Have you ever reached the start of a traﬃc jam fearing the worst—road works, an accident, a fallen tree—to later discover no clear reason for the delay? Then you fell victim to one of the strangest traﬃc phenomena: the phantom traffic jam. This ghostly foe may seem supernatural, but can actually be predicted through the theory of shockwaves.
We can model traﬃc by treating it as a compressible ﬂuid, the equations for which are above. In this case, x is the distance along a road (of inﬁnite length), ρ(x) is the density of cars at any given point and u(x) is the speed. A car braking a little too hard somewhere will result in a small decrease in the steady ¯ and the traﬃc density state solution, where all cars are travelling at the same speed u is constant. The car behind then has to brake slightly, as does the car behind that , and the one behind that, and so on, resulting in a soliton-like wave—aptly named a jamiton—travelling backwards along the road at a speed of around 12 mph (as calculated by traﬃc monitoring agencies). Spooky, huh?