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Journal of Hydrodynamics

, Volume 18, Issue 1, pp 300–305 | Cite as

The evolution equation for the second-order internal solitary waves in stratified fluids of finite depth

  • Youliang Cheng
  • Zhongyao Fan
Session A5

Abstract

The evolution equation for the second-order internal solitary waves in stratified fluids, that are income-pressible inviscid ones of mixed stratification in which the density varies with depth in lower layer but keeps uniform in upper layer with a finite depth, is derived directly from the Euler equation by using the balance between the nonlinearity and the dispersion with the scaling μ=O(ε), and by introducing the Gardner-Morikawa transformation, asymptotic expansions and the matching of the solutions for the upper and lower layers of fluid via the computer algebraic operation. The evolution equation and its solution derived for the first-order wave amplitude are consistent with the classical ones, and the desired equation governing the second-order amplitude is reduced as follows where \(g(\xi) = - {1 \over {2\pi}}\int_{- \infty}^\infty {k\,\coth (kl){e^{jk\xi}}dk} \) and G2 (f1, c1, κ1; c0, ϕ) is the inhomogeneous term, f1 is the first-order wave amplitude.

Key words

finite depth internal solitary waves stratified fluids perturbation methods 

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References

  1. [1]
    Joseph, R.I. Solitary Waves in a Finite Depth Fluid [J], J. Phys. A. Math. and Gne., 1977, 11Google Scholar
  2. [2]
    Whitham, G.B. Linear and Non-Linear Waves, John Wiley and Sons, inc., N.Y., 1974 Kubota, T., Ko, D. R. S., Dobbs, L., Weakly-Nonlinear, Long Internal Gravity Waves in Stratified Fluids of Finite Depth [J], J. Hydronautics, 1978, 12: 157–165MATHGoogle Scholar
  3. [3]
    Joseph, R.I., Egri, R. Multi-soliton solutions in a finite depth fluid [J], J. Phys. A: Math. Gen., 1978, 11Google Scholar
  4. [4]
    Chen H. H., Lee Y.C. Internal-Wave Solitons of Fluids With Finite Depth [J], Physical Review Letters, 1979, 43(4): 264–266MathSciNetCrossRefGoogle Scholar
  5. [5]
    Nakamura, A., Matsuno, Y. Exct one and two-periodic wave solutions of fluids of finite depth [J], J. Phys. Soc. Japan, 1980, 48: 653CrossRefGoogle Scholar
  6. [6]
    Miloh, Touvia. On periodic and solitary wavelike solutions of the intermediate long-wave equation [J], J. Fluid Mech., 1990, 211: 617–627MathSciNetCrossRefGoogle Scholar
  7. [7]
    Yang T.S., Akylas T.R. Radiating Solitary Waves of A Model Evolution Equation in Fluids of Finite Depth [J], Physic D, 1995, 82(4): 418–425MathSciNetCrossRefGoogle Scholar
  8. [8]
    Koop, C.G., Butler, G. An Investigation of Internal Solitary Waves in a Two-Fluid System [J], J. Fluid Mech., 1981, 112: 225–251MathSciNetCrossRefGoogle Scholar
  9. [9]
    Zhou Q.F. The 2nd-order Solitary Waves in Stratified Fluids with Finite Depth [J]. Mathematical Physical Astronomical & Technical Sciences, 1985, 28(2): 159–171MathSciNetMATHGoogle Scholar
  10. [10]
    Zhou Q.F. High-Order Thory of Internal Solitary Waves with a Free Surface in Two-Layer Fluid System of Finite Depth [J], Appl. Maths & Mech., 1987, 8(1): 69–77Google Scholar
  11. [11]
    Segur, H., Hammack, J. L. Solitary Models of Long Internal Waves [J]. J. Fluid Mech., 1982, 118: 285–304MathSciNetCrossRefGoogle Scholar
  12. [12]
    Ono H. Algebraic Solitary Waves in Stratified Fluid [J], J. Phys. Soc. Japan, 1975, 39(4): 1082–1091MathSciNetCrossRefGoogle Scholar
  13. [13]
    Grimshaw, R. Theory of solitary waves in shallow fluids [M], Encyclopedia of Fluid Mechanics, vol. 2, Gulf Publishing Co. 1986: 1–25Google Scholar
  14. [14]
    Maslowe, S.A., Redekopp, L.G., Long Nonlinear Waves in Stratified Shear Flows [J], J. Fluid Mech., 101: 321–348, 1980MathSciNetCrossRefGoogle Scholar
  15. [15]
    Grimshaw, R. Evolution Equations for Long Nonlinear Internal Waves in Stratified Shear Flows [J], Studies Appl. Mech., 1981, 65: 159–1MathSciNetCrossRefGoogle Scholar

Copyright information

© China Ship Scientific Research Center 2006

Authors and Affiliations

  • Youliang Cheng
    • 1
  • Zhongyao Fan
    • 1
  1. 1.Department of Power EngineeringNorth China Electric Power UniversityBaoding, HebeiChina

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