Advertisement

Kinematic Waves and Real-World Nonlinear Problems

  • Lokenath Debnath

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.

Keywords

Shock Wave Traffic Flow Traffic Density Lorenz System Simple Wave 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Bibliography

  1. Burton, C.V. (1893). On plane and spherical sound waves of finite amplitude, Philos. Mag. 35, 317–333. Google Scholar
  2. Cornish, V. (1934). Ocean Waves and Kindred Geophysical Phenomena, Cambridge University Press, Cambridge. Google Scholar
  3. Courant, R. and Friedrichs, K.O. (1948). Supersonic Flow and Shock Waves, Interscience, New York and London MATHGoogle Scholar
  4. Debnath, L. (1995). Integral Transforms and Their Applications, CRC Press, Boca Raton. MATHGoogle Scholar
  5. Dressler, R.F. (1949). Mathematical solution of problem of roll waves in inclined open channels, Commun. Pure Appl. Math. 2, 149–194. MathSciNetMATHCrossRefGoogle Scholar
  6. Haight, F.A. (1963). Mathematical Theories of Traffic Flow, Academic Press, New York. MATHGoogle Scholar
  7. Jeffrey, A. (1976). Quasilinear Hyperbolic Systems and Waves, Research Notes in Mathematics, Vol. 5, Pitman Publishing Company, London. MATHGoogle Scholar
  8. Jeffrey, A. and Taniuti, T. (1964). Nonlinear Wave Propagation, Academic Press, New York. Google Scholar
  9. Kynch, G.F. (1952). A theory of sedimentation, Trans. Faraday Soc. 48, 166–176. CrossRefGoogle Scholar
  10. Lax, P.D. (1954a). Weak solutions of nonlinear hyperbolic equations and their numerical computation, Commun. Pure Appl. Math. 7, 159–193. MathSciNetMATHCrossRefGoogle Scholar
  11. Lax, P.D. (1954b). The initial-value problem for nonlinear hyperbolic equations in two independent variables, Ann. Math. Stud. (Princeton) 33, 211–299. MathSciNetMATHGoogle Scholar
  12. Lax, P.D. (1957). Hyperbolic systems of conservation law, II, Commun. Pure Appl. Math. 10, 537–566. MathSciNetMATHCrossRefGoogle Scholar
  13. Lax, P.D. (1973). Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, Society for Industrial and Applied Mathematics, Philadelphia. MATHGoogle Scholar
  14. Le Veque, R.J. (1990). Numerical Methods for Conservation Laws, Birkhäuser Verlag, Boston. Google Scholar
  15. Lighthill, M.J. (1956). Viscosity effects in sound waves of finite amplitude, in Surveys in Mechanics (eds. G.K. Batchelor and R.M. Davies), Cambridge University Press, Cambridge, 250–351. Google Scholar
  16. Lighthill, M.J. and Whitham, G.B. (1955). On kinematic waves: I. Flood movement in long rivers; II. Theory of traffic flow on long crowded roads, Proc. R. Soc. Lond. A229, 281–345. MathSciNetGoogle Scholar
  17. Lorenz, E.N. (1963a). Deterministic nonperiodic flow, J. Atmos. Sci. 20, 130–141. CrossRefGoogle Scholar
  18. Lorenz, E.N. (1963b). The predictability of hydrodynamic flow, Trans. N. Y. Acad. Sci. Ser. II 25, 409–432. Google Scholar
  19. Lorenz, E.N. (1963c). The mechanics of vacillation, J. Atmos. Sci. 20, 448–464. CrossRefGoogle Scholar
  20. Nye, J.F. (1960). The response of glaciers and ice-sheets to seasonal and climatic changes, Proc. R. Soc. Lond. A256, 559–584. MathSciNetGoogle Scholar
  21. Nye, J.F. (1963). The response of a glacier to changes in the rate of nourishment and wastage, Proc. R. Soc. Lond. A275, 87–112. Google Scholar
  22. Pack, D.C. (1960). A note on the breakdown of continuity in the motion of a compressible fluid, J. Fluid Mech. 8, 103–108. MathSciNetMATHCrossRefGoogle Scholar
  23. Richards, P.I. (1956). Shock waves on the highway, Oper. Res. 4, 42–51. MathSciNetCrossRefGoogle Scholar
  24. Riemann, B., (1858). Uber die Fortpflanzung ebener Luftwellen von endlicher Schwingungsweite, Göttingen Abhandlunger, Vol. viii, p. 43 (Werke, 2te Aufl., Leipzig, 1892, p. 157). Google Scholar
  25. Seddon, J.A. (1900). River hydraulics, Trans. Am. Soc. Civ. Eng. 43, 179–243. Google Scholar
  26. Smoller, J. (1994). Shock Waves and Reaction–Diffusion Equations, 2nd edition, Springer, New York. MATHGoogle Scholar
  27. Stoker, J.J. (1957). Water Waves, Interscience, New York. MATHGoogle Scholar
  28. Whitham, G.B. (1974). Linear and Nonlinear Waves, Wiley, New York. MATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Texas, Pan AmericanEdinburgUSA

Personalised recommendations