Citation

Newell G. Transp. Traffic Theory 1999; 14: 125-146.

Copyright

(Copyright © 1999, Publisher varies)

DOI

unavailable

PMID

unavailable

Abstract

Suppose that the local capacity of a highway is a smooth function of location, approximated by a parabolic function with a minimum value at some location (the bottleneck). The flow approaching the bottleneck increases approximately linearly with time as it exceeds the capacity of the bottleneck. This paper presents an analytic solution for the resulting flow pattern upstream of the bottleneck as predicted by the theory of Lighthill and Whitham (1955) for two different types of analytic forms for the relation between flow and density. While in both cases the formulation of the problem contains seven parameters, it is shown that, by appropriate linear transformation of variables, the flow pattern can be described in terms of a single dimensionless pattern. In each case, a shock first forms at some point upstream of the bottleneck with an amplitude which increases proportional to the square root of the time from its beginning.