Norman W. Garrick

Time-Distance Diagrams of Traffic Flow

Vehicle 2u2 = 30 mph (constant)

Vehicle 1u1 = 50 mph (constant)

Distance

TimeFix Point in Time

Slope = speed

s

h

Fix Position

Norman W. Garrick

Time-Distance Diagrams of Traffic Flow

Distance

Time

Norman W. Garrick

Time-Distance Diagrams of 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 are associated with a particular vehicle in the stream slowing down or stopping

A shock wave might be associated with the pressure being released and a traffic jam dissipating

Norman W. Garrick

Example of a Shock WaveAt a Stop

Traffic is flowing normalFlow, q = 500 veh/hr Conc, k = 10 veh/mi

T = t1 sec

Norman W. Garrick

Example of a Shock WaveAt a Stop

T = t2 sec

Flagman stops first vehicle in the queue

Shockwave

Norman W. Garrick

Example of a Shock WaveAt a Stop

T = t3 secMore vehicles have joined the queue

The shockwave have moved backwards

Shockwave 1

On either side of the shockwave there are two different state of flow

State 1q = 500 veh/hrk = 10 veh/mi

State 2q = 0 veh/hrk = 260 veh/mi

Norman W. Garrick

Example of a Shock WaveAt a Stop

T = t4 secFlagman releases queue

Shockwave 1

There is now a second shockwave and a third state of flow - the flow state for traffic released from the queue

Shockwave 2

State 3q = 1000 veh/hrk = 110 veh/mi

State 1q = 500 veh/hrk = 10 veh/mi

State 2q = 0 veh/hrk = 260 veh/mi

Norman W. Garrick

Distance-Time Diagram for Shock Wave

Distance

Time

XShockwave 1

Shockwave 2

Norman W. Garrick

Calculation of Shockwave Travel

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

usw = (q2-q1) / (k2-k1)

Shockwave 1

State 1q = 500 veh/hrk = 10 veh/mi

State 2q = 0 veh/hrk = 260 veh/mi

Norman W. Garrick

Calculation of Shockwave Travel

Shockwave 1

State 1q = 500 veh/hrk = 10 veh/mi

State 2q = 0 veh/hrk = 260 veh/mi

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 minutesLength = u*t = 2 mph * 3/60 hr = 0.1 mile

How many vehicles are in the queue after 3 minutesno. of vehicles = k * L = 250 *0.1 = 25 vehicles

Norman W. Garrick

Calculation of Shockwave Travel

usw2 = (q3-q2) / (k3-k2)(1000-0) / (110-260) = - 6.67 mph

Shockwave 2 is moving upstream at 6.67 mph

Shockwave 1 Shockwave 2

State 3q = 1000 veh/hrk = 110 veh/mi

State 1q = 500 veh/hrk = 10 veh/mi

State 2q = 0 veh/hrk = 260 veh/mi

Norman W. Garrick

Calculation of Shockwave Travel

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 mphTime to dissipate a 0.1 mile queue is L/speed0.1 mile / 4.67 mph = 0.021 hr = 12.6 minutes

Shockwave 1 Shockwave 2

State 3q = 1000 veh/hrk = 110 veh/mi

State 1q = 500 veh/hrk = 10 veh/mi

State 2q = 0 veh/hrk = 260 veh/mi