. A natural question is: how fast does that front travel? You cannot read it off from the differential equation, which has stopped making sense at the jump. The Rankine-Hugoniot condition is the rule that answers it — it ties the speed of a shock to the heights and fluxes on its two sides.
It comes straight from conservation. The same bookkeeping that gave u_t + f(u)_x = 0 must still hold across the jump in its integral form (mass is conserved even where it is discontinuous). Track the amount of conserved quantity that the moving front sweeps up: a shock moving at speed s, with state u_L on the left and u_R on the right, must satisfy s (u_R - u_L) = f(u_R) - f(u_L). In words, speed times jump-in-density equals jump-in-flux. Writing [u] = u_R - u_L for the jumps, this is the memorable form s [u] = [f]. For Burgers' equation, f(u) = u^2/2, so s = (u_L + u_R)/2 — a shock moves at the average of the two heights it separates. For systems the same condition reads s (U_R - U_L) = f(U_R) - f(U_L), a set of equations the jump and speed must jointly satisfy.
Why it matters: Rankine-Hugoniot is what makes a discontinuous function a legitimate weak solution — it is exactly the condition under which the integral conservation law holds across the jump. But it is only half the story. Many different jumps can satisfy it (a discontinuity and its time-reverse both obey the same algebra), so Rankine-Hugoniot alone does NOT pick a unique physical solution. That selection is the job of the entropy condition.
Burgers' equation with u_L = 2 on the left and u_R = 0 on the right: the jump is u_R - u_L = -2, the flux jump is f(u_R) - f(u_L) = 0 - 2 = -2, so the shock speed is s = (-2)/(-2) = 1 — equivalently the average (2 + 0)/2 = 1. The discontinuity travels rightward at speed 1, separating the value 2 from the value 0.
Speed times the jump in density equals the jump in flux: s [u] = [f].
Rankine-Hugoniot is necessary but not sufficient: it admits unphysical 'rarefaction shocks' that run time backward. A discontinuity satisfying R-H is an admissible shock only if it ALSO satisfies the entropy condition; otherwise the correct solution is a smooth rarefaction, not a jump.