Norman W. Garrick Time-Distance Diagrams of Traffic Flow Vehicle 2 u 2 = 30 mph (constant) Vehicle 1 u 1 = 50 mph (constant) Distance Time Fix 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 Wave At a Stop Traffic is flowing normal Flow, q = 500 veh/hr Conc, k = 10 veh/mi T = t 1 sec

Norman W. Garrick Example of a Shock Wave At a Stop T = t 2 sec Flagman stops first vehicle in the queue Shockwave

Norman W. Garrick Example of a Shock Wave At a Stop T = t 3 sec More 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 1 q = 500 veh/hr k = 10 veh/mi State 2 q = 0 veh/hr k = 260 veh/mi

Norman W. Garrick Example of a Shock Wave At a Stop T = t 4 sec Flagman 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 3 q = 1000 veh/hr k = 110 veh/mi State 1 q = 500 veh/hr k = 10 veh/mi State 2 q = 0 veh/hr k = 260 veh/mi

Norman W. Garrick Distance-Time Diagram for Shock Wave Distance Time X Shockwave 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 u sw = (q 2 -q 1 ) / (k 2 -k 1 ) Shockwave 1 State 1 q = 500 veh/hr k = 10 veh/mi State 2 q = 0 veh/hr k = 260 veh/mi

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 u sw1 = (q 2 -q 1 ) / (k 2 -k 1 ) (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 = 250 *0.1 = 25 vehicles

Norman W. Garrick Calculation of Shockwave Travel u sw2 = (q 3 -q 2 ) / (k 3 -k 2 ) (1000-0) / ( ) = mph Shockwave 2 is moving upstream at 6.67 mph Shockwave 1 Shockwave 2 State 3 q = 1000 veh/hr k = 110 veh/mi State 1 q = 500 veh/hr k = 10 veh/mi State 2 q = 0 veh/hr k = 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 u sw1 = - 2 mphu sw2 = 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 = hr = 12.6 minutes Shockwave 1 Shockwave 2 State 3 q = 1000 veh/hr k = 110 veh/mi State 1 q = 500 veh/hr k = 10 veh/mi State 2 q = 0 veh/hr k = 260 veh/mi