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 and infuriating slowdown of the cars ahead, causing 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. Pure gridlock.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 the first cars started coming 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 driving habits when there are just too many cars on the road.
The simplest explanation for why traffic waves happen is that 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 harder you have to brake to compensate and keep a safe distance. The same goes for the car behind you, which has to brake even harder than you did in order to slow down faster. And so on down the road, in a domino-like effect.
To illustrate this concept, programmer Lewis Lehe, a civil engineering graduate student at UC Berkeley, created this visualization. Select a car from the bunch, click “Hit the Brakes” to slow down your highlighted car, and then wait until a traffic wave forms. The red bars represent deceleration levels (braking) and the green, acceleration (speeding up). Mouse over a car to slow everything down see its velocity and acceleration at any given point during the wave (assuming all the cars are in the same single lane).
The equation used in the car circle above is relatively complex. Known as the Intelligent Driver Model, it was first proposed in 2000 by researchers at Germany’s Dresden University of Technology. The creators made this Java applet demonstration. Formal equations to 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. The simplest such formula:
where a is the car’s acceleration, Δv is the difference in velocity compared with the car behind it, T 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.”
So, put really 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 lower and 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 speed evens out.
Over time, congestion researchers have developed more complex models of traffic behavior that include more realistic conditions or incorporate additional data collected from traffic detectors. 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 fact that cars can slow down much faster than they can speed up. 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.