The movement of cars along a freeway may seem to involve so many variables as to defy analysis, but since the 1950's several methods of examination have been introduced and gradually refined. The methods liken the flow of cars to the flow of a fluid, focusing on the average properties of the traffic and ignoring the detailed and random fluctuations.

One early study was published in 1955 by Michael J. Lighthill and Gerald B. Whitham of the University of Manchester. They proposed that small variations in the flow sweep through the traffic as "kinematic waves." If waves traveling at much different speeds happen to meet, a "shock wave" develops, and it is at the site of the shock wave that drivers must suddenly brake. Depending on circumstances, a shock wave can travel in the direction of traffic (downstream) or in the opposite direction (upstream), or it can even be stationary.

Two of the possibilities are shown in Figure 1: two series of overhead views of cars moving along a single lane. The three columns on the left demonstrate a downstream shock wave, the three on the right an upstream shock wave. In each group the first column is an early view, and the next two columns are progressively later views; in each group speedy and widely spaced cars near the bottom catch up with a pack of slower-moving and more closely spaced cars near the top. The shock wave is at the car that has just been braked hard to avoid a collision with the trailing slow car in the pack. Notice that in the left-hand group the shock wave travels downstream, that is, each successive car goes a little farther before having to brake; in the right-hand group the shock wave travels upstream. After outlining the recipe developed by Lighthill and Whitham for studying these situations, I shall explain how you can put their analysis to the test with only a stopwatch, a video camera and a videotape player.

The analysis involves measurements that are tedious to make, but it leads to a simple and yet powerful technique for predicting how kinematic waves and shock waves develop and move. The measurements allow you to calculate the flow, concentration and mean speed of the traffic. First, two lines are mentally drawn across the lanes in which the traffic is moving in one direction. The lines are separated by some short distance L, perhaps 40 meters or so. For a certain time period T, say 60 seconds, you count the number of cars that move through L, and you also record the time each car takes to move through L. (That last k measurement, of course, is impossible if you simply stand at the side of the road and watch the traffic, but the chore becomes feasible if you record the traffic on videotape, which can be reversed and rerun at will.)

The flow of traffic through L is defined as the number of cars you count divided by the time T. To calculate the concentration of cars, you add up their transit times through L and divide the sum by the product of T and L. To get the mean speed, you divide the flow by the concentration. (The equation can be rearranged into a form that will be useful later: the flow is equal to the product of the concentration and the mean speed.) For example, suppose that 20 cars pass through an L of 40 meters in 60 seconds and the sum of their transit times is 36 seconds. The flow through L is then 1/3 car per second, the concentration.015 car per meter and the mean speed 22.2 meters per second.

These results are, of course, only samples, because from moment to moment the number of cars and their transit times through L vary. If you repeat the measurements, the samples reveal more definitively how much traffic travels through L. I should particularly emphasize that the computed speed is only a mean. If the traffic is especially light or heavy, most cars travel at the mean speed, but for other concentrations none of them may travel at that speed. Some of the cars may even go considerably faster, particularly on a multiple-lane expressway, where an aggressive driver can (at some risk of collision) switch lanes when approaching a slower driver.

Imagine that you review a videotape of expressway traffic as it gradually builds from a low concentration to the greatest possible concentration: a traffic jam in which all the cars stop. At first the rare car that passes through L travels at the speed limit. Once the concentration increases somewhat, the separation of the cars decreases to the point where the drivers begin to "interact," that is, they reduce their speed because of proximity. The reduction may result from a concern for safety, each driver slowing because he (or she) senses that the car in front might suddenly stop. The reduction may also arise because the increased concentration lessens the possibility of a lane change, so that cars begin to pile up behind a slow driver. As the concentration continues to increase on the videotape, the mean speed continues to fall, for either or both reasons. At the limit, the concentration reaches its greatest value, and the cars are stationary, bumper-to-bumper.

What happens to the flow during the buildup in concentration? Recall that flow can be expressed as the product of the concentration and the mean speed and that the former increases at the expense of the latter. Initially the flow increases owing to the increase in concentration and in spite of the slowing of the cars. Later on the tendency is reversed: the flow decreases because of the slowing and in spite of the increase in concentration. The results can be displayed by a "flow graph" in which the flow is plotted against the concentration [Figure 2]. Notice that at some intermediate concentration the flow is maximum; such a state is rare for expressway traffic. Before rush hour the relevant part of the graph is on the left side. During rush hour the right side of the graph dominates. The transition between the two traffic states is usually rapid.

Once a flow graph has been constructed from observations and measurements, the mean speed of the cars for any given concentration can be derived. Draw a straight line from the origin of the graph to the point on the curve corresponding to the concentration in question. The slope of the line is the mean speed for that concentration. Note that if you first consider a point on the left side of the curve (such as A) and then consider a point farther to the right (such as B), the slope of the line decreases, as does the mean speed. If the point is at the extreme right of the curve, the line has zero slope, and the cars are frozen in place in a traffic jam.

So far the analysis provides little predictive capability, but Lighthill and Whitham noted a more powerful aspect of the graph: it shows how the kinematic waves travel through the traffic. The waves result from adjustments the drivers make in response to small variations in concentration. Suppose that you drive through moderate traffic. When your speed matches the speed of the car ahead of you and you do not want to pass, you tend to maintain your speed. If, however, the driver ahead of you slows down somewhat, the distance between your cars narrows and the concentration there increases, and so you too slow down. Your reaction is not instantaneous; it takes about a second. After another second, the driver behind you begins to slow down as well. Thereafter, the process of slowing travels back through the line of cars in your lane as a kinematic wave.

A similar wave is created if the driver in front of you speeds up a bit, increasing the space in front of you and decreasing the concentration. After a second or so, you too speed up. After another second, the driver behind you speeds up, and so on. In both examples the change in speed approximately offsets the change in concentration, and so the flow (which is the product of the two factors) remains constant. For this reason kinematic waves are said to be waves of constant flow: their existence keeps the flow constant in spite of inevitable small fluctuations in the speed of individual drivers.

How fast kinematic waves travel through the traffic depends on the overall, or average, concentration, and it can be determined from the flow graph [Figure 3]. Find the point on the curve corresponding to the concentration, and draw a tangent to the curve at that point. The velocity of the waves (with respect to the road or to an observer alongside the road) is equal to the slope of the tangent. If the point is on the left side of the graph, the slope is positive, which means that the waves travel downstream. If the point is on the right side, the slope is negative, and the waves travel upstream. When the point is at the top of the curve, the tangent has no slope, and the waves are stationary.

In all cases the waves travel more slowly than the mean speed of the cars and so must travel back through the traffic. A wave is probably invisible to anyone watching the traffic from the roadside because the adjustments associated with it are so slight. It is invisible even when you drive through it; you perceive only the distance to the car in front of you and the adjustments you make in your own speed.

The information about the mean speed and the velocity of the kinematic waves becomes more useful when it is transferred from the flow graph to a "distance graph," in which distance along the expressway is plotted against time [see Figure 5]. The lower part of the new graph pertains to the beginning of a stretch of highway, and the upper part pertains to a more distant segment. The left side reflects traffic conditions early in the period of observation, and the right side reflects conditions later on. Suppose that the concentration and flow are the ones represented by A on the flow graph. Draw a straight line from the origin to A, and then draw on the distance graph a line that has the same slope. This second (broken) line represents the progress of a car that happens to move at the mean speed of the traffic. The car enters the picture at some early time and then moves steadily along the expressway as time passes. Additional lines can be added to represent other cars that enter the picture at other times and that move at the mean speed.

The information about kinematic waves can also be transferred between the graphs. Draw a tangent to A on the flow graph, and then, on the distance graph, draw a series of lines that have the same slope as the tangent. Their spacing is immaterial, but their slope reveals how the waves travel along the stretch of highway being studied. A wave that first appears at the near end of the highway gradually moves downstream (toward the top of the graph). You can do the same thing for the concentration associated with point B on the flow graph. In this case, the lines representing the mean speed and the kinematic waves all have less slope than in the preceding example.

If the concentration is dense, the relevant part of the flow graph might be point C. The mean speed is lower than it is in the previous examples, and the kinematic waves are now represented by lines that have a negative slope. The slope indicates that a wave begins far down the expressway and moves steadily toward the nearby segment.

The graphs help one to picture how a shock wave moves along an expressway. A wave might develop when something increases the concentration suddenly and thus requires the mean speed to decrease dramatically. For example, a shock wave can be created when a heavy truck pulls onto the expressway and travels slower than the prevailing mean speed. As the cars in the truck's lane approach the truck, they suddenly slow, and their concentration increases. The changes in speed and concentration travel back through the approaching traffic as a shock wave.

Suppose that before the truck appears the flow conditions are represented by A on the flow graph and that afterward they are represented by B. The slope of a line connecting A and B equals the speed of the shock wave. It is also about equal to the average of the slopes of the tangents at A and B. The line has a positive slope, which means the shock wave travels downstream. To follow its motion, transfer the line to a distance graph [see Figure 5]. Complete the graph by adding lines to represent the kinematic waves and the travel of a car at mean speed for conditions associated with points A and B on the flow graph. In front of the shock wave (higher on the graph), the conditions are those associated with point B. Behind the shock wave (lower on the graph), the conditions are those associated with point A. Notice that the shock wave is the line along which the kinematic waves having one speed intersect with kinematic waves having a much different speed. In this example, the intersection produces a shock wave that moves downstream.

Suppose that you were to drive through such a shock wave. Initially you are traveling at some modest speed in moderately concentrated traffic. Then you suddenly realize that the car ahead of you has slowed abruptly, and you hit the brakes to avoid a collision. Thereafter, you travel slower and in traffic with greater density than before. If it was a slow truck that produced the shock wave and if you are well behind it, the whole affair will be puzzling because you do not pass anything that accounts for the drama or the delay. If the truck gradually picks up speed, the traffic near it also picks up speed and thins out. The changes in speed and concentration travel back through the pack of cars, and eventually, perhaps much later, they reach you.

The shock wave I have described travels downstream because the line connecting A and B on the flow graph has a positive slope. You might investigate situations in which the line connecting two points on the flow graph has a negative slope, in which case the shock wave moves upstream or has no slope (the shock wave is stationary with respect to the road).

I decided to study the traffic flow on a section of expressway near Cleveland State University in order to see how it changes from midafternoon to the time when rush-hour traffic partially clogs the three outbound lanes. From a bridge spanning the highway I aimed a video camera to look upstream at the approaching traffic, and I ran tape for two hours. The camera had a convenient digital-clock feature that superposed the hour and minute on the recorded image of the highway. (It is important to mount the camera over the middle of the traffic flow being recorded; if the view is from one side, a large truck or bus in one lane can hide small cars in a farther lane.

Later, at home, I studied the tape and followed Lighthill and Whitham's recipe for constructing a flow graph. To establish an arbitrary distance L, I affixed narrow strips of masking tape across my monitor, one near the top and one near the bottom. To establish a time period T, I advanced the tape to a region of interest and let it run until the superposed minute symbol changed. That change marked the beginning of T, and the next change in the minute figure marked its end; during the 60-second interval many vehicles moved from the top strip to the bottom one on the monitor. I rewound the tape to the beginning of the interval, and when the first vehicle appeared I measured its strip-to-strip transit time with a stopwatch. Then I backed up the tape to time the next vehicle, and so on. When the minute symbol finally changed at the end of the measuring interval, the vehicle I was timing just then was the last to be included in the data for that interval.

To measure L, I could have noted particular features of the highway near the strips of tape on the monitor (skid marks, litter and so forth) and then returned to the scene to pace off the distance, but I chose a safer technique. When a school bus appeared on the videotape, I froze the frame and with a ruler measured the length of the bus on the monitor and also the distance L. By later measuring the actual length of a bus at a local school, I converted the measurements from the monitor to get L.

I took measurements for only the middle lane of traffic. That created a complication. When a vehicle changes lanes somewhere along L, should it be included in the data or not? If the flow in adjacent lanes is much different from that in the middle lane, the progress of such a vehicle will be different than if it had not changed lanes. I decided to include the data if the car was in the middle lane for at least half the length of L.

After taking data a number of times in the course of the two-hour videotape, I plotted a flow graph. Although the data points are scattered, a curve drawn through them roughly resembles the idealized flow graph of Lighthill and Whitham. The points reflecting the traffic flow before rush hour fall on the left side of the graph, and those associated with rush-hour traffic fall on the right side. One point high on the graph is about midway along the concentration axis and apparently represents the flow when it was near its maximum.

I spent additional hours watching the videotape without taking measurements (much to the amusement of my family). Although I saw cars speed up and slow down, I failed to detect distinct kinematic waves. They are just too subtle. I did easily spot shock waves, however. Before rush hour they usually moved downstream, although they sometimes moved upstream when for a moment the concentration increased markedly. In rush hour they always moved upstream. Every time a shock wave appeared—and particularly when it was accompanied by the sound of squealing tires—I was grateful that I had only to watch the wave, not to drive through it.

You might extend my analysis by studying the "traffic hump" that is created when there is a large influx of cars onto a highway where the flow has been moderate-say after a sporting event lets out and the fans head for home. Or you might dissect the flow associated with a "bottleneck," where a specific obstacle, such as an accident, blocks one lane or more. A hump can be short-lived, but a bottleneck can sometimes constrain traffic flow for an hour or more after the obstacle is removed.

Bibliography

ON KINEMATIC WAVES: II. A THEORY OF TRAFFIC FLOW ON LONG CROWDED ROADS. M. J. Lighthill and G. B. Whitham in Proceedings of the Royal Society of London, Series A: Mathematical and Physical Sciences, Vol. 229, No. 1178, pages 317-345; May 10, 1955.

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