Flows upstream of a highway bottleneck

Suppose that the local capacity of a highway is a smooth function of locati on, approximated by a parabolic function with a minimum value at some locat ion (the bottleneck). The flow approaching the bottleneck increases approxi mately linearly with time as it exceeds the capacity of the bottleneck. We present here an analytic solution for the resulting flow pattern upstream o f the bottleneck as predicted by the theory of Lighthill and Whitham (1955) for two different types of analytic forms for the relation between flow an d density. Although, in each of the two cases, the formulation of the problem contains seven parameters, it is shown that, by appropriate linear transformation o f variables, the flow pattern can be described in terms of a single dimensi onless pattern. In each case, a shock first forms at some point upstream of the bottleneck with an amplitude which increases proportional to the squar e root of the time from its beginning.