Kinematic wave theory of bottlenecks of varying capacity

Abstract

This paper discusses a problem in traffic flow by the method of kinematic waves developed by Lighthill and Whitham(1,2). The theory of kinematic waves introduced by Lighthill and Whitham has been extended here to include the case when the flow q varies with the time, and it is seen that the expression for the wave velocity is of the same form as before, namely, ∂q/∂k, where q is the flow (quantity passing a given point in unit time) and k is the concentration (quantity per unit distance). The theory is applied to the problem of estimating how a uniform oncoming flow behaves on entering a bottleneck, the capacity of which varies with time. This capacity has initially a higher value than the oncoming flow but falls at a uniform rate to a lower value, where it remains constant for a time, and again rises at a uniform rate to the original value. A shock wave is found to move back from the bottleneck, and later forward again and through it, much as in the case of a bottleneck of constant capacity with varying oncoming flow studied by Lighthill and Whitham.