Section 2: The Gate Keepers and the Green Wave.

Question 17

We are going to further investigate what happens at the front of the race. Let’s start with the river flow. Imagine now we have a gate every \(200 \mathrm{m}\) (like traffic signals at intersections).

Model Answer

The problem is easy as there is no dispersion at the front. We only need to have an offset of \(\frac{200 \mathrm{m}}{4.1 \mathrm{m/s}} = 48.8 \mathrm{s}\).

You have just created a green wave for river flow (even though not really useful as there are no opposite flows passing during red time).

Question 18

Model Answer

\[\begin{split} \begin{align} x &= 200 \mathrm{m} \\ t_2 - t_1 &= 18.2 - 15.4 \\ &= 2.8 \mathrm{s} \end{align} \end{split}\]

Think \(s = v \cdot t\). We know \(s\) and \(v_{min}\) and \(v_{max}\), so we can calculate \(t_2 - t_1\).

Similarly, for the rest,

\[\begin{split} \begin{align} &x = 400 \mathrm{m}, &&t_2 - t_1 = 5.6 \mathrm{s} \\ &x = 600 \mathrm{m}, &&t_2 - t_1 = 8.4 \mathrm{s} \\ &x = 800 \mathrm{m}, &&t_2 - t_1 = 11.2 \mathrm{s} \\ &x = 1000 \mathrm{m}, &&t_2 - t_1 = 14 \mathrm{s} \end{align} \end{split}\]

Question 19

By using similar (but more advance as \(T\) is now in the middle of the rarefaction waves) technique as in Q5, it is possible to calculate the time required for one vehicle (cumulative flow value equal to 1) to completely cross each intersection after the front wave has passed. The values are \(4.1 \mathrm{s}, 5 \mathrm{s}, 5.7 \mathrm{s}, 6.4 \mathrm{s}\) for the intersections at \(x = 400\), \(600\), \(800\), and \(1000 \mathrm{m}\) respectively.

Model Answer

A rarefaction fan represents the dispersion of the front wave. The further we are from the initial discontinuity, the smoother the transition in flow is.

Question 20

Model Answer

You want to minimize the total delay in the system (collective optimum). If dispersion is observed it is useless to provide green immediately to the very first vehicles as the total outflow will not be maximum. It is better to stop them and wait for the main vehicle pack, especially if the possible green time is limited.