Nonlinear Geometrical Optics

  • John K. Hunter
Part of the The IMA Volumes in Mathematics and Its Applications book series (IMA, volume 29)


Using asymptotic methods, one can reduce complicated systems of equations to simpler model equations. The model equation for a single, genuinely nonlinear, hyperbolic wave is Burgers equation. Reducing the gas dynamics equations to a Burgers equation, leads to a theory of nonlinear geometrical acoustics. When diffractive effects are included, the model equation is the ZK or unsteady transonic small disturbance equation. We describe some properties of this equation, and use it to formulate asymptotic equations that describe the transition from regular to Mach reflection for weak shocks. Interacting hyperbolic waves are described by a system of Burgers or ZK equations coupled by integral terms. We use these equations to study the transverse stability of interacting sound waves in gas dynamics.


Sound Wave Burger Equation Weak Shock Eikonal Equation Shock Strength 
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  1. [1]
    Ablowitz, M.J., and Segur, H., Solitons and the Inverse Scattering Transform, SIAM, Philadelphia (1981).MATHGoogle Scholar
  2. [2]
    Bamberger, A., Enquist, B., Halpern, L., and Joly, P., Parabolic wave equations and approximations in heterogeneous media, SIAM J. Appl. Math., 48 (1988), pp. 99–128.MathSciNetMATHCrossRefGoogle Scholar
  3. [3]
    Cates, A., Nonlinear diffractive acoustics, fellowship dissertation, Trinity College, Cambridge, unpublished, (1988).Google Scholar
  4. [4]
    Chang, T., and Hsiao, L., the Riemann Problem and Interaction of Waves in Gas Dynamics, Longman, Avon (1989).MATHGoogle Scholar
  5. [5]
    Cole, J.D., and Cook, L.P., Transonic aerodynamics, Elsevier, Amsterdam (1986).MATHGoogle Scholar
  6. [6]
    Cramer, M.S., and Seebass, A.R., Focusing of a weak shock at an arête, J. Fluid Mech, 88 (1978), pp. 209–222.MATHCrossRefGoogle Scholar
  7. [7]
    Crighton, D.G., Model equations for nonlinear acoustics, Ann. Rev. Fluid Mech., 11 (1979), pp. 11–13.CrossRefGoogle Scholar
  8. [8]
    Crighton, D.G., Basic theoretical nonlinear acoustics, in Frontiers in Physical Acoustics, Proc. Int. School of Physics “Enrico Fermi”, Course 93 (1986), North-Holland, Amsterdam.Google Scholar
  9. [9]
    Hamilton, M.F., Fundamentals and applications of nonlinear acoustics, in Nonlinear Wave Propagation in Mechanics, ed. T.W. Wright, AMD-77 (1986), pp. 1–28.Google Scholar
  10. [10]
    Harabetian, E., Diffraction of a weak shock by a wedge, Comm. Pure Appl. Math., 40 (1987), pp. 849–863.MathSciNetMATHCrossRefGoogle Scholar
  11. [11]
    Hornung, H., Regular and Mach reflection of shock waves, Ann. Rev. Fluid Mech., 18 (1986), pp. 33–58.CrossRefGoogle Scholar
  12. [12]
    Hunter, J.K., Transverse diffraction and singular rays, SIAM J. Appl. Math., 75 (1986), pp. 187–226.MathSciNetMATHGoogle Scholar
  13. [13]
    Hunter, J.K., Hyperbolic waves and nonlinear geometrical acoustics, in Transactions of the Sixth Army Conference on Applied Mathematics and Computing, Boulder CO (1989), pp. 527–569.Google Scholar
  14. [14]
    Hunter, J.K., and Keller, J.B., Weakly nonlinear high frequency waves, Comm. Pure Appl. Math., 36 (1983), pp. 547–569.MathSciNetMATHCrossRefGoogle Scholar
  15. [15]
    Hunter, J.K., and Keller, J.B., Caustics of nonlinear waves, Wave motion, 9 (1987), pp. 429–443.MathSciNetMATHCrossRefGoogle Scholar
  16. [16]
    Hunter, J.K., Majda, A., and Rosales R.R., Resonantly interacting weakly nonlinear hyperbolic waves, II: several space variables, Stud. Appl. Math., 75 (1986), pp. 187–226.MathSciNetMATHGoogle Scholar
  17. [17]
    Kadomtsev, B.B., and Petviashvili, V.I., On the stability of a solitary wave in a weakly dispersing media, Sov. Phys. Doklady, 15 (1970), pp. 539–541.MATHGoogle Scholar
  18. [18]
    Keller, J.B., Rays, waves and asymptotics, Bull. Am. Math. Soc, 84 (1978), pp. 727–750.MATHCrossRefGoogle Scholar
  19. [19]
    Kodama, Y., Exact solutions of hydrodynamic type equations having infinitely many conserved densities, IMA Preprint # 478 (1989).Google Scholar
  20. [20]
    Kodama, Y., and Gibbons, J., A method for solving the dispersionless KP hierarchy and its exact solutions II, IMA Preprint # 477 (1989).Google Scholar
  21. [21]
    Kuznetsov, V.P., Equations of nonlinear acoustics, Sov. Phys. Acoustics 16 (1971), pp. 467–470.Google Scholar
  22. [22]
    Lighthill, M.J., On the diffraction of a blast I, Proc. R. Soc. London Ser. A 198 (1949), pp. 454–470.MathSciNetCrossRefGoogle Scholar
  23. [23]
    Ludwig, D., Uniform asymptotic expansions at a caustic, Comm. Pure Appl. Math. 19 (1966), pp. 215–250.MathSciNetMATHCrossRefGoogle Scholar
  24. [24]
    Majda, A., Nonlinear geometrical optics for hyperbolic systems of conservation laws, in Oscillation Theory, Computation, and Methods of Compensated Compactness, Springer-Verlag, New York, IMA Volume 2 (1986), pp. 115–165.CrossRefGoogle Scholar
  25. [25]
    Majda, A. and Rosales, R.R., Resonantly interacting hyperbolic waves, I: a single space variable, Stud. Appl. Math. 71 (1984), pp. 149–179.MathSciNetMATHGoogle Scholar
  26. [26]
    Majda, A., Rosales, R.R., and Schonbek, M., A canonical system of integro-differential equations in resonant nonlinear acoustics, Stud. Appl. Math., 79 (1988), pp. 205–262.MathSciNetMATHGoogle Scholar
  27. [27]
    Nayfeh, A., A comparison of perturbation methods for nonlinear hyperbolic waves, in Singular Perturbations and Asymptotics, eds. R. Meyer and S. Parter, Academic Press, New York (1980), pp. 223–276.Google Scholar
  28. [28]
    Nimmo, J.J.C., and Crighton, D.C., Nonlinear and diffusive effects in nonlinear acoustic propagation over long ranges, Phil. Trans. Roy. Soc. London Ser. A 384 (1986), pp. 1–35.Google Scholar
  29. [29]
    Pego, R., Some explicit resonating waves in weakly nonlinear gas dynamics, Stud. Appl. Math., 79 (1988), pp. 263–270.MathSciNetMATHGoogle Scholar
  30. [30]
    Sturtevant, B., and Kulkarny, V.A., The focusing of weak shock waves, J. Fluid Mech., 73 (1976), pp. 1086–1118.CrossRefGoogle Scholar
  31. [31]
    Timman, R., in Symposium Transsonicum, ed. K. Oswatitsch, Springer-Verlag, Berlin, 394 (1964).Google Scholar
  32. [32]
    Whitham, G.B., Linear and Nonlinear Waves, Wiley, New York (1974).Google Scholar
  33. [33]
    Zabolotskaya, E.A., and Khokhlov, R.V., Quasi-plane waves in the nonlinear acoustics of confined beams, Sov. Phys.-Acoustics, 15 (1969), pp. 35–40.Google Scholar
  34. [34]
    Zahalak, G.I., and Myers, M.K., Conical now near singular rays, J. Fluid. Mech., 63 (1974), pp. 537–561.MATHCrossRefGoogle Scholar
  35. [35]
    Johnson, R.S., Water Waves and Korteweg-deVries Equations, J. Fluid. Mech., 97 (1980), pp. 701–719.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag New York, Inc. 1991

Authors and Affiliations

  • John K. Hunter
    • 1
  1. 1.Department of MathematicsColorado State UniversityFort CollinsUSA

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