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Diffraction effects in weakly nonlinear detonation waves

Hyperbolic P.D.E. Theory

Part of the Lecture Notes in Mathematics book series (LNM,volume 1402)

Abstract

In the limit of small heat release, large activation energy and weak nonlinearity, the propagation of detonation waves obeys a Geometrical Optics approximation. These equations develop caustic singularities, where the approximation fails. Here we present a derivation of a modified set of equations for weakly nonlinear detonation waves incorporating lateral diffraction effects. The modified set of equations does not fail at caustics.

Keywords

  • Wave Front
  • Shock Front
  • Detonation Wave
  • Triple Point
  • Geometrical Optic

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.

This work was performed in part while the author was visiting the Department of Mathematics at Stanford University, Stanford, California. The author was partially supported by grants from the AFOSR, NSF and the Wade Foundation.

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References

  1. Buchal, R. N. and Keller, J. B., “Boundary Layer Problems in Diffraction Theory,” Comm. Pure Appl. Math., vol. 13, pp. 85–114, 1960.

    CrossRef  MathSciNet  MATH  Google Scholar 

  2. Choquet-Bruhat, Y., “Ondes Asymptotiques et Approchées pour des systèmes d'Équations aux Dérivées Partielles non Linéaires,” J. Math. Pures et Appl., vol. 48, pp. 117–158, 1969.

    MathSciNet  MATH  Google Scholar 

  3. Cole, J. D., “Modern Developments in Transonic Flow,” SIAM J. Appl. Math., vol. 29, pp. 763–787, 1975.

    CrossRef  MathSciNet  MATH  Google Scholar 

  4. Cramer, M. S. and Seebass, A. R., “Focusing of Weak Shock Waves at an Arête,” J. Fluid Mech., vol. 88, pp. 209–222, 1978.

    CrossRef  MATH  Google Scholar 

  5. Fickett, W. and Davis, W. C., Detonation, Univ. of California Press, Berkeley, 1979.

    Google Scholar 

  6. Hunter, J. K., “Transverse Diffraction of Nonlinear Waves and Singular Rays,” SIAM J. Appl. Math., vol. 48, pp. 1–37, 1988.

    CrossRef  MathSciNet  MATH  Google Scholar 

  7. Hunter, J. K. and Keller, J. B., “Weakly Nonlinear, High-frequency Waves,” Comm. Pure Appl. Math., vol. 36, pp. 547–569, 1983.

    CrossRef  MathSciNet  MATH  Google Scholar 

  8. Hunter, J. and Keller, J. B., “Caustics of Nonlinear Waves,” Wave Motion, vol. 9, pp. 429–443, 1987.

    CrossRef  MathSciNet  MATH  Google Scholar 

  9. Hunter, J. K., Majda, A. and Rosales, R., “Resonantly Interacting Weakly Nonlinear Hyperbolic Waves. II. Several Space Variables,” St. Appl. Math., vol. 75, pp. 187–226, 1986.

    CrossRef  MathSciNet  MATH  Google Scholar 

  10. Keller, J. B., “Rays, Waves and Asymptotics,” Bull. Am. Math. Soc., vol. 84, pp. 727–750, 1978.

    CrossRef  MathSciNet  MATH  Google Scholar 

  11. Kevorkian, J. and Cole, J. D., Perturbation Methods in Applied Mathematics, Springer-Verlag, New York, 1980.

    MATH  Google Scholar 

  12. Ludwig, D., “Uniform Asymptotic Expansions at a Caustic,” Comm. Pure Appl. Math., vol. 19, pp. 215–250, 1966.

    CrossRef  MathSciNet  MATH  Google Scholar 

  13. Majda, A. and Rosales, R., “Resonantly Interacting Weakly Nonlinear Hyperbolic Waves. I. A Single Space Variable,” St. Appl. Math., vol. 71, pp. 149–179, 1984.

    CrossRef  MathSciNet  MATH  Google Scholar 

  14. Rosales, R. R. and Majda, A., “Weakly Nonlinear Detonation Waves,” SIAM J. Appl. Math., vol. 43, pp. 1086–1118, 1983.

    CrossRef  MathSciNet  MATH  Google Scholar 

  15. Sturtevant, B. and Kulkarny, V. A., “The Focusing of Weak Shock Waves,” J. Fluid Mech., vol. 73, pp. 651–671, 1976.

    CrossRef  Google Scholar 

  16. Whitham, G. B., Linear and Nonlinear Waves, John Wiley and Sons, New York, 1974.

    MATH  Google Scholar 

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© 1989 Springer-Verlag

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Rosales, R.R. (1989). Diffraction effects in weakly nonlinear detonation waves. In: Carasso, C., Charrier, P., Hanouzet, B., Joly, JL. (eds) Nonlinear Hyperbolic Problems. Lecture Notes in Mathematics, vol 1402. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0083879

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  • DOI: https://doi.org/10.1007/BFb0083879

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  • Print ISBN: 978-3-540-51746-7

  • Online ISBN: 978-3-540-46800-4

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