Traffic Waves

Equations

We consider a single lane of road, and we measure distance along the road with the variable [math]\displaystyle{ x }[/math] and [math]\displaystyle{ t }[/math] is time. We define the following variables

[math]\displaystyle{ \begin{matrix} &\rho(x,t) &: &\mbox{car density (cars/km)} \\ & v(\rho) &: &\mbox{car velocity (km/hour)} \\ & q(x,t) =\rho v &: &\mbox{car flow rate (cars/hour)} \\ \end{matrix} }[/math]

If we consider a finite length of road [math]\displaystyle{ x_1\leq x \leq x_2 }[/math] then the net flow of cars in and out must be balanced by the change in density. This means that

[math]\displaystyle{ \frac{\partial}{\partial t} \int_{x_1}^{x_2} \rho(x,t) dx = -q(x_2,t) + q(x_1,t) }[/math]

We now consider continuous densities (which is obviously an approximation) and set [math]\displaystyle{ x_2 = x_1 + \Delta x }[/math] and we obtain

[math]\displaystyle{ \frac{\partial}{\partial t} \rho(x_1,t) = -\frac{q(x_2,t) + q(x_1,t)}{\Delta x} }[/math]

and if we take the limit as [math]\displaystyle{ \Delta x \to 0 }[/math] we obtain the differential equation

[math]\displaystyle{ \frac{\partial \rho}{\partial t} + \frac{\partial q}{\partial x} = 0 }[/math]

Note that this equation has been derived purely from the need to conserve cars and it currently is not possible to solve until we have derived a connection between [math]\displaystyle{ \rho }[/math] and [math]\displaystyle{ q }[/math].

Relationship between [math]\displaystyle{ \rho }[/math] and [math]\displaystyle{ q }[/math]

The simplest relationship between [math]\displaystyle{ \rho }[/math] and [math]\displaystyle{ q }[/math] is derived from the following assumptions

• When the density [math]\displaystyle{ \rho = 0 }[/math] the speed is [math]\displaystyle{ v=v_0 }[/math]
• When the density is [math]\displaystyle{ \rho = \rho_{\max} }[/math] the speed is [math]\displaystyle{ v=0 }[/math]
• The speed is a linear function of [math]\displaystyle{ \rho }[/math] between these two values.

This gives us

[math]\displaystyle{ v(\rho) = v_0\frac{\rho_{\max} - \rho}{\rho_{\max}} }[/math]