# 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:
Visualization with the TU-Dresden 3D Traffic Simulator
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

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 $C_{1}$ to $C_{2}$, 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 $k_{1}$, below capacity ($k_{opt}$). 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 $q_{1}$ and $q_{2}$, then $q_{1}=k_{1}v_{1}$ and $q_{2}=k_{2}v_{2}$.
$v_{w}={\frac {q_{2}-q_{1}}{k_{2}-k_{1}}}\,\!$

### Relative speed

With $v_{1}$ equal to the space mean speed of vehicles in area 1, the speed relative to the line $w$ is:
$v_{r1}=v_{1}-v_{w}\,\!$
The speed of vehicles in area 2 relative to the line w is $v_{r2}=v_{2}-v_{w}\,\!$

### Boundary crossing

The number of vehicles crossing line 2 from area 1 during time period $t$ is
$N_{1}=v_{r1}k_{1}t=(v_{1}-v_{w})k_{1}t\,\!$
and similarly
$N_{2}=v_{r2}k_{2}t=(v_{2}-v_{w})k_{2}t\,\!$
By conservation of flow, the number of vehicles crossing from left equals the number that crossed on the right
$N_{1}=N_{2}\,\!$
so:
$v_{2}k_{2}-v_{1}k_{1}=v_{w}(k_{2}-k_{1})\,\!$
or
$q_{2}-q_{1}=v_{w}(k_{2}-k_{1})\,\!$
which is equivalent to
$v_{w}={\frac {q_{2}-q_{1}}{k_{2}-k_{1}}}\,\!$

## Examples
### Example 1

The traffic flow on a highway is $q_{1}=2000veh/hr$ with speed of $v_{1}=80km/hr$. As the result of an accident, the road is blocked. The density in the queue is $k_{2}=275veh/km$. (Jam density, vehicle length = 3.63 meters).
* (A) What is the wave speed ($v_{w}$)?
* (B) What is the rate at which the queue grows, in units of vehicles per hour ($q$)?
(A) At what rate does the queue increase?
1. Identify Unknowns:
$k_{1}=q/v_{1}=2000/80=25veh/km\,\!$
$v_{2}=0,q_{2}=k_{2}v_{2}=0\,\!$
2. Solve for wave speed ($v_{w}$)
$v_{w}={\frac {q_{2}-q_{1}}{k_{2}-k_{1}}}={\frac {0-2000}{275-25}}=-8km/hr\,\!$
Conclusion: the queue grows against traffic
(B) What is the rate at which the queue grows, in units of vehicles per hour?
${\begin{array}{l}N_{1}=\left({v_{1}-v_{w}}\right)k_{1}t=\left({v_{2}-v_{w}}\right)k_{2}t=N_{2}\\dropping\quad t\;\left({let\;t=1}\right)\\v_{1}k_{1}-v_{w}k_{1}=v_{2}k_{2}-v_{w}k_{2}\\q_{1}-v_{w}k_{1}=q_{2}-v_{w}k_{2}\\2000-(-8)*25=0-(-8)*275\\2200veh/hr=2200veh/hr\\\end{array}}\,\!$

## 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.

## Sample Problem

* Problem (Solution)

## Additional Questions

* Additional Questions
* Homework

## Variables

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

## Key Terms

* Shockwaves
* Time lag, space lag

## Videos

* Shockwaves
* Shockwave Formation and Dissipation