## 2.5 Kinematic Wave Model

An extension of the fundamental relationships is to consider speed, flow and density as functions of time (t) and space (x), and the three parameters are not independent of one another. For example, flow is a function of density k, which is a function of time t. A model that considers the traffic process in time and space is the *kinematic* wave model of Lighthill and Whitham (1955), which is more suitable for high density conditions and therefore has its place in analysing flow breakdowns.

The kinematic model assumes that high density traffic will behave like a continuous fluid (hence also called a *continuum* model). Consider the flow in and out of a short length of road \(\mathrm{\partial x}\). The condition of *continuity* requires that if the density of vehicles has increased it must have been due to a difference in the amounts flowing in at one end and out at the other, or

\[\frac{\mathrm{\partial k}}{\mathrm{\partial t}}\mathrm{+}\frac{\mathrm{\partial q}}{\mathrm{\partial x}}\mathrm{ = 0}\] | 2.10 | |

where | ||

q is the flow (veh/h) | ||

k is the density (veh/km) | ||

x is distance (km) | ||

t is time (h) to travel a distance of x km |

With q as a function of density k, Lighthill and Whitham developed Equation 2.10 further into the LW model as follows:

\[\frac{\mathrm{\partial k}}{\mathrm{\partial t}}\mathrm{+}\frac{\mathrm{\partial q}}{\mathrm{\partial k}}\frac{\mathrm{\partial k}}{\mathrm{\partial x}}\mathrm{ = 0}\] | 2.11 |

Define below a *wave speed* U that represents the speed of waves carrying continuous changes of vehicle flow in a traffic stream:

\[\mathrm{U}\mathrm{ }\mathrm{=}\mathrm{ }\frac{\mathrm{\partial q}}{\mathrm{\partial k}}\] | 2.12 | |

then | \[\frac{\mathrm{\partial k}}{\mathrm{\partial y}}\mathrm{ +}\mathrm{ }\mathrm{U}\frac{\mathrm{\partial k}}{\mathrm{\partial x}} = 0\] | 2.13 |

Because q = v k from Equation 2.3, the wave speed:

\[\mathrm{U = }\frac{\mathrm{\partial(}\mathrm{\text{vk}}\mathrm{)}}{\mathrm{\partial k}}\] | ||

\[\mathrm{ = v + k }\frac{\mathrm{\partial v}}{\mathrm{\partial k}}\] | 2.14 |

Because speed decreases with density, \(\frac{\mathrm{\partial v}}{\mathrm{\partial k}}\) is always negative (Figure 2.4) and the wave speed U is therefore always less than the space mean speed v.

The relationship between space mean speed (v) and wave speed (U) are illustrated in the flow‑density diagram in Figure 2.5, which also shows the shock wave speed (USW). The following observations can be made (Wohl & Martin 1967):

- At low densities when vehicle-to-vehicle interactions are minimal, \(\frac{\mathrm{\partial v}}{\mathrm{\partial k}} \)is almost zero and the wave speed is similar to the space mean speed. The wave moves forward relative to the road.
- At the maximum flow and critical density, the wave is stationary. At densities higher than the critical density (k
_{c}), the wave moves backward relative to the road. - The wave speed changes with density according to Equation 2.14 and a traffic stream can have different densities on different sections of a freeway. A section of light traffic could follow a section of high density due to a decrease in lanes, an accident or on-ramp traffic. The wave in the low density traffic moves forward (relative to the freeway) at a speed faster than the wave in the high density traffic.
- When the two waves meet, a new wave called a
*shock wave*will be formed. All three waves move forward for the situation shown in Figure 2.5. The shock wave speed U_{SW}is given by:

\[\mathrm{U}_{\mathrm{\text{sw}}}\mathrm{ = }\frac{\mathrm{q}_{\mathrm{2}} - \mathrm{ }\mathrm{q}_{\mathrm{1}}}{\mathrm{k}_{\mathrm{2}} - \mathrm{ }\mathrm{k}_{\mathrm{1}}}\] | 2.15 |

Figure 2.6 illustrates the case for a negative shock wave speed due to capacity decrease at a bottleneck (e.g. lane drop) on a freeway. Two fundamental diagrams are required. The inner diagram represents the characteristics of the bottleneck with capacity q_{b} less than the approach section. If the approach flow is larger than q_{b}, a complex queuing situation occurs at the entry to the bottleneck.

The density at the bottleneck entry suddenly increases from the density at C to the density at E in Figure 2.6. The wave speed at E is negative with respect to the freeway and will be reflected from the bottleneck back to the approach section. The reflected wave will meet the oncoming wave corresponding to the slope at C. A shock wave of negative speed relative to the freeway is formed. The effect of the bottleneck will be reflected along the entire approach section if the arrival flow remains constant, with a consequent loss of maintaining capacity flow (q_{m}).

Edie and Foote (1958) reported how shock waves were generated at an upgrade leading to the Holland Tunnel exit in New York. The shock waves propagated backward towards the tunnel entry with inefficient traffic flow. The solution was to control the entry of vehicles into the tunnel so that the entry flow did not exceed the capacity of the bottleneck section. The vehicles entered in short platoons of about 40 veh every 2 min with a 10 s gap between platoons.

The kinematic model can be solved using the finite difference (or finite element) method and has continued to be an interesting area of research (see, e.g. Leo & Pretty 1992, Michalopolous 1988, Ngoduy, Hoogendoorn & Van Zuylen 2006, Papageorgiou 1983, Payne 1971). At the University of Queensland, Leo and Pretty were able to model the propagation of congested density upstream in a freeway lane drop situation. They also modelled the platoon movements in a pair of coordinated signals at very small, discrete levels of time (0.5 to 1 s) and space (about 15 m).

The LW model is a first order model with limitations such as (Papageorgiou 1998):

- Assume that vehicle speeds can change instantaneously, i.e. large values of acceleration and deceleration rates are assumed possible at a bottleneck (E in Figure 2.6).
- Predict that the tail-end of a platoon on arterial roads will speed up to catch up with the main platoon when it is more common to observe a dispersed tail-end.

Assume that outflow (q_{b}) at a freeway bottleneck is best achieved with some congestion at the bottleneck entry. This is equivalent to assuming that the outflow cannot be increased by avoiding mainline congestion, i.e. no control. The reality is that some control of a bottleneck (if possible) can improve throughput.

Second order LW models have been proposed (Daganzo 2006, Papageorgiou, Blosseville & Hadj‑Salem 1990, Payne 1971, Schonhof and Helbing 2007) to overcome these limitations.

The use of Kinematic models in understanding freeway flow breakdowns is discussed further in Section 7.5.4.