Module 1 : Traffic Stream Characteristics
Lecture 03 : Traffic Stream Models
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Shock waves

The flow of traffic along a stream can be considered similar to a fluid flow. Consider a stream of traffic flowing with steady state conditions, i.e., all the vehicles in the stream are moving with a constant speed, density and flow. Let this be denoted as state A (refer figure 1. Suddenly due to some obstructions in the stream (like an accident or traffic block) the steady state characteristics changes and they acquire another state of flow, say state B. The speed, density and flow of state A is denoted as $ v_A$, $ k_A$, and $ q_A$, and state B as $ v_B$, $ k_B$, and $ q_B$ respectively.
Figure 1: Shock wave: Stream characteristics
\begin{figure} \centerline{\epsfig{file=t16-shockwave-stream-char.eps,width=8cm}} \end{figure}
The flow-density curve is shown in figure 2.
Figure 2: Shock wave: Flow-density curve
\begin{figure} \centerline{\epsfig{file=t18-shockwave.eps,width=8cm}} \end{figure}
The speed of the vehicles at state A is given by the line joining the origin and point A in the graph. The time-space diagram of the traffic stream is also plotted in figure 3.
Figure 3: Shock wave : time-distance diagram
\begin{figure} \centerline{\epsfig{file=t17-shockwave-distance-time-graph.eps,width=8cm}} \end{figure}
All the lines are having the same slope which implies that they are moving with constant speed. The sudden change in the characteristics of the stream leads to the formation of a shock wave. There will be a cascading effect of the vehicles in the upstream direction. Thus shock wave is basically the movement of the point that demarcates the two stream conditions. This is clearly marked in the figure 2. Thus the shock waves produced at state B are propagated in the backward direction. The speed of the vehicles at state B is the line joining the origin and point B of the flow-density curve. Slope of the line AB gives the speed of the shock wave (refer figure 2). If speed of the shock-wave is represented as $ \omega_{AB}$, then

$\displaystyle \omega_{AB} = \frac{q_A-q_B}{k_A-k_B}$ (1)

The above result can be analytically solved by equating the expressions for the number vehicles leaving the upstream and joining the downstream of the shock wave boundary (this assumption is true since the vehicles cannot be created or destroyed. Let $ N_A$ be the number of vehicles leaving the section A. Then, $ N_A=q_B~t$. The relative speed of these vehicles with respect to the shock wave will be $ v_A-\omega_{AB}$. Hence,

$\displaystyle N_A=k_A~(v_A-\omega_{AB})~t$ (2)

Similarly, the vehicles entering the state B is given as

$\displaystyle N_B=k_A~(v_B-\omega_{AB})~t$ (3)

Equating equations 2 and 3, and solving for $ \omega_{AB}$ as follows will yield to:
$\displaystyle N_A$ $\displaystyle =$ $\displaystyle N_B$  
$\displaystyle k_A~(v_A-\omega_{AB})~t$ $\displaystyle =$ $\displaystyle k_B~(v_B-\omega_{AB})~t$  
$\displaystyle k_A~v_A~t-k_A~\omega_{AB}~t$ $\displaystyle =$ $\displaystyle k_B~v_B~t-k_B\omega_{AB}~t$  
$\displaystyle k_A\omega_{AB}~t-k_B\omega_{AB}~t$ $\displaystyle =$ $\displaystyle k_A~v_A-k_B~v_B$  
$\displaystyle \omega_{AB}~(k_A-k_B)$ $\displaystyle =$ $\displaystyle q_A-q_B$  

This will yield the following expression for the shock-wave speed.

$\displaystyle \omega_{AB} = \frac{q_A-q_B}{k_A-k_B}$ (4)

In this case, the shock wave move against the direction of traffic and is therefore called a backward moving shock wave. There are other possibilities of shock waves such as forward moving shock waves and stationary shock waves. The forward moving shock waves are formed when a stream with higher density and higher flow meets a stream with relatively lesser density and flow. For example, when the width of the road increases suddenly, there are chances for a forward moving shock wave. Stationary shock waves will occur when two streams having the same flow value but different densities meet.