Group

Participating Group Members

MONASH UNIVERSITY

DEPARTMENT OF CIVIL ENGINEERING

CIV2282: Transport and Traffic Engineering

Practical Class 3: Shock Waves and the Time-Space Diagram

With Answers

The exercises in this practical class draw on the material presented in the lecture slides “Traffic Flow Theory (3) – Shock Waves”

You will probably need a spreadsheet such as Excel to complete these exercises,
remember to bring a computer with spreadsheet software into the practical class.

Please complete this exercise sheet and hand it in at the end of the practice class. Remember to include the names of all group members who participated in solving this exercise.

Exercise 1 – Shock waves with time-dependent bottleneck

An arterial road having three lanes in each direction carries traffic quite well in the evening peak period until a car illegally parks, blocking one of the three lanes and forcing traffic to merge into the other two lanes. A traffic queue quickly builds up behind the parked car.

The fundamental diagram for this road can be modelled using a triangular-shaped flow-density relationship, having a capacity (maximum flow) of 4500 vehicles per hour, a critical density of 100 vehicles per kilometre, and a jam density of 300 vehicles per kilometre.

Note that previously we had been using a parabolic-shaped fundamental diagram, this example uses a triangular shape. Note also that these flows and densities may seen much higher than previously, remember this road has three lanes of traffic.

Plot (to a fixed scale) the fundamental diagram of the three-lane traffic stream on the flow/density axes provided on page 4.

See last page

At the bottleneck site, where traffic has merged from three into two lanes, the traffic stream can once again be modelled using a triangular-shaped flow-density relationship. Here the capacity has dropped to 3200 vehicles per hour, the critical density is still 100 vehicles per kilometre, and the jam density is now 250 vehicles per kilometre.

On the same flow/density axes, using a different colour, plot the fundamental diagram at the bottleneck site.

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Traffic approaches the bottleneck at a flow of 3500 vehicles per hour, assumed to be uniformly distributed across the three lanes.

A backwards forming shock wave is created due to the arrival flow being greater than the capacity of the road at the bottleneck. This shock wave indicates the position at which vehicles must slow from their arrival speed to join the queue of slow moving traffic approaching the bottleneck.

What are the flow (q) and density (k) conditions in region A (approaching traffic) and in region B (queued traffic slowly approaching bottleneck)? Mark regions A and B on the fundamental diagram (flow-density diagram).

q_A = 3500 veh/h, k_A = 77.8 veh/km, q_B= 3200 veh/h, k_B = 157.8 veh/km

At what speed are vehicles travelling in region A and in region B?

v_A = q_A / k_A= 45 km/h, v_B = q_B / k_B = 20.3 km/h

Draw the shock wave between regions A and B on the flow-density diagram.

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At what speed is this shock wave moving?

w_AB = (q_A–q_B) / (k_A–k_B) = -3.75 km/h = -1.04 m/s

In front of the parked car, the road returns to three lanes wide, but the flow of traffic is limited to that which can go through the two-lane bottleneck. Let’s call this region C.

What are the flow, density, and speed conditions in region C? Mark region C on the flow-density diagram.

q_C = 3200 veh/km, k_C = 3200/45 = 71 veh/km, v_C = 45 km/h

There are three regions A, B and C, and hence there would be three shock waves between them. We have already found one of those shock waves, between regions A and B.

Draw the other two shock waves (AC and BC), and calculate their speeds.

wAC = (qA-qC) / (kA-kC) = 45 km/h = 12.5 m/s (Note: this is not shock-wave since and C belong to the free-flow regime)

wBC = (qB-qC) / (kB-kC) = 0 km/h

Describe what these two shock waves represent.

AC represents the reduction in flow in front of the parked car after it blocks one lane.

Its location moves along at the speed of the vehicles (at free-flow speed). This sometimes called free-flow wave.

BC represents the speeding up of vehicles once they pass the parked car. Its location does not change

We now are able to draw what happens around and behind the parked car in a trajectory diagram (time-space diagram).

Plot these three regions and the shock waves between them to scale on a time-space diagram, and sketch some vehicle trajectories before, and after the bottleneck.

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Moving back to the rear of the queue of slow-moving vehicles (i.e. shock wave AB)…

How far would the shock wave at the rear of the queue of slow-moving traffic have travelled in a time of 30 minutes after the parked car blocks one of the lanes

Shock wave speed is w_AB = -3.75 km/h. In 30 minutes, it travels 3.75/2 = 1.875 km

How many vehicles would be in the slow-moving queue at this time of 30 minutes after the blockage?

In the slow-moving queue (region B), the density is k_B = 157.8 veh/km. So, in a distance of 1.875 km, there would be 157.8 × 1.875 = 296 vehicles

After 30 minutes, the driver of the illegally parked car returns to their vehicle and drives off, removing the bottleneck. Having removed the restriction, vehicles quickly move off from the front of the queue of slow moving vehicles at the highest possible flow that the three-lane road can handle.

The vehicles departing from the front of the queue of slow moving vehicles are in region D. Mark this on the flow-density diagram

q_D = Capacity Flow = 4500 veh/h, k_D = Critical Density = 100 veh/km

At what speed are those vehicles in region D travelling?

v_D = q_D / k_D = 45 km/h

These departing vehicles form a second shock wave at the front of the slow-moving queue of vehicles. Is this shock wave travelling forwards or backwards, and at what speed?

w_BD = (q_B–q_D)/(k_B–k_D) = -22.5 km/h = -6.25 m/s

Between the four regions, there are two other shock waves which we have not yet described. Where are those shock waves, what do they represent, and what speeds are they travelling at? Show them on the flow-density diagram and time-space diagram.

CD represents the increase in flow in front of the bottleneck after it is removed. It moves forwards at the speed of the traffic, 45 km/h

AD represents the last of the vehicles that had been in the queue of vehicles. After this, vehicles can pass through without being impeded. It moves forward at the speed of the traffic, 45 km/h. Again, A and D belong to the free-flow regime, and AD is called free-flow wave.

In order to estimate the effects of that one illegally parked car, we need to know how much delay is caused, since vehicles are still slowly moving in region B after the parked car moves off.

Because vehicles leave the front of the queue faster than they arrive at the back, eventually there will be no more vehicles in the queue and region B would cease to exist.

Thinking of the shock waves at the front and the rear of Region B, how can we determine how long and how far the queue reaches?

Find where the shock waves AB and BD intersect.

How long after the parked car arrives does the queue of slow-moving vehicles take to dissipate? (Hint: You may need to solve some simultaneous equations)

Let t = time after car parks in seconds, x = distance behind in metres.

Shock wave BD moves at 6.25 m/s from t = 1800 seconds x = 6.25 (t – 1800)

Shock wave AB moves at 1.04 m/s from t = 0 x = 1.04 t

Solve for t = 2160 seconds = 36 minutes

How far does this queue of vehicles reach behind where the car was parked?

Substitute t = 2160 into either of above equations gives x = 2250 metres

Exercise 2 – Shock waves with time-dependent arrivals

This exercise is very similar to the previous one. However, the parked car remains, and after a period of 30 minutes, the arrival flow decreases from 3500 vehicles per hour to 2500 vehicles per hour (as the evening peak period finishes).

Regions A, B, and C remain as before, however there is no region D (vehicles quickly leaving from the front of the queue once the bottleneck is removed). Instead we have a region E, representing the lower flow of arriving vehicles. Since the arrival flow in region E is now below the flow of slow-moving queued vehicles (region B), the rear of the queue would start moving forwards (that is, shock wave BE would be forwards-moving).

Plot Regions A, B, C and E to scale on a new flow-density diagram, and indicate the six shock waves between those four regions.

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Calculate the speeds of the six shock waves.

Region	Flow, q (veh/h)	Density, k (veh/km)
A	3500	77.78
B	3200	157.78
C	3200	71.11
E	2500	55.55

Shock wave	Speed (km/h)	Speed (m/s)
AB	-3.75	1.04
AC	45	12.5
AE	45	12.5
BC	0	0
BE	6.84	1.902
CE	45	12.5

Draw these six shock waves on a new time-space diagram, and sketch some indicative vehicle trajectories.

See last page

How far would the queue of slow-moving vehicles reach in the 30 minutes taken until the arrival flow decreases? (This should be the same as in question 11).

In 30 minutes, shock wave AB moves 3.75 / 2 = 1.875 kilometres

How long would it take the queue of slow moving vehicles to dissipate after the arrival flow decreases?

Shock wave BE recovers this 1.875 km in a time of 1.875 / 6.84 = 0.274 h

= 16.42 minutes = 16 minutes, 26 seconds = 986 seconds

Exercise 1 – Time dependent bottleneck

Exercise 2 – Time dependent arrivals at bottleneck