Shock Waves

Whenever a stream of traffic flowing under certain stream conditions (say, speed = $u_A$, density = $k_A$, and flow = $q_A$) meets another stream flowing under different conditions (say, speed = $u_B$, density = $k_B$, and flow = $q_B$) a shock wave is started. The shock wave is basically the movement of the point that demarcates the two stream conditions. This demarcation point may move forward or backward or stay at the same place with respect to the road. The rate at which this demarcation point moves (the direction of motion of the vehicles is taken as the positive direction) is referred to as the speed of the shock wave.

In order to see the generation and movement of shock waves consider the distance-time graph shown in Figure 1. This figure is drawn for the following situation. A slow moving vehicle with speed $u_B$ (whose distance-time plot is shown as a dotted line) enters a traffic stream originally moving at $u_A$, $k_A$, $q_A$. This slow moving vehicle slows the traffic and creates another flow condition denoted by $u_B$, $k_B$, $q_B$. From the graph it can be seen that there exists a line (a bold line marked Shock wave 1) that is the locus of the point that demarcates the two flow conditions at any given time. After a while at point Q the slow moving vehicle leaves the traffic stream and the congested condition created by the slow moving vehicle is released say at some other stream condition (denoted by $u_C$, $k_C$, $q_C$). Obviously, the point that demarcates the flow conditions $u_B$, $k_B$, $q_B$ from the flow conditions $u_C$, $k_C$, $q_C$ also causes a shock wave. The locus of this point is marked as Shock wave 2 in the figure. At some time $t$ these two shock waves meet signalling the end of the flow conditions $u_B$, $k_B$, $q_B$. But this time the flow conditions $u_A$, $k_A$, $q_A$ meet the flow conditions $u_C$, $k_C$, $q_C$ starting Shock wave 3 (see figure). Interestingly, the vacant area in the figure bounded by the line denoting the motion of last vehicle to go without being caught behind the slow moving vehicle, the dotted line denoting the motion of the slow moving vehicle, and the line denoting the motion of the first vehicle to get released after the departure of the slow moving vehicle, also represent a flow condition; namely the free flow conditions. In the figure this zone is indicated as a stream with conditions $u=u_f$ (the free flow speed), $k=0$, and $q=0$. Hence when the stream condition $u_B$, $k_B$, $q_B$ meets the stream condition $u=u_f$, $k=0$, and $q=0$ a shock wave should and does emanate. The only difference here is that the point which demarcates the two conditions is the same as the point representing the slow moving vehicle; hence, the shock wave that emanates here moves with the slow moving vehicle and the time-distance diagram of this shock wave is the same as the time-distance diagram of the slow moving vehicle. Similarly, the time-distance diagram of the first vehicle that is released is the time-distance diagram of the shock wave that emanates when flow condition $u_C$, $k_C$, $q_C$ meets $u=u_f$, $k=0$, and $q=0$.

図 1: Distance time diagram illustrating the creation and movement of shock waves.

From the above discussion it is clear that the key parameters of a shock wave are (i) the point at which it starts, (ii) the point at which it ends, and (iii) the speed of the shock wave. As will be apparent later, of these, the only parameter which needs to be studied in details is the speed of the shock wave. This is topic of the next section.