Shock Waves

A slow moving truck drives along the roadway at 10 MPH. The existing conditions on the roadway before the truck enters are shown at point 1 below: 40 mph, flow of 1000 vehicles per hour, and density of 25 vehicles per mile. The truck enters the roadway and causes a queue of vehicles to build, giving the characteristics of point 2 below: flow of 1200 vehicles per hour and a density of 120 vehicles per mile. Using the information provided below, find the velocity of the shockwave at the front and back of the platoon.

Point 1: Normal flow ( u_s = 40 MPH, k=25 veh/mi, q= 1000 vph.

Point 2: Slow Truck: ( u_s = 10 MPH, k=120 veh/mi, q= 1200 vph.

[Solution Shown Below]

Solution

Figures 3.6.2 and 3.6.3, shown below, illustrate the behavior of the vehicles that are impacted by the shockwave.

The speed of the shockwave in front of the truck at point A-A ( q_b= 0, k_b = 0) can be found by substituting the correct values into the general shockwave equation. Upon substitution, as shown below, we find that the shockwave is moving at the same speed as the truck, or 10 MPH downstream with reference to a stationary point on the roadway.

Solving for the speed of the shockwave at the end of the platoon (B-B) is accomplished by substituting the correct values into the general shockwave equation.

q_a= 1000 vph, k_a=25 vpm

q_b= 1200 vph, k_b =120 vpm

The (+) sign indicates that the shockwave is moving downstream with respect to a fixed observer.

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

Problem adapted from:
Papacostas, C.S., and Prevedourous P.D., Transportation Engineering and Planning, 2nd Edition, Prentice Hall, pages 151-157