Nonlinear Geometrical Optics
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.
KeywordsEntropy Attenuation Soliton Crest Boulder
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- Cates, A., Nonlinear diffractive acoustics, fellowship dissertation, Trinity College, Cambridge, unpublished, (1988).Google Scholar
- 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
- 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
- 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
- Kodama, Y., Exact solutions of hydrodynamic type equations having infinitely many conserved densities, IMA Preprint # 478 (1989).Google Scholar
- Kodama, Y., and Gibbons, J., A method for solving the dispersionless KP hierarchy and its exact solutions II, IMA Preprint # 477 (1989).Google Scholar
- Kuznetsov, V.P., Equations of nonlinear acoustics, Sov. Phys. Acoustics 16 (1971), pp. 467–470.Google Scholar
- 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
- 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
- Timman, R., in Symposium Transsonicum, ed. K. Oswatitsch, Springer-Verlag, Berlin, 394 (1964).Google Scholar
- Whitham, G.B., Linear and Nonlinear Waves, Wiley, New York (1974).Google Scholar
- 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