Traffic flow

Stop-and-go traffic

The little blobs are cars moving along a highway. They stop and go according to the traffic signal. Cars in motion are drawn in green. Those that are stopped are drawn in red. The animation is produced according to the traffic model of Chapter 3.

The rest of this Web page illustrate various solutions of the traffic flow IVP: \begin{align*} &\frac{\partial \rho}{\partial t} + c(\rho) \frac{\partial \rho}{\partial x} = 0 &&-\infty < x < \infty, \quad t>0, \\ &\rho(x,0)=f(x) && -\infty < x < \infty, \end{align*} where the traffic velocity is a linear function of the traffic density: \[ v = \sigma(\rho) = \vmax \Bigl(1 - \frac{\rho}{\rhomax}\Bigr), \] and consequently \[ c(\rho) = \vmax \Bigl(1 - \frac{2\rho}{\rhomax} \Bigr). \]

Example 1: A rarefaction wave

We take $\vmax=1$, $\rhomax=1$, and \[ f(x) = \begin{cases} 1 & \text{if } x <0, \\ 0 & \text{if } x >0. \end{cases} \] Here is how the initial density evolves over the time interval $0 \le t \le 3$.

Example 2: A rarefaction wave

We take $\vmax=1$, $\rhomax=8$, and \[ f(x) = \begin{cases} 5 & \text{if } x <0, \\ 2 & \text{if } x >0. \end{cases} \] Here is how the initial density evolves over the time interval $0 \le t \le 8$.

Example 3: A rarefaction wave

We take $\vmax=4$, $\rhomax=8$, and \[ f(x) = \begin{cases} 5 & \text{if } x <0, \\ 5 - 2x &\text{if } 0 < x < 1, \\ 3 & \text{if } x > 1. \end{cases} \] Here is how the initial density evolves over the time interval $0 \le t \le 2.5$.

Example 4: A shock wave

We take $\vmax=3$, $\rhomax=6$, and \[ f(x) = \begin{cases} 2 & \text{if } x <0, \\ 5 & \text{if } x > 0. \end{cases} \] Here is how the initial density evolves over the time interval $0 \le t \le 2.5$.

Example 5: A shock wave

We take $\vmax=3$, $\rhomax=6$, and \[ f(x) = \begin{cases} 2 & \text{if } x <0, \\ 2+x & \text{if } 0 < x < 3, \\ 5 & \text{if } x > 3. \end{cases} \] Here is how the initial density evolves over the time interval $0 \le t \le 9$.

Shock waves in transonic flight

The phrase “shock wave” originates in gas dynamics. When an object/projectile moves faster than the speed of sound in air (or any other gas-like medium,) it develops a shock wave ahead of it. Here is a rather simplistic explanation of the phenomenon. The object pushes the air in its direction of motion, slightly compressing it. The compressed air will normally expand and run away from the projectile, traveling in air at the speed of sound, that is the speed at which small disturbances propagate in the medium. But if the projectile is moving faster than the speed of sound, the compressed air has no chance to run away since it is chased by even a faster moving projectile. The compressed air accumulates behind a sharp interface, the shock, which is pushed forward by the projectile. The air pressure, and its density and temperature, change abruptly across the shock. The photograph below is taken in a NASA wind-tunnel of an aircraft model speeding through air at 1.2 times the speed of sound. The light and dark shades of gray indicate variations of density. Multiple shock waves are visible surrounding the model.

If an actual aircraft flies overhead at faster than speed of sound, the shock wave is perceived on the ground as a sonic boom. Why? As the shock wave sweeps over a ground observer, the pressure in the observer's ear drum changes abruptly from the normal atmospheric pressure to the higher pressure that exists behind the shock. That abrupt change in pressure is perceived as a loud popping sound, not unlike what one hears near a popping balloon. The animation below is meant to convey that event.