### Congestion Redux

James M. Greenberg

U.S. Office of Naval Research

London, England

and

Carnegie Mellon University

Department of Mathematical Sciences

In this paper we analyze a class of 2nd order traffic models and show
these models support stable oscillatory traveling waves typical of the
waves observed on a congested roadway. The basic model has trivial or
constant solutions where cars are uniformly spaced and travel at a
constant equilibrium velocity that is determined by the car
spacing. The stable traveling waves arise because there is an interval
of car spacing for which the constant solutions are unstable. These
waves consist of a smooth part where both the velocity and spacing
between successive cars are increasing functions of a Lagrange mass
index. These smooth portions are separated by shock waves that travel
at computable negative velocity.