Why Traffic Jams Drift Backward

Cars obey a conservation law: density changes only when more cars flow in than out. A small bump in density rides along characteristic lines at a speed set by the traffic itself, and in heavy traffic that speed is negative: the jam moves backward while every car moves forward.

How it works

First-order PDEs and characteristics

In one sentence

A first-order PDE $u_t + c\,u_x = 0$ is constant along the characteristic lines $x - ct = \text{const}$ , so $u(x,t) = f(x - ct)$ . For traffic, conservation gives $\rho_t + q(\rho)_x = 0$ with flux $q(\rho) = \rho\,v_{\max}(1 - \rho/\rho_{\max})$ ; small disturbances on a background density $\rho_0$ travel at $c = q'(\rho_0) = v_{\max}(1 - 2\rho_0/\rho_{\max})$ .

General formula

{\htmlData{sym=0}{\rho}}_t + {\htmlData{sym=1}{c}}\,{\htmlData{sym=0}{\rho}}_x = 0, \qquad {\htmlData{sym=1}{c}} = {\htmlData{sym=2}{v_{\max}}}\left(1 - \frac{2{\htmlData{sym=3}{\rho_0}}}{{\htmlData{sym=4}{\rho_{\max}}}}\right), \qquad {\htmlData{sym=0}{\rho}}(x, t) = {\htmlData{sym=3}{\rho_0}} + {\htmlData{sym=5}{f}}(x - {\htmlData{sym=1}{c}}t)

Information about density travels along characteristics. In dense traffic each driver reacts to the car ahead, so a slowdown passes backward from car to car even as everyone drives forward.

What the symbols mean

$\rho$
Car density
Cars per kilometer of road.
$c$
Wave speed
How fast a density bump travels: negative means backward.
$v_{\max}$
Free-flow speed
How fast drivers go on an empty road: 108 km/h here.
$\rho_0$
Background density
The steady traffic density the bump sits on.
$\rho_{\max}$
Jam density
Bumper to bumper: 150 cars per kilometer here.
$f$
Initial bump
The shape of the slowdown at time zero.

Conservation of cars: the density $\rho$ changes only when more cars flow into a stretch than out, $\rho_t + q_x = 0$ , where the flow $q = \rho v$ counts cars per hour.
Drivers slow down in denser traffic, $v = v_{\max}(1 - \rho/\rho_{\max})$ , so $q = \rho\,v_{\max}(1 - \rho/\rho_{\max})$ .
For a small bump on a background density $\rho_0$ this becomes the transport equation $\rho_t + c\,\rho_x = 0$ with $c = q'(\rho_0)$ : the bump keeps its shape and slides at speed $c$ along the characteristic lines $x - ct = \text{const}$ .
Above half the jam density, $c < 0$ : the bump moves backward. That's the "phantom jam" drivers run into with no visible cause.

Learn the math

First-Order PDEs and CharacteristicsPDE · Lesson 2, Foundations D'Alembert's SolutionPDE · Lesson 11, Separation of Variables