Shockwave example

1. Om Namo Narayanaya Namah
TFT 23: Shockwave Examples

2. Learning Objectives
• Analyze the queue formation behind a slow
moving truck on a single lane road
• Determine shockwave velocities
• Determine queue performance measures

3. Next Red Phase Q-K diagram
density
q
Kj = kb
A
kc
ka
qa
wac
k0
wcb
wab

4. What will happen at the next red
phase? Case 1
t
Ua, ka
Ub, kb
k0
wAB
wA0
WB0
w0C
wBC
t1 t2
t3
wAC
wC0
w0B
B
C
A

5. What will happen at the next red
phase? Case 2
t
Ua, ka
Ub, kb
k0
wAB
wA0
WB0
w0C
wBC
t1 t2
t3
wCB
wAB
wC0
w0B
B
C
A

6. • Slow moving truck drives along highway at 10
mph.
• Determine the speed at which shockwave moves
at the front and back of the platoon
• Initial flow: Ka = 25 veh/mi, Ua = 40 mph

7. Flow-density profile

8. • Point 1: Normal flow ( us = 40 MPH,
k=25 veh/mi, q= 1000 vph.
• Point 2: Slow Truck: ( us = 10 MPH, k=120
veh/mi, q= 1200 vph.

9. •
Shockwave in front of Truck at point A-A
( qb= 0, kb = 0)
• U = (0 – 1200) / (0 – 120) = 10 kmph
• Same as speed of truck!

10. Behind the truck
• Platoon of vehicles build behind Truck
• qa= 1000 vph, ka=25 vpm (1) at approach to
end of platoon
• qb= 1200 vph, kb =120 vpm (2) Platoon conditions
• Wab = (1000 – 1200)/(25-120) = 2.1 mph
• This indicates that as the traffic is slowed down behind
the truck the flow increases!
• The (+) sign indicates that the shockwave is moving
downstream with respect to a fixed observer.

11. Platoon behaviour
• A-A moves forward relative to the
roadway at 10 MPH
• B-B moves forward relative to the
roadway at 2.1 MPH
• Platoon Growth: 10-2.1 = 7.9 MPH

13. • Given: The truck leaves the roadway
after 10 minutes. The vehicles are
released from the front of the platoon at
a rate of 20 MPH, and k=70 vpm.
• Find: The length of the platoon at this point,
number of vehicles in the platoon just before the
truck exits, and the time for the platoon to
dissipate.

14. Queue Performance
• Length of the platoon: vrel x time = 7.9
MPH x 10/60 hr
• = 1.32 miles
• Number of vehicles in platoon = k x
length of platoon
• k=120 veh/mi ( platoon conditions )
• Number of vehicles = 120 vpm x 1.32
miles = 159 vehicles

15. • To find the time for the platoon to dissipate,
we need to find the velocity of the shockwave
that occurs at the front of the platoon when
the vehicles are released.
• Platoon conditions: qa= 1200 vph, ka =120
vpm
• Release conditions: qb= 1400 vph, kb =70
vpm
• Wab = (1200 – 1400)/(120-70) = -4 MPH

16. • The negative sign indicates that the platoon is
moving upstream.
• The shockwave at the rear of the platoon is
still moving downstream at 2.1 MPH and the
shockwave at the front of the platoon is
moving upstream at 4.0 MPH.
• The relative speed of the two shockwaves is :
• 2.1-(-4.0) = 6.1 MPH

17. • Since that platoon was 1.32 miles long
the time to dissipate is
•
= 1.32m / 6.1mph
• Or = 12.98 minutes

18. Distance time plot
0
1.67
0.8
0
0.35
0.8
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
0 10 22.98
Platoon Distance
Front
Platoon Distance
Rear

19. • The truck slows down the speed on the
roadway for 10 minutes, and the front of the
platoon moves with the truck downstream at
10 MPH for 10 minutes. What is happening
to the platoon during this time?
• Front of the platoon travels: 10 MPH x 10/60
Hr = 1.67 miles
• Rear of the platoon travels: 2.1 MPH x 10/60
Hr = 0.35 miles

20. • Maximum Length of the Platoon (1.67-
0.35 = 1.32 miles long)
• At time = 10+12.98 minutes the platoon
has dissipated, and how far has the rear
of the platoon traveled?
• 2.1 MPH x 22.98/60 Hr = 0.80 miles