Kinematic Waves and Real-World Nonlinear Problems

Abstract

This chapter deals with the theory and applications of kinematic waves to several real-world problems, which include traffic flow on highways, flood waves in rivers, glacier flow, roll waves in an inclined channel, chromatographic models, and sediment transport in rivers. The general ideas and essential features of these problems are of wide applicability. Other applications of conservation laws include various chromatographic models in chemistry and the movement of pollutants in waterways. The propagation of traffic jams is almost similar to the shock waves that cause noise pollution near airports and spaceports. Kinematic wave phenomena also play an important role in traveling detonation and combustion fronts, the wetting water fronts observed in soils after rainfall, and the clanking of shunting trains. All of these problems are essentially based on the theory of kinematic waves developed by Lighthill and Whitham (Proc. R. Soc. Lond. A229:281–345, 1955). Many basic ideas and important features of hyperbolic waves and kinematic shock waves are found to originate from gas dynamics, so specific nonlinear models which describe Riemann’s simple waves with Riemann’s invariants and shock waves in gas dynamics are discussed. Considerable attention is also given to nonlinear hyperbolic systems and Riemann’s invariants, generalized simple waves, and generalized Riemann’s invariants.

… as Sir Cyril Hinshelwood has observed … fluid dynamicists were divided into hydraulic engineers who observed things that could not be explained and mathematicians who explained things that could not be observed.

James Lighthill

In every mathematical investigation, the question will arise whether we can apply our mathematical results to the real world.

V.I. Arnold