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]

Equation for [math]\displaystyle{ \rho }[/math] only

If we substitute the expression for [math]\displaystyle{ q }[/math] into our differential equation we obtain

[math]\displaystyle{ \frac{\partial \rho}{\partial t} + \frac{\partial }{\partial x} \left(v(\rho)\rho\right) = 0 }[/math]

which gives us

[math]\displaystyle{ \frac{\partial \rho}{\partial t} + \left(v^{\prime}(\rho)\rho + v(\rho)\right) \frac{\partial \rho }{\partial x} = 0 }[/math]

or

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

where [math]\displaystyle{ c(\rho) = \left(v^{\prime}(\rho)\rho + v(\rho)\right) }[/math] is the kinematic wave speed. Note that this is not the speed of the cars, but the speed at which disturbances in the density travel.

Small Amplitude Disturbances

We can linearise the model by assuming that the variation in density is small so that we can write

[math]\displaystyle{ \rho = \rho_0 + \tilde{\rho} }[/math]

where we assume that [math]\displaystyle{ \tilde{\rho} }[/math] is small. This allows us to write the equations as

[math]\displaystyle{ \frac{\partial \tilde{\rho}}{\partial t} + c(\rho_0) \frac{\partial \tilde{\rho}}{\partial x} = 0 }[/math]

where the main difference between this and the full equation is that we assume that [math]\displaystyle{ c }[/math] is a constant. This linearises the equation.

Under these assumptions the solution to the equation is

[math]\displaystyle{ \tilde{\rho} = f(x - c(\rho_0)t) }[/math]

where [math]\displaystyle{ f }[/math] is determined by the initial condition. This represents disturbances which travel with speed [math]\displaystyle{ c(\rho_0) }[/math] in the positive [math]\displaystyle{ x }[/math] direction.

We now consider the characteristic curves which are curves along which the density [math]\displaystyle{ \rho }[/math] is a constant. These are give by

[math]\displaystyle{ x = X(t) = x_0 + c(\rho_0) t. }[/math]

Nonlinear Initial Value Problem

We wish to solve

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

subject to the initial conditions

[math]\displaystyle{ \rho(x,0) = \rho_0(x) }[/math]

It turns out that the concept of characteristic curves is very important for this problem.

If we want [math]\displaystyle{ \rho(X(t),t) }[/math] to be a constant then we require

[math]\displaystyle{ \frac{d}{dt}\rho(X(t),t) = \frac{d X}{dt} \frac{\partial \rho}{\[partial t }[/math]