Flow of car traffic is a classical example of a system demonstrating shock waves. Earliest models go back to the middle of 20th century, when Lighthill and Whitham introduced their continuum model of traffic and explained the basics of traffic-jam formation in terms of shock-wave solutions of their model.
Our work involves building continuum models and understanding their properties by analytical and numerical means. Recently, we have shown how a two-equation model (comprised of the continuity and momentum equations) exhibits instability that saturates as a self-sustained traveling shock wave, we call jamiton. We found a simple analytical solution of the model that compares nicely against direct numerical solution of the governing system.
The theoretical solution is based on the observation that the traffic jam is analogous to a self-sustained detonation wave in that it also consists of a shock followed by a transonic flow. Similar to that of the ZND model of detonation, the existence of a sonic point allows one to find the jamiton speed.
Our recent preprints:
M. R. Flynn, A. R. Kasimov, J.-C. Nave, R.R. Rosales, B. Seibold, Self-sustained nonlinear
waves in traffic flow, 2008 (Submitted for publication, arXiv:0810.2820) (PDF)
M. R. Flynn, A. R. Kasimov, J.-C. Nave, R.R. Rosales, B. Seibold, On “jamitons,” self-sustained
nonlinear traffic waves, 2008, arXiv:0809.2828v2. (PDF).