the traffic-flow equation

Why does a traffic jam appear out of nowhere on a busy highway with no accident, and then drift backward against the direction of travel like a ghost? You can understand this with a single first-order PDE that treats traffic as a flowing fluid. The traffic-flow equation models cars not one by one but as a continuous density that obeys a conservation law — cars are neither created nor destroyed.

Let rho(x, t) be the density of cars (cars per metre) at position x and time t. Conservation of cars says rho_t + q_x = 0, where q is the flux (cars per second passing a point). The modelling step is to assume the flux depends on density through the local speed: q = rho v(rho), where the speed v decreases as the road gets more crowded (empty road, drive fast; jammed road, crawl). A simple choice v(rho) = v_max (1 - rho/rho_max) gives a quadratic flux and the equation rho_t + (q(rho))_x = 0 — a scalar conservation law, exactly the structure of Burgers' equation. You solve it by the method of characteristics: information travels along curves whose speed depends on the density.

Here is the payoff and the honest subtlety. Because the wave speed depends on density, characteristics can cross — denser traffic behind catches up with sparser traffic ahead — and when they cross, a shock forms: a sharp jump in density, which is precisely the back of a traffic jam, and it can travel backward even though every car moves forward. After a shock forms you can no longer have a smooth solution; you need a weak solution plus an entropy condition (the Rankine-Hugoniot jump and the Lax entropy condition) to single out the physically correct one — a clean everyday illustration of why nonlinear conservation laws need shocks and entropy.