Nonlinear effects and shock formation in the focusing of a spherical acoustic wave

Numerical simulations and experiments in liquid helium
  • C. AppertEmail author
  • C. Tenaud
  • X. Chavanne
  • S. Balibar
  • F. Caupin
  • D. d’Humiéres


The focusing of acoustic waves is used to study nucleation phenomena in liquids. At large amplitude, nonlinear effects are important so that the magnitude of pressure or density oscillations is difficult to predict. We present a calculation of these oscillations in a spherical geometry. We show that the main source of nonlinearities is the shape of the equation of state of the liquid, enhanced by the spherical geometry. We also show that the formation of shocks cannot be ignored beyond a certain oscillation amplitude. The shock length is estimated by an analytic calculation based on the characteristics method. In our numerical simulations, we have treated the shocks with a WENO scheme. We obtain a very good agreement with experimental measurements which were recently performed in liquid helium. In addition, the comparison between numerical and experimental results allows us to calibrate the vibration of the ceramic used to produce the wave, as a function of the applied voltage.


Helium Experimental Measurement Applied Voltage Acoustic Wave Large Amplitude 
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  1. 1.
    H. Lambaré, P. Roche, S. Balibar, H.J. Maris, O.A. Andreeva, C. Guthman, K.O. Keshishev, E. Rolley, Eur. Phys. J. B 2, 381 (1998)CrossRefGoogle Scholar
  2. 2.
    F. Caupin, S. Balibar, Phys. Rev. B 64, 064507 (2001)CrossRefGoogle Scholar
  3. 3.
    X. Chavanne, S. Balibar, F. Caupin, Phys. Rev. Lett. 86, 5506 (2001)CrossRefGoogle Scholar
  4. 4.
    M.G. Sirotyuk, Sov. Phys. Acoustics 8, 165 (1962)Google Scholar
  5. 5.
    R.A. Roy, S.I. Madanshetty, R.E. Apfel, J. Acoust. Soc. Am. 87, 2451 (1990)Google Scholar
  6. 6.
    S.K. Nemirovskii, Sov. Phys. Usp. 33, 429 (1990)Google Scholar
  7. 7.
    H.P. Greenspan, A. Nadim, Phys. Fluids A 5, 1065 (1993)CrossRefzbMATHGoogle Scholar
  8. 8.
    H.J. Maris, Phys. Rev. Lett. 66, 45 (1991)CrossRefGoogle Scholar
  9. 9.
    X. Chavanne, S. Balibar, F. Caupin, C. Appert, D. d’Humiéres, J. Low Temp. Phys. 126, 643 (2002)CrossRefGoogle Scholar
  10. 10.
    C. Appert, X. Chavanne, S. Balibar, D. d’Humiéres, C. Tenaud, in 4es Rencontres du Non-Linéaire, March 2001 (Univ. Paris Sud Paris XI Editions, 2001)Google Scholar
  11. 11.
    G.W. Waters, D.J. Watmough, J. Wilks, Phys. Lett. A 26, 12 (1967)CrossRefGoogle Scholar
  12. 12.
    M.F. Hamilton, D.T. Blackstock, Nonlinear Acoustics (Academic Press, 1998), p. 26Google Scholar
  13. 13.
    A. Jeffrey, T. Taniuti, Nonlinear wave propagation (Academic Press, 1964)Google Scholar
  14. 14.
    L. Landau, E. Lifshitz, in Course of Theoretical Physics, Vol. 6, Fluid Mechanics (Pergamon Press, 1959), p. 266Google Scholar
  15. 15.
    C. Canuto, M.Y. Hussaini, A. Quarteroni, T.A. Zang, Spectral methods in fluid dynamics (Springer Verlag, New-York, 1988)Google Scholar
  16. 16.
    C. Bernardi, Y. Maday, Approximations spectrales de problémes aux limites elliptiques, Collection Mathématiques & Applications, edited by J.M Ghidaglia, P. Lascaux (Springer Verlag, 1992)Google Scholar
  17. 17.
    R. Peyret, Spectral methods with application to incompressible viscous flow (Springer Verlag, 2002)Google Scholar
  18. 18.
    S.K. Lele, J. Comp. Phys. 103, 16-42 (1992)MathSciNetzbMATHGoogle Scholar
  19. 19.
    K. Mahesh, J. Comp. Phys. 145, 332-358 (1998)CrossRefMathSciNetzbMATHGoogle Scholar
  20. 20.
    C.W. Shu, S. Osher, J. Comp. Phys. 77, 439 (1988)MathSciNetzbMATHGoogle Scholar
  21. 21.
    C.W. Shu, S. Osher, J. Comp. Phys. 83, 32 (1989)MathSciNetzbMATHGoogle Scholar
  22. 22.
    G.-S. Jiang, C.H. Shu, J. Comp. Phys. 126, 202 (1996)CrossRefMathSciNetzbMATHGoogle Scholar
  23. 23.
    C. Tenaud, E. Garnier, P. Sagaut, Int. J. Numerical Methods Fluids 33, 249 (2000)CrossRefzbMATHGoogle Scholar
  24. 24.
    X. Chavanne, S. Balibar, F. Caupin, J. Low Temp. Phys. 125, 155 (2001)CrossRefGoogle Scholar
  25. 25.
    X. Chavanne, S. Balibar, F. Caupin, J. Low Temp. Phys. 126, 615 (2002)CrossRefGoogle Scholar
  26. 26.
    R.M. Corless, G.H. Gonnett, D.E.G. Hare, D.J. Jeffrey, D.E. Knuth, Adv. Comp. Math. 5, 329 (1996)MathSciNetzbMATHGoogle Scholar
  27. 27.
    P.L. Roe, J. Comp. Physics 43, 367 (1981)Google Scholar

Copyright information

© Springer-Verlag Berlin/Heidelberg 2003

Authors and Affiliations

  • C. Appert
    • 1
    Email author
  • C. Tenaud
    • 2
  • X. Chavanne
    • 1
  • S. Balibar
    • 1
  • F. Caupin
    • 1
  • D. d’Humiéres
    • 1
  1. 1.Laboratoire de Physique Statistique de l’ENSassocié au CNRS et aux Universités Paris 6 et 7Paris Cedex 05France
  2. 2.Laboratoire d’Informatique pour la Mécanique et les Sciences de l’IngénieurUPR CNRS 3251Orsay CedexFrance

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