ρ_t + j’(ρ)ρ_x = 0where subscripts indicate partial differentiation and j’ indicates differentiation with respect to ρ. If we make j(ρ)=4ρ(2-ρ) and start with a discontinuous initial density distribution like
1 if x <= 1
1/2 if 1< x <=3
2/3 if x > 3
Such persisting discontinuities are called shock waves and appear as lines across which the density changes discontinuously. For example, in this figure we have lines of constant density intersecting at x = 3. The philosophical question is "what are we to make of this representation of a given traffic system?" That is, what does the system have to be like for the representation of a shock wave to be correct? My suggestion is that we need only a thin strip around x = 3 where the density changes very quickly, i.e. so quickly that a driver crossing it would have to decelerate to zero speed. Then, on the other side of the strip, the driver experiences a dramatic drop off in density, and so can accelerate again. Still, there is something a bit strange in talking about shock waves in traffic cases where the number of objects involved is so small, as opposed to fluid cases where many more fluid particles interact across a shock wave. Here, then, I would suggest that we have a case where the mathematics works, but we are less than sure what it is representing in the world.
See this New Scientist article (and amusing video) for the claim that shock waves can be observed in actual traffic experiments (summarizing this 2008 article).