Jan 1998 William Beaty, Electrical Engineer

(note: wait for animation to load)
[traffic wave animation, five lane highway packed with a grid of cars in a jam, with waves or 'holes' moving backwards through the grid as each car jumps forward into any empty space.]

[traffic wave animation]


I live in Seattle and my two daily commutes last about 45 minutes. (That's when I'm lucky; sometimes it's more like two hours each.) This has given me an immense amount of time for watching the interesting patterns in the cars. Boredom led me to fantasize about the traffic being like a flowing liquid, with cars acting as giant water molecules. Over many months I slowly realized that this was not just a fantasy. Why had I never noticed all the "traffic fluid dynamics" out there? Once my brain became sensitized to it, I started seeing quite a variety of interesting things occurring. Eventually I started using my car to poke at the flowing traffic. Observation eventually leads to experimentation, no? There are amazing things you can do as an "amateur traffic dynamicist." You can drive like an "anti-rubbernecker" and erase slowdowns created by other drivers. But first, some basic phenomena.


Have you ever been driving on an interstate highway when traffic suddenly slows to a crawl? You inch along for many minutes while waiting to see the accident which must have caused the jam. At the same time you also curse the "rubberneckers" who are causing the whole problem. But then all the cars ahead of you take off at high speed. The jam is over, but no accident, no police cars, nothing. WHAT THE HECK WAS THAT?!! A traffic jam with no cause? In the rear-view mirror you see all the poor saps behind you still stuck in the jam. But why? If all those people could just speed up at the same time, the whole traffic jam would evaporate. Why don't they ever do that? What caused the mysterious slowdown in the first place?

After experiencing many of these "invisible accidents", I came up with the following explanation. To best understand this, imagine that you look down on traffic from an aerial view point. Pretend you're in a Traffic Reporter's helicopter looking downwards.

[View from above: car wreck on highway, row of 
stopped cars, more cars approaching and having 
to stop too.  A long line builds up.]

figure 1.  Cars lining up behind an accident

Above in fig. 1 I've drawn a one-lane road, an accident, and a row of cars stuck behind the wreck. Other cars are approaching from the left and stopping too. Suppose that the "wrecked" car (the red one above) has simply become temporarily stuck. Maybe it spun out on ice. What will happen when the red car moves and unplugs the flow?

[Four figures in time sequence: the wreck is cleared, 
and a 'wave' of spaces moves backwards as cars begin
to leave, yet more cars stack up behind.  A 'wave' of
 stopped cars moves backwards along the highway.
The cars are unmoving, yet the wave itself moves.]

figure 2.  A wave of 'condensed' traffic creeps backwards

Refer to fig. 2 above. In the top row at 2a, the flow is suddenly unplugged. But not all the cars can move, since most cars are stuck behind drivers who are stopped. Figure 2B shows the traffic a few moments later, and figure 2C shows it a few moments after that. Notice the orange car in 2A, and see how it eventually becomes unjammed in 2D and begins moving. At the same time the red car in 2A approaches the jam and is swallowed up.


After the wreck is removed, there seems to be no reason for the traffic jam to persist. Yet it does. The reason for this is sensible: if I am stuck behind a car that is stopped, then I have to stop too, and so does the car behind me. All the cars in the jam are in this situation. Even though the wreck is gone, they remain locked at standstill because if they want to move, they ALL have to move at once. They never do, because each driver is waiting for the car ahead to move. If I am in the traffic jam, I'm not going to move forward because I have no room to do so. I'd bump the car ahead of me. We all think like this, so none of us can move.

When the car in front of me leaves, I still cannot accelerate instantly, so I will remain stopped for a moment. I must delay leaving for a moment. If I started up instantly, I'd stay too close to the car ahead of me, and that would not be safe. Each departing car must delay in the same way, and this causes the jam to "evaporate" starting from the forward downstream end. It evaporates in a wave which begins at the forward end of the jam, (near the wreck). The wave eats into the jam from right to left, yet new cars are piling onto the back end of the jam.

Starting at figure 2A, the cars depart from the jam in sequence. In 2B the wave of "evaporation" has moved away from the wreck site, and in 2C and 2D it is far from the wreck. But notice an interesting thing: even though the CARS THEMSELVES are moving from left to right, the "wave of evaporation" moves in the opposite direction. It moves leftwards as it eats into the traffic jam.

There is a second important thing to notice. While some cars are still jammed, more cars are piling up behind them at the trailing end of the jam. Even after the wreck is removed, more cars are still "condensing" onto the back of the jam. The traffic jam is like a solid object whose front end is evaporating and whose back end is growing like a crystal. Cars move left to right, yet look at the the group of stopped cars. The stoppage is creeping slowly upstream, in the opposite direction to the moving cars. The accident is gone, but a "moving wave" of stopped cars remains behind. It's not a traffic jam, it's a shock wave which propagates through the "automotive material". It's a traffic-clot in the blood vessel. It's a traveling wave of traffic-condensation.


These sorts of travelling waves are common during heavy traffic conditions. An accident isn't needed to create them, sometimes they are caused by near-misses, by people cutting each other off, by merging lanes at a construction site, or simply by extra cars entering from an on-ramp. In traffic engineering lingo, they can be caused by "incidents" on the highway. A single "rubbernecker" could cause one by momentarily stopping to look at something interesting. Whenever you slow way down in order to merge across a lane to get to your upcoming exit, YOU could create one.

Sometimes the traffic waves have have no real cause at all. They arise from nothing because tiny random motions can trigger large results. They are like sand ripples and sand dunes, and they just build up for no clear reason. They are like ocean waves caused by the steady breeze, or like the waves which move along a flapping flag. They just "emerge" spontaneously from the writheing lines of traffic. In the science of Nonlinear Dynamics this is called an EMERGENT PHENOMENON."

How long will the "traffic wave" last after the accident is cleared? Its lifetime depends upon the amount of traffic, and on the number of cars trapped in the jam, but sometimes these things can persist for many hours. When traffic is slight, the jam might shrink rapidly to nothing. But if traffic remains heavy, then there's no reason for the travelling wave to ever dissipate at all. Also, if the conditions are just right (if the "condensation" happens faster than the "evaporation",) then even a tiny wave could grow large and larger. Sort of like dropping a tiny seed crystal into a supersaturated solution. When traffic is heavy and unstable, slight braking by any driver can cause the traffic to freeze into a gigantic crystal. Like Kurt Vonnegut's end of the world story CAT'S CRADLE it's the "Ice Nine" of the highways.

So, next time you are commuting and you approach a stoppage, don't think of it as a stupid f@#$% traffic jam. Think of it as a pressure wave which has approached your car and engulfed it. Think of it as a simple living thing which is composed of cars rather than molecules. Stay hopeful that the Crystalline Amoeba poops your car out soon. Take an aerial viewpoint, and visualize the wave which is moving backwards as you move forwards. next page... >


     LOTS MORE READING at Univ. ETH Zurich: popular press and research papers

Also see my Frequently asked questions and visitor comments

NEXT, PAGE 2: TRAFFIC EXPERIMENTS and a possible cure for certain types of traffic congestion



Note that traffic physicists have a different name for traffic waves: the pinch effect.


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a great little program, $25, try their free eval version.
[FYI SEE: idleworm, how to animate]

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