Waves, not cars - modelling traffic as a fluid

A transcript of this video with snippets of animation is available at https://matsvermeeren.xyz/waves-not-c...

We give a visual introduction to the Lighthill-Whitham-Richards model of traffic flow and the method of characteristics. Through this model, we study examples of light and heavy motorway traffic, traffic lights, and the use of variable speed limits.

00:00 Intro
00:35 Traffic as a fluid
01:51 Speed and flow
03:25 Derivation of traffic equation
05:17 Method of characteristics
08:22 Reconstructing the density plot
11:14 Motorway examples
13:32 A traffic light
15:04 Active traffic management

COPYRIGHT CC-BY-SA

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BIBLIOGRAPHICAL NOTES

The Lighthill-Whitham-Richards model is also known as the kinematic wave model of traffic flow (https://en.wikipedia.org/wiki/Traffic....

The original paper by Lighthill and Whitham is surprisingly readable:
Lighthill, Whitham (1955) On kinematic waves II. A theory of traffic flow on long crowded roads. https://doi.org/10.1098/rspa.1955.0089

A textbook reference we found useful in making this video is:
Richard Haberman (1977) Mathematical Models: Mechanical Vibrations, Population Dynamics, and Traffic Flow. Prentice-Hall, New Yersey.

Another useful textbook, with more mathematical detail, is
Sandro Salsa, Gianmaria Verzini (2022) Partial Differential equations in Action. Fourth edition. Springer, Cham.

Optimising traffic flow through measures like variable speed limits is called active traffic management: https://en.wikipedia.org/wiki/Active_...

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ACKNOWLEDGMENTS

This video is supported by the Engineering and Physical Sciences Research Council (EP/Y006712/1) and Loughborough University as part of my outreach work. Any mistakes or opinions in this video are my own.

Animations created with help of Matthew Truman.

Additional media:
Motorway photos by David Dixon (https://www.geograph.org.uk/photo/370...) and Simon Green (https://commons.wikimedia.org/wiki/Fi...)
Molecular dynamics animation by Christopher Rowley (https://commons.wikimedia.org/wiki/Fi...)
Wave dynamics animation by Kraaiennest (https://commons.wikimedia.org/wiki/Fi...)

Music: "Oxalis-1 (Excerpt)" by Alex Bainter (https://play.generative.fm/generator/...)

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NOTES ON THE ANIMATIONS

In theory, a shock will never disappear, but when a shock gets so weak that the difference in density on either side is negligible, we may for all practical purposes ignore it, which is why we stop drawing some of them on the space-time plot.

Manim code is available at https://github.com/mvermeeren/SoME4

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#SoME4 #Mathematics #Traffic #LWRModel #KinematicWaveModel #Characteristics #some