In this chapter we consider various physical systems in which shock waves arise. These systems can be studied in terms of characteristic curves, on which information from the boundary and initial conditions propagates. However, this approach usually only gives a valid solution for a finite time, after which the solution at some points becomes multi-valued. This difficulty can be dealt with by inserting discontinuities in the solution, which represent shock waves.

Traffic Waves

Traffic flow modelling has developed rapidly over the last forty years, and sophisticated models are used in the planning of new roads and analysis of existing road networks. The type of model that we will discuss is the simplest possible and was one of the first to be postulated. In spite of this, it manages to capture many of the qualitative and quantitative features of real traffic flows. It is an excellent way of introducing the mathematics and physics of shock waves, and the solutions can be readily interpreted in terms of our everyday experience of road travel.

Derivation of the Governing Equation

We begin by stating our main assumptions.

1. — There is only one lane of traffic and no overtaking. This may seem restrictive, but the inclusion of several lanes with traffic switching between lanes, along with a model for overtaking, is a difficult business. Moreover, the model that we will develop has been shown to be in reasonable agreement with observations, even for multi-lane roads (see however Kerner (1999) for an introduction to more complex phenomena on multi-lane roads).

2. […]