The Growth of Instability Waves in a Slightly Nonuniform Medium

  • Marten T. Landahl
Conference paper
Part of the International Union of Theoretical and Applied Mechanics book series (IUTAM)


Kinematic wave theory, extended to include weak dissipation, higher-order amplitude dispersion, and weak nonlinearity, is used to calculate the long-time evolution of instability waves in a three-dimensional boundary layer. An unstable wave on a convected inhomogeneity reaches criticality when propagating at the velocity of the inhomogeneity; it then becomes absolutely unstable in that moving reference frame. A trapped such a wave is found to be governed by the Ginzburg-Landau equation within an inner region of size ε−172 times the wave lengt where ε << 1 measures the non-uniformity of the wave system or the background medium.


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  1. 1.
    Landahl, M.T.: The application of kinematic wave theory to wave trains and packets with small dissipation. Phys. Fluids 25 (1982) 1512–1516CrossRefMATHADSMathSciNetGoogle Scholar
  2. 2.
    Whitham, G.B. Linear and nonlinear waves. New York, London, Sydney, Toronto: John Wiley & Sons 1974.MATHGoogle Scholar
  3. 3.
    Hayes, W.D.: Conservation of action and modal wave action. Proc. Roy. Soc. A320 (1970) 187–208.CrossRefADSGoogle Scholar
  4. 4.
    Chu, V.H.; Mei, C.C.: On slowly varying Stokes waves. j. Fluid Mech. 41 (1970) 873–887.CrossRefMATHADSMathSciNetGoogle Scholar
  5. 5.
    Nordstrom, J.; Landahl, M.T.: The growth of instability waves in a three-dimensional boundary layer (to be published)Google Scholar
  6. 6.
    Landahl, M.T.: Wave mechanics of breakdown. J. Fluid Mech. 56 (1972) 775–802.CrossRefMATHADSGoogle Scholar
  7. 7.
    Stewartson, K.: Some aspects of nonlinear stability theory. Polish Academy of Sciences, Fluid Dynamic Transaction 7 (1977) 101–128.ADSGoogle Scholar
  8. 8.
    Bers, A.: Linear waves and instabilities. In Physique des Plasmas (ed. C. DeWitt & J. Peyraud) New York: Gordon and Breach 1975.Google Scholar
  9. 9.
    Hocking, L.M.; Stewartson, K. Stuart, J.T.: A nonlinear instability burst in plane Poiseullle flow. J. Fluid Mech. 51 (1972) 705–735.CrossRefMATHADSGoogle Scholar
  10. 10.
    Moon, H.T.; Huerre, P.; Redekopp, L.G.: Transition to chaos in the Ginzgurg-Landau equation. Physica 7D (1983) 135–150.MathSciNetGoogle Scholar

Copyright information

© Springer-Verlag, Berlin, Heidelberg 1985

Authors and Affiliations

  • Marten T. Landahl
    • 1
    • 2
  1. 1.Massachusetts Institute of TechnologyCambridgeUSA
  2. 2.Royal Institute of TechnologyStockholmSweden

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