Traffic Flow Distance-time Diagram and Shockwaves
Lecture 14
CE 2710
Norman W. Garrick
Time-Distance Diagrams of Traffic Flow
Vehicle 1
u1 = 50 mph
(constant)
Vehicle 2
u2 = 30 mph
(constant)
Distance
Slope = speed
s
h
Fix Position
Time
Fix Point in Time
Norman W. Garrick
Time-Distance Diagrams of Traffic Flow
Distance
Time
Norman W. Garrick
Time-Distance Diagrams
Actual Traffic Flow
Ref: Papacostas and Prevedouros
Norman W. Garrick
Shock Waves
A shock wave occur when there is a change in the travel condition
on the roadway that affect the stream flow.
For example, a shock wave occur when drivers slow down to look
at an accident (rubberneck) - this can cause a traffic jam that
is seemingly more dramatic than one would expect given the
nature of the act that caused it.
Shock waves can be associated with a particular vehicle in the
stream slowing down or stopping
A shock wave is also associated with the traffic pressure being
released and a traffic jam dissipating
Norman W. Garrick
Example of a Shock Wave
At a Stop
T = t1 sec
Traffic is flowing normal
Flow, q = 500 veh/hr Conc, k = 10 veh/mi
Norman W. Garrick
Example of a Shock Wave
At a Stop
T = t2 sec
Flagman stops first vehicle in the queue
Shockwave
Norman W. Garrick
Example of a Shock Wave
At a Stop
T = t3 sec
More vehicles have joined the queue
The shockwave have moved backwards
Shockwave 1
State 1
q = 500 veh/hr
k = 10 veh/mi
State 2
q = 0 veh/hr
k = 260 veh/mi
On either side of the shockwave there are two different state of flow
Norman W. Garrick
Example of a Shock Wave
At a Stop
T = t4 sec
Flagman releases queue
Shockwave 1
State 1
q = 500 veh/hr
k = 10 veh/mi
Shockwave 2
State 2
q = 0 veh/hr
k = 260 veh/mi
State 3
q = 1000 veh/hr
k = 110 veh/mi
There is now a second shockwave and a third state of flow the flow state for traffic released from the queue
Norman W. Garrick
Distance-Time Diagram for Shock Wave
Distance
X
Shockwave 1
Shockwave 2
Time
Norman W. Garrick
Calculation of Shockwave Travel
usw = (q2-q1) / (k2-k1)
Shockwave 1
State 1
q = 500 veh/hr
k = 10 veh/mi
State 2
q = 0 veh/hr
k = 260 veh/mi
The speed of the shockwave can be calculated using the above equation
The sign is important so remember to number the travel states from upstream to downstream
If the sign is +ve it means that the shockwave is moving downstream
Norman W. Garrick
Calculation of Shockwave Travel
Shockwave 1
State 1
q = 500 veh/hr
k = 10 veh/mi
State 2
q = 0 veh/hr
k = 260 veh/mi
Speed of Shockwave
usw1 = (q2-q1) / (k2-k1)
(0-500) / (260-10) = - 2 mph
Shockwave 1 is moving upstream at 2 mph
What is the length of the queue after 3 minutes
Length = u*t = 2 mph * 3/60 hr = 0.1 mile
How many vehicles are in the queue after 3 minutes
no. of vehicles = k * L = 260 *0.1 = 26 vehicles
Norman W. Garrick
Calculation of Shockwave Travel
Shockwave 1
State 1
q = 500 veh/hr
k = 10 veh/mi
Shockwave 2
State 2
q = 0 veh/hr
k = 260 veh/mi
State 3
q = 1000 veh/hr
k = 110 veh/mi
Speed of shockwave 2
usw2 = (q3-q2) / (k3-k2)
(1000-0) / (110-260) = - 6.67 mph
Shockwave 2 is moving upstream at 6.67 mph
Norman W. Garrick
Calculation of Shockwave Travel
Shockwave 1
State 1
q = 500 veh/hr
k = 10 veh/mi
Shockwave 2
State 2
q = 0 veh/hr
k = 260 veh/mi
State 3
q = 1000 veh/hr
k = 110 veh/mi
How long will it take to clear the queue if the flagman held the
queue for 3 minutes
Length after 3 minutes = u*t = 2 mph * 3/60 hr = 0.1 mile
usw1 = - 2 mph
usw2 = - 6.67 mph
Therefore the queue will dissipate at rate of 4.67 mph
Time to dissipate a 0.1 mile queue is L/speed
0.1 mile / 4.67 mph = 0.021 hr = 12.6 minutes
Norman W. Garrick