Traffic: pile up

Source code for images: code.wl
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Problem

\gdef \Vmax {V_\mathrm{max}} \gdef \Nmax {N_\mathrm{max}} \gdef \Fmax {F_\mathrm{max}} \gdef \Vsh {V_\mathrm{shock}} \gdef \Vb {V_\mathrm{before}} \gdef \Va {V_\mathrm{after}} \gdef \Nb {N_\mathrm{before}} \gdef \Na {N_\mathrm{after}} \gdef \Fb {F_\mathrm{before}} \gdef \Fa {F_\mathrm{after}}

Cars travelling at 60% maximum density (40% of the speed limit) are faced with a denser & slower region ahead, at 80% maximum density (20% of the speed limit). In symbols,

\eval{N}_{t = 0} = \begin{cases} 0.6 \Nmax, & x < 0 \\ 0.8 \Nmax, & x > 0 \end{cases}.

What happens?

Animation

Animation for pile up

Characteristics

We have the following:

Qty Before After
N 0.6 \Nmax 0.8 \Nmax
V 0.4 \Vmax 0.2 \Vmax
F 0.24 \Nmax\Vmax 0.16 \Nmax\Vmax
\frac{\td F}{\td N} -0.2 \Vmax -0.6 \Vmax

Thus we have characteristics…

We have a problem: the 'after' characteristics (slow & dense) have a higher signal speed than the 'before' characteristics (fast & sparse), so eventually they will collide. What happens?

Shockwave

\gdef \shock #1 {\colr{#1}}

Unlike the traffic light example where we had a transition fan (moving from slow & dense to fast & sparse), here we have a shockwave at the interface where fast & sparse abruptly becomes slow & dense.

Suppose the shockwave interface travels at velocity \Vsh. The relative velocity of cars is \Vb - \Vsh on the 'before' side and \Va - \Vsh on the 'after' side. Since flux (density times velocity) must be conserved, we have

\Nb (\Vb - \Vsh) = \Na (\Va - \Vsh)

and therefore the shockwave speed is

\begin{aligned} \shock{\Vsh} &= \frac{\Na \Va - \Nb \Vb}{\Na - \Nb} \\[\tallspace] &= \frac{\Fa - \Fb}{\Na - \Nb} \end{aligned}

Note that this is the slope of the secant line through the points (\Nb, \Fb) and (\Na, \Fa) on the flux–density parabola.

For the current problem we get \shock{\Vsh = -0.4 \Vmax}, and so the shockwave interface is the line x = \shock{-0.4 \Vmax} t. This separates the 'before' region (fast & sparse) and the 'after' region (slow & dense).

Result

N (x, t) = \begin{cases} 0.6 \Nmax, & x < -0.4 \Vmax t \\ 0.8 \Nmax, & x > -0.4 \Vmax t \end{cases}

Remarks

Animation for pile up (again)

In real life we obviously don't brake instantly, but abrupt reductions in speed do propagate backward in a shockwave, e.g. when an idiot doesn't get up to speed before doing a merge.

END

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