Waves of Finite Amplitude

  • Seán Prunty
Part of the Shock Wave and High Pressure Phenomena book series (SHOCKWAVE)


The propagation of waves of finite amplitude and the change in wave profile leading to the formation of shock waves is the subject matter of this chapter. Simple examples of piston motion are presented to illustrate the formation of a normal shock wave and the time and place for its formation are also discussed. A brief introduction to Riemann invariants and the method of characteristics is presented and some examples to illustrate the method of characteristics in solving partial differential equations are outlined. The solution of nonlinear equations that result in the characteristics intersecting and the formation of shock waves are illustrated. The chapter concludes with some examples to illustrate the application of the method of characteristics and Riemann invariants to simple flow problems involving piston motion.


Waves of finite amplitude Shock waves Riemann invariants Method of characteristics Nonlinear distortion 


  1. 1.
    W. Band, Introduction to Mathematical Physics (Van Nostrand Company, Inc., Princeton, NJ, 1959)zbMATHGoogle Scholar
  2. 2.
    S. Temkin, Elements of Acoustics (Wiley, New York, 1981), Section 3.7Google Scholar
  3. 3.
    J.D. Anderson, Modern Compressible Flow with Historical Perspective, 3rd edn. (McGraw-Hill, New York, 2003), Chapter 7Google Scholar
  4. 4.
    L.D. Landau, E.M. Lifshitz, Fluid Mechanics (Pergamon Press, London, 1966), p. 366Google Scholar
  5. 5.
    N. Curle, H.J. Davies, Modern Fluid Dynamics, vol 2 (Van Nostrand Reinhold Company, London, 1971), p. 68zbMATHGoogle Scholar
  6. 6.
    O.V. Rudenko, S.I. Soluyan, Theoretical Foundations of Nonlinear Acoustics (Consultants Bureau, A Division of Plenum Publishing Company, New York, 1977), Chapter 1CrossRefGoogle Scholar
  7. 7.
    D. Mihalas, B. Weibel-Mihalas, Foundations of Radiation Hydrodynamics (Dover Publications, Inc., New York, 1999), p. 227zbMATHGoogle Scholar
  8. 8.
    W. Band, G.E. Duvall, Physical nature of shock propagation. Am. J. Phys. 29, 780 (1961)CrossRefGoogle Scholar
  9. 9.
    W.C. Griffith, W. Bleakney, Shock waves in gases. Am. J. Phys. 22, 597 (1954)CrossRefGoogle Scholar
  10. 10.
    N. Curle, H.J. Davies, Modern Fluid Dynamics, vol 2 (Van Nostrand Reinhold Company, London, 1971), Section 3.3.2zbMATHGoogle Scholar
  11. 11.
    L.D. Landau, E.M. Lifshitz, Fluid Mechanics (Pergamon Press, London, 1966), Chapter 9Google Scholar
  12. 12.
    H.W. Liepmann, A. Roshko, Elements of Gas Dynamics (Dover Publications Inc., Mineola, NY, 1956), Section 3.9zbMATHGoogle Scholar
  13. 13.
    G.B. Whitham, Linear and Nonlinear Waves (Wiley, New York, 1999)CrossRefGoogle Scholar
  14. 14.
    R. Courant, K.O. Friedrichs, Supersonic Flow and Shock Waves (Interscience Publishers, Inc., New York, 1956)zbMATHGoogle Scholar
  15. 15.
    A.R. Paterson, A First Course in Fluid Dynamics (Cambridge University Press, London, 1983), Chapter 14CrossRefGoogle Scholar
  16. 16.
    J.D. Logan, Applied Mathematics, 2nd edn. (Wiley, New York, 1977), Chapter 6zbMATHGoogle Scholar
  17. 17.
    A.J. Chorin, J.E. Marsden, A Mathematical Introduction to Fluid Mechanics (Springer, New York, 1979), Chapter 3CrossRefGoogle Scholar
  18. 18.
    F.H. Harlow, LA-2412 Report (Los Alamos Scientific Laboratory of the University of California, Los Alamos, New Mexico, 1960), Chapter 3Google Scholar
  19. 19.
    M.A. Saad, Compressible Fluid Flow (Prentice-Hall, Inc., Englewood Cliffs, NJ, 1985), Chapter 9Google Scholar
  20. 20.
    J. Billingham, A.C. King, Wave Motion (Cambridge University Press, New York, 2000), Chapter 7zbMATHGoogle Scholar
  21. 21.
    O. Regev, O.M. Umurhan, P.A. Yecko, Modern Fluid Dynamics for Physics and Astrophysics (Springer, New York, 2016), Chapter 6CrossRefGoogle Scholar
  22. 22.
    J.H.S. Lee, The Gas Dynamics of Explosions (Cambridge University Press, New York, 2016), Chapter 1CrossRefGoogle Scholar
  23. 23.
    W.A. Strauss, Partial Differential Equations: An Introduction (Wiley, New York, 1992), Chapter 14zbMATHGoogle Scholar
  24. 24.
    P.J. Olver, Introduction to Partial Differential Equations (Springer, Cham, 2014)CrossRefGoogle Scholar
  25. 25.
    S. Salsa, Partial Differential Equations in Action: From Modelling to Theory (Springer, Milan, Italy, 2008), Chapter 4zbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Seán Prunty
    • 1
  1. 1.BallincolligIreland

Personalised recommendations