# Fundamentals of Transportation/Shockwaves

Shockwaves are byproducts of traffic congestion and queueing. They are transition zones between two traffic states that move through a traffic environment like, as their name states, a propagating wave. On the urban freeway, most drivers can identify them as a transition from a flowing, speedy state to a congested, standstill state. However, shockwaves are also present in the opposite case, where drivers who are idle in traffic suddenly are able to accelerate. Shockwaves are one of the major safety concerns for transportation agencies because the sudden change of conditions drivers experience as they pass through a shockwave often can cause accidents.

## Visualization

While most people have probably experienced plenty of traffic congestion first hand, it is useful to see it systematically from three different perspectives: (1) That of the driver (with which most people are familiar), (2) a birdseye view, and (3) a helicopter view. Some excellent simulations are available here, please see the movies:

This movie shows traffic jams without an "obvious source" such as an on-ramp, but instead due to randomness in driver behavior: Shockwave traffic jams recreated for first time

## Analysis of shockwaves

Shockwave Diagram
Shockwave Diagram

Shockwaves can be seen by the cascading of brake lights upstream along a highway. They are often caused by a change in capacity on the roadways (a 4 lane road drops to 3), an incident, a traffic signal on an arterial, or a merge on freeway. As seen above, just heavy traffic flow alone (flow above capacity) can also induce shockwaves. In general, it must be remembered that capacity is a function of drivers rather than just being a property of the roadway and environment. As the capacity (maximum flow) drops from ${\displaystyle C_{1}}$ to ${\displaystyle C_{2}}$, optimum density also changes. Speeds of the vehicles passing the bottleneck will of course be reduced, but the drop in speed will cascade upstream as following vehicles also have to decelerate.

The figures illustrate the issues. On the main road, far upstream of the bottleneck, traffic moves at density ${\displaystyle k_{1}}$, below capacity (${\displaystyle k_{opt}}$). At the bottleneck, density increases to accommodate most of the flow, but speed drops.

## Shockwave Math

### Shockwave speed

If the flow rates in the two sections are ${\displaystyle q_{1}}$ and ${\displaystyle q_{2}}$, then ${\displaystyle q_{1}=k_{1}v_{1}}$ and ${\displaystyle q_{2}=k_{2}v_{2}}$.

${\displaystyle v_{w}={\frac {q_{2}-q_{1}}{k_{2}-k_{1}}}\,\!}$

### Relative speed

With ${\displaystyle v_{1}}$ equal to the space mean speed of vehicles in area 1, the speed relative to the line ${\displaystyle w}$ is:

${\displaystyle v_{r1}=v_{1}-v_{w}\,\!}$

The speed of vehicles in area 2 relative to the line w is ${\displaystyle v_{r2}=v_{2}-v_{w}\,\!}$

### Boundary crossing

The number of vehicles crossing line 2 from area 1 during time period ${\displaystyle t}$ is

${\displaystyle N_{1}=v_{r1}k_{1}t=(v_{1}-v_{w})k_{1}t\,\!}$

and similarly

${\displaystyle N_{2}=v_{r2}k_{2}t=(v_{2}-v_{w})k_{2}t\,\!}$

By conservation of flow, the number of vehicles crossing from left equals the number that crossed on the right

${\displaystyle N_{1}=N_{2}\,\!}$

so:

${\displaystyle v_{2}k_{2}-v_{1}k_{1}=v_{w}(k_{2}-k_{1})\,\!}$

or

${\displaystyle q_{2}-q_{1}=v_{w}(k_{2}-k_{1})\,\!}$

which is equivalent to

${\displaystyle v_{w}={\frac {q_{2}-q_{1}}{k_{2}-k_{1}}}\,\!}$

## Examples

### Example 1

Problem:

The traffic flow on a highway is ${\displaystyle q_{1}=2000veh/hr}$ with speed of ${\displaystyle v_{1}=80km/hr}$. As the result of an accident, the road is blocked. The density in the queue is ${\displaystyle k_{2}=275veh/km}$. (Jam density, vehicle length = 3.63 meters).

• (A) What is the wave speed (${\displaystyle v_{w}}$)?
• (B) What is the rate at which the queue grows, in units of vehicles per hour (${\displaystyle q}$)?
Solution:

(A) At what rate does the queue increase?

1. Identify Unknowns:

${\displaystyle k_{1}=q/v_{1}=2000/80=25veh/km\,\!}$

${\displaystyle v_{2}=0,q_{2}=k_{2}v_{2}=0\,\!}$

2. Solve for wave speed (${\displaystyle v_{w}}$)

${\displaystyle v_{w}={\frac {q_{2}-q_{1}}{k_{2}-k_{1}}}={\frac {0-2000}{275-25}}=-8km/hr\,\!}$

Conclusion: the queue grows against traffic

(B) What is the rate at which the queue grows, in units of vehicles per hour?

${\displaystyle {\begin{array}{l}N_{1}=\left({v_{1}-v_{w}}\right)k_{1}t=\left({v_{2}-v_{w}}\right)k_{2}t=N_{2}\\dropping\quad t\;\left({let\;t=1}\right)\\v_{1}k_{1}-v_{w}k_{1}=v_{2}k_{2}-v_{w}k_{2}\\q_{1}-v_{w}k_{1}=q_{2}-v_{w}k_{2}\\2000-(-8)*25=0-(-8)*275\\2200veh/hr=2200veh/hr\\\end{array}}\,\!}$

## Thought Question

Problem

Shockwaves are generally something that transportation agencies would like to minimize on their respective corridor. Shockwaves are considered a safety concern, as the transition of conditions can often lead to accidents, sometimes serious ones. Generally, these transition zones are problems because of the inherent fallibility of human beings. That is, people are not always giving full attention to the road around them, as they get distracted by a colorful billboard, screaming kids in the backseat, or a flashy sports car in the adjacent lanes. If people were able to give full attention to the road, would these shockwaves still be causing accidents?

Solution

Yes, but not to the same extent. While accidents caused by driver inattentiveness would decrease nearly to zero, accidents would still be occurring between different vehicle types. For example, in a case where conditions change very dramatically, a small car (say, a Beetle) would be able to stop very quickly. A semi truck, however, is a much heavier vehicle and would require a longer distance to stop. If both were moving at the same speed when encountering the shockwave, the truck may not be able to stop in time before smashing into the vehicle ahead of them. That is why most trucks are seen creeping along through traffic with very big gaps ahead of them.

## Variables

• ${\displaystyle q}$ - flow
• ${\displaystyle c}$ - capacity (maximum flow)
• ${\displaystyle k}$ - density
• ${\displaystyle v}$ - speed
• ${\displaystyle v_{r}}$ - relative speed (travel speed minus wave speed)
• ${\displaystyle v_{w}}$ - wave speed
• ${\displaystyle N}$ - number of vehicles crossing wave boundary

## Key Terms

• Shockwaves
• Time lag, space lag