[attachment=12674]
Shock Wave in Traffic Flow
Background
Traffic waves work very similar to shock waves
In a traffic jam, traffic wave moves backward
How Do We Determine Shock Wave Speeds?
Usw = speed of shock wave (km/h or mph)
qi = density (vehicles per hour)
ki = flow (vehicles per kilometer(or miles))
Positive usw = shock wave moving in the same direction as flow
Negative usw = shock wave moving opposite
Example
traffic instantaneously changes from state B to state A where state B is up stream and has a volume of 1200 vph and 40 mph and state A is 2000 vph at 20 mph. The resulting shock wave (the difference between the speed of the forward moving traffic and the backward moving platoon of traffic is defined by the equation
Apply the equation:
Qa = 2000 vph, ka = 2000 vph/20 mph = 100 vph
Qb = 1200 vph, kb = 1200 vph/40 mph = 30 vph
Usw = (2000 – 1200)/(100 - 30) = 11.4 mph
Greenshield’s Model
Bruce D. Greenshields (1935)
Linear relationship between speed and density
Example
a traffic stream on the approach to a signal has an approach speed of 40 mph and density of 45 vpm and a jam density of 130 vpm. Find a speed of the shock wave.
k = 45 vpm
kj = 130 vpm
u = 40 mph
40 = uf(1 – 45/130)
uf = 61.2 mph
u = q/ k
q = 45 * 40 = 1800 vph
usw = (0 – 1800)/(130 – 45) = -21.2 mph
Problems with Greenshield’s Model
Does not consider human factors or look at individual cars
Based on single lane roads