There has been a lot of talk on the tubes lately about the traffic flow problem, specifically a part of this problem that we’re all familiar with: complete stoppages that seem to have no explanation. Some recent links on the popularized tubes (aka, not the science tubes), seem to indicate that there has been some incredible breakthrough in our understanding on this subject. For example:

Slashdot: Scientists solve the mystery of traffic jams

This is fine and well, but unfortunately these people fail to mention the most important work on the subject which initially came from the theory of nonlinear wave equations, and was more or less solved in 1974. It was summed up in a classic text on linear and nonlinear waves so titled and written G. B. Whitham. The book is out of print but it’s around on Amazon as well as other stores and any self-respecting science library should have this book sitting on the shelves. The main problem is one of wave propagation leading to “shock fronts” in traffic. If one person brakes for no reason, shock waves develop and travel backwards (for most flow problems) relative to the moving frame of the cars. Consider a velocity function for cars as a function of the density.

It’s quite simple to assume that must be a decreasing function of which starts from some maximum value at and decreases to zero as , and the maximum density flow occurs at some specific value of . Guess what? Actual observations peg the value of at about 255 vehicles per mile and the maximum flow density at about 80 (or 1500 vehicles per hour). Amazingly these values scale in a near linear fashion as lanes are added to the flow on a simple highway. It turns out the maximum flow rate is actually achieved at about 20 miles per hour. If we then develop a simple expression for the propagation velocity:

Since the derivative of the velocity function is less than 0, propagation of shock waves in a traffic flow travel backwards, and according to Whitham, “warn the drives of disturbances ahead”. Unfortunately this has some pretty negative consequences for you and I, the driver, who will inevitably be fed up with random stoppages in the road for no particular reason. Whitham continues to make some elementary arguments on the status of a wave near the stoppage density of traffic on a road. It turns out that the second derivative of the density flow function is less than zero, which means that a local increase of density propagates backwards, and shock forms somewhere behind the initial disturbance.

Now I’m sure that people have made some improvements in the mathematical description of this problem since the pioneering work of Whitham, but don’t be fooled: pretty much everything you read about “new developments” in this area in the popular media have been solved for more than 4 decades.

Cheers.