Select a car from the bunch, click “Hit the Brakes” to slow it down and watch the traffic wave form. The red bars show deceleration levels (braking) and the green, acceleration (speeding up). Mouse over any car to see its velocity and acceleration at any given point during the wave (assuming all the cars are in the same single lane).

[If the graphic doesn’t load, refresh the page with this link]


Who doesn’t love sitting in traffic?

Especially when there’s no apparent reason for it: no crashes, no tolls, no flaming mattresses.

Just a sudden, infuriating slowdown that forces you to slam on the brakes, spill coffee all over yourself and slow to a glacial crawl, usually when you’re already late for something important — a job interview, for instance.

And then, when all hope seems lost, the congestion breaks as seemingly spontaneously as it began. And you’re on your way again … for a good 2 minutes before the whole thing repeats itself.

Welcome to the world of traffic waves, a phenomenon that’s been exasperating drivers since cars started rolling off Ford’s assembly line a century ago.

On average, Americans spend upwards of 40 hours a year stuck in traffic, according to Texas A&M’s annual mobility study. That figure rises to more than 60 hours in some of the most congested metro areas, like Los Angeles, Washington D.C. and — yup, you guessed it — San Francisco. And, contrary to popular belief, much of this congestion is NOT because of major impediments, but simply a result of annoying driving habits that arise when there are just too many cars on the road.

The simplest explanation for why traffic waves happen is that we drivers have relatively slow reaction times: if the car in front of you suddenly slows down, it’ll likely take you a second or so to hit the brakes. The slower your reaction time, the more you have to brake to keep a safe distance. Same deal for the car behind you, which has to brake even harder than you did in order to slow down that much faster. And so on down the road, like a domino-like effect.

To illustrate this concept, Lewis Lehe, a civil engineer and programmer, and  created the above visualization.

The equation he uses is known as the  Intelligent Driver Model, which was first proposed in 2000 by researchers at Germany’s Dresden University of Technology (the designers created this Java applet demonstration).

The formal equations that explain these traffic patterns in terms of individual behavior are called car following models. They were first developed by researchers at General Motors in the 1950s. This is the simplest such formula:

lambda formula

Here, is the car’s acceleration, Δv is the difference in velocity compared with the car behind it,is reaction time and ƛ is some constant that researchers estimate from data. The equation says, “At time t, you accelerate at a rate proportional to the difference in speed between your car and the speed of the car you’re following, but with a gap of T seconds.”

Put simply, if you’re going faster than the car in front of you, then you slow down. And if you’re going slower, you speed up.

This equation produces the graph below. At the 10-second mark, the grey car slows down, and the cars that brake later have to slow down to subsequently lower minimum speeds. Each line shows the history of the speed of a different car. Drag the slider to graphically see a traffic wave unfold. Note how the cars at the bottom of the chart get closer together with time, as speeds even out.

Over time, congestion researchers have developed more complex models of traffic behavior that include realistic conditions and incorporate additional traffic data. For example, our “simple” equation assumes that the car in front of you will impact your behavior even if it’s a mile away. Some of the first improvements to the equation added terms for the size of that gap and the understanding that cars can slow down much faster than they can speed up. You can read more about the history of car-following models here.

Lewis Lehe is a PhD student in Civil Engineering at the UC Berkeley, where he researches electronic road tolling and runs the Visualizing Urban Data idealab.

  • Since I don’t follow the math: It it akin to PIO? Sounds like it since it’s a feedback loop issue, having to do with the delayed reaction time and delayed system response times.

  • QueensWatcher

    Fun to see the math confirming what I’ve been observing for decades. The solution is to allow more space between you and the car in front of you when in slow traffic and void the temptation to quickly speed up when traffic seems to start moving again. The added distance grants you more reaction time and you do not need to hit the brakes so hard; you can interrupt the wave.

    • MB

      I always try to do that and, inevitably, someone cuts right in front of me because my cushion to not have to brake so much JUST fits their car. Frustrating. I still leave space…but it’s a case of more people needing to get on board. It’s just not human nature seemingly.

      • Josh G

        This is what you want, in order to make the traffic better. Aim to leave more than just enough room, and when someone slips in, realize you’ve done a little bit of needed good to the traffic flow and, once again, open up some more space.
        see e.g.
        and others…

        • Brian H

          So you establish a personal wave in which you move further and further back in line, and get slower and slower until you stop? 😉

          • wbeaty

            Doesn’t happen. Try it.

            On the other hand, habitual tailgaters are locked in an Ego battle, where it totally ruins their day if ONE PERSON cuts in and slows them by 0.5 seconds. But in heavy congestion, cars are 0.5 – 1.0 seconds apart! If 100 cars cut in front of you, it has no impact on your 40-min daily commute (totally insignificant, less than 1/2 min slowing. If 30 sec is that important, then set your morning alarm for 30sec earlier!!!)

            If you succeed in blocking merges, you’ll sit at 0MPH, because traffic is like gear teeth. If you can block anyone from “invading your space,” the gears jam and everything halts.

    • PrettyPenny

      Precisely! This, combined with people trying to avoid the inevitable slowdown by switching lanes (causing even more braking) makes traffic worse for everyone. I truly believe if everyone stayed in their lane and gave ample room between cars during peak traffic hours we would see a reduction in the overall traffic time.

    • Calmeilles

      I like to leave plenty of space between myself and the vehicle in front but all too often some other driver sees it as a gap that needs filling by changing lanes.

      • wbeaty

        If yours is the only gap in a close-packed lane of tailgaters then of course any desperate driver is finally going arrive at your gap and make their needed lane change. But if your true goal is to block all merging, then you need to tailgate aggressively.

        Be like a pro trucker. Open up lots of space and encourage merging.

  • rex

    so basically the wingnut in front of you texting is cause all the traffic

    • DFinMA

      Yes, but no. Any differential in speed causes the problem. Curves and hills are a big problem along with solar glare, on and off ramps, lane changing, natural variation, human behavior, merging freeways like where 80, 580 and 880 are all tangled up.

      Road design is a big one. Merging doesn’t work, cloverleaf interchanges don’t work and offensive drivers certainly don’t help.

    • keenplanner

      No. Everyone involved is traffic

  • Andy Burlingame

    There’s no loss term, here. The wave keeps propagating indefinitely. Really great work, and I don’t mean to be a downer, but we’re missing some interesting dynamics.

    • Andy Burlingame

      The lossless sim is better than the overly-lossy sim, actually. 🙂

  • Steven

    Great Job. I would love to see this with mobile texting factored into the equation.

    • Mark

      It already is. See lambda or T.

  • Val

    What happens when we add Google cars? I figure a driverless car equipped with a million sensors (and maybe even communicating with other driverless cars up ahead in traffic) could react to changes in speed much sooner and smoother than humans do. My guess is that traffic at current volumes would be more efficient with driverless vehicles. I just wonder how driverless cars would deal with ice. I have driven at temperatures below freezing on dry roads and ice glazed roads. Can a driverless car anticipate reduced traction better than a human?

    • Lewis Lehe

      Actually, you don’t need completely driverless cars for that. As adaptive cruise control becomes smarter, and the share of vehicles with adaptive cruise control rises, we should see higher traffic flows. The current rule of thumb is about 1800-2000 vehicles per lane-miles per hour on a US freeway, but some people project this could go much higher. Here is Honda’s analysis:

      I am not sure about ice, but that is a very important concern.

    • richensf

      Modern drivetrain sensors can detect loss of traction at the wheels at a much lower threshold than humans by calculating expected response (rate of acceleration, deceleration, lateral force) from a given stimulus (acceleration, braking, steering), comparing those measurements to actuals, and making the necessary corrections at safer and sharper increments than a human who would otherwise detect loss of traction later, tend to overcorrect, and make slower adjustments in response.

      In theory a relatively cheap laser or radar could be pointed at the road surface from the front of the car at adjustable distances based on speed to detect light diffraction on the road surface, which differs between dry, wet, and ice conditions, and apply a safe approach over that patch far ahead of its point of contact.

  • phoneranger

    Does anyone have congestion-based speed limits? Seems to me if everyone on the 401 during rush hour was limited to max 35mph they’d all get where they’re going faster.

    • Calmeilles

      It’s used on parts of London’s orbital motorway, the M25. But it does seem to be reactive rather than predictive.

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  • Ben Witz

    Simple solution… we have an overpopulation in this world and most of the ppl that make up this overpopulation are stupid, believably asinine idiots that should never have a driver’s license to begin with. Stop that.

  • DFinMA

    Gridlock is [very] different than congestion on a highway.

  • richensf

    A bad driver reacts only to the car in front of him. A good driver reacts to the cars in front of the car in front of the car in front of him. The best driver on the road is the one that uses his brakes and lane changes the least and thus minimizes his cost per mile (less fuel, tire wear, brake wear, suspension wear, electrical wear, etc).

    This is accomplished by perfecting the art of pacing and spacing, which is a matter of timing acceleration and braking to occur early and in soft increments to buy time for the wave to expand ahead of you and to absorb and decouple the wave effect for those behind.

    The guy that accelerates hard, brakes often, and changes lanes frequently is seen as an immature driver, still trying to figure out life’s basics.

  • Alex

    How can we stop having so many traffic jams.

  • Carol Patterson

    Too often I’ll find the entire traffic tie up is caused by 2 cars, in side by side lanes, going the exact speed limit and not allowing anyone to pass them thereby hogging the highway and slowing everyone down . They become the annoying unofficial “pace setters”. A big problem here in Minnesota.


Matthew Green

Matthew Green produces and edits The Lowdown, KQED’s multimedia news education blog, an online resource for educators and the general public. He previously taught journalism at Fremont High School in East Oakland, and has written for numerous local publications, including the Oakland Tribune and San Francisco Chronicle. Email:; Twitter: @KQEDlowdown


Lewis Lehe

Lewis Lehe is a PhD student in Civil Engineering at the UC Berkeley, where he researches electronic road tolling and runs the VUDlab (Visualizing Urban Data Idealab). He also creates data visualizations under the brand “Setosa” at

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