Shock Wave Analysis Definition: Flow-speed-density states change over space and time. When these changes of state occur, a boundary is established that separates the time-space domain of one flow state from another. This boundary is referred to as a shock wave.

Shock Wave Analysis Chapter 7 Shock wave concept Types of shock wave Qualitative analysis

Shock Waves at signalized Intersection Length=? Speed=? Time=?

Shock Wave Analysis Chapter 11 Quantitative analysis Shock wave speeds Time durations Queue lengths … A flow-Density Relationship Needed. Demand-capacity process is deterministic.

Historical Perspective Shock wave analysis first appeared in the literature in the late 1950s (Richards, 1956). The earlier works involved highway traffic analysis and were typically based on changes in density. Later applications involved studies of traffic flows in tunnels. More recently, shock wave analysis has been applied at signalized intersections. Pipes (1965) used hydrodynamic theories to describe traffic flow wave phenomenon. Lighthill and Whitham’s work considered monumental.

Shock Wave Speed Equation Continuity: # leaving state B = # entering state A. Speed just upstream/downstream of the shockwave boundary relative to the shock wave are:

Shock Wave Speed Calculation

What affect Distance Wave Travels? Incident Case Example: Upstream: 1800vph, 50 vpm downstream 1224vph, 170vpm Duration 15 minutes  Queue length = 1.2 mile upstream

Shock Wave Analysis Example: Signalized Intersection Assumptions: Starting with: departure=arrival Same arrival during red interval r (t1 to t2) Maximum departure flow (saturation) after green resumes May have residual queue at the start of next green

Shock Wave @ Signalized Intersection

Shock Wave @ Signalized Intersection

Two-lane Highway Example

Traffic Wave and Bottleneck Bottleneck downstream limit flow upstream Upstream flow can not reach capacity. Max flow limited by downstream bottleneck capacity

Traffic Wave and Bottleneck

Flows around Bottleneck Cap A Cap B q q k k

Measured flow Measured flow = the demand when uncongested Measured flow depends on the capacity of the system, determined by the bottleneck, and also depends on the measurement location: at the point of observation at a point upstream, or at a point downstream

Shock Wave Application Example Example: incident Detecting incident by measuring backward forming queue TRR 461

Shock Wave Example: Incident

Shock Wave Example: Incident

Shock Wave Example: Incident

Shock Wave Example: Incident

Shock Wave Example: Incident

Shock Wave Example: Incident

uf kj u = uf - k k = kj - u Wu1 = Speed of shock wave = uf kj qB - qA kB - kA

Wd1 = Speed of clearing wave (the first Wd1 = Speed of clearing wave (the first wave) moving downstream from the incident. Wu2 = Speed of the capacity boundary wave moving upstream. Wd2 = Speed of the capacity boundary wave moving downstream. Wd3 = Speed of the final clearing wave moving downstream. un = Normal Speed uq = Speed in congested queue.

Wu1 = = - uf + un + uq Wd1 = un - uq Wu2 = - + uq Wd2 = - uq uf qq - qn Wu1 = = - uf + un + uq Wd1 = un - uq Wu2 = - + uq Wd2 = - uq Wd3 = - + un kq - kn uf 2 uf 2 uf 2

uq = Speed in Congested Queue = 1 - 1 - = 1 - 1 - where q = flow under incident conditions qm = available capacity under normal conditions uf q qm 2

Assumptions/Limitations of Shock Wave Theory (1) The capacity over the length of the study section is either constant or changes instantaneously to specific constant values at prespecified points along the study section, (2) The capacity at a location over the entire time duration of the study is either constant or changes instantaneously to specific constant values at prespecified points in time, (3) The demand for service over the length of the study section is constant and there is only one entrance and one exit, (4) The demand for service over the entire time duration of the study is either constant or changes instantaneously to specific constant values at prespecified points in time,

Assumptions/Limitations of Shock Wave Theory (5) A single flow-density relationship is specified for the entire length of the study section, (6) The selected flow-density relationship does not vary over the time duration of the study, (7) Only a single bottleneck is studied and the possibility of queue collisions and queue splits are not considered, (8) All vehicles travel at exactly the same speed for a specific condition on the flow-density relationship, and (9) Drivers do not anticipate changes in downstream flow conditions and are assumed to change their speeds instantaneously only at shock wave boundaries.

Shockwave Applications Crashes/Incidents Congestions Work Zone Limitations; cannot model dynamic flow changes, such as speed change for arriving vehicles. Related issue: variable speed limit (VSL) can improve flow condition and safety (realtime control)

VSL Improve safety More efficient use of highway Less burdened justice system Responsive to dynamic conditions Provide real time information

VSL Evaluation(Effectiveness) Speed limit compliance Credibility of speed limits Safety improvements Traffic flow improvements