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 increase approximatley linearly with time as it exceeds the capacity of the bottleneck. We present here an analytic solution for the resulting flow pattern upstream of the bottleneck as predicted by the theory of Lighthill and whitham (1955) for two differnet types of analytic forms for the relation between flow and density. Although, in each of the two cases, the formulation of the problem contains seven parameters, it is shown that, by appropriate linear transformation of variable, the flow pattern can be described in terms of a single dimensionelss 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.
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