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 G 2 (f 1, c 1, κ 1; c 0, ϕ) is the inhomogeneous term, f 1 is the first-order wave amplitude.
Similar content being viewed by others
References
Joseph, R.I. Solitary Waves in a Finite Depth Fluid [J], J. Phys. A. Math. and Gne., 1977, 11
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: 157–165
Joseph, R.I., Egri, R. Multi-soliton solutions in a finite depth fluid [J], J. Phys. A: Math. Gen., 1978, 11
Chen H. H., Lee Y.C. Internal-Wave Solitons of Fluids With Finite Depth [J], Physical Review Letters, 1979, 43(4): 264–266
Nakamura, A., Matsuno, Y. Exct one and two-periodic wave solutions of fluids of finite depth [J], J. Phys. Soc. Japan, 1980, 48: 653
Miloh, Touvia. On periodic and solitary wavelike solutions of the intermediate long-wave equation [J], J. Fluid Mech., 1990, 211: 617–627
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): 418–425
Koop, C.G., Butler, G. An Investigation of Internal Solitary Waves in a Two-Fluid System [J], J. Fluid Mech., 1981, 112: 225–251
Zhou Q.F. The 2nd-order Solitary Waves in Stratified Fluids with Finite Depth [J]. Mathematical Physical Astronomical & Technical Sciences, 1985, 28(2): 159–171
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): 69–77
Segur, H., Hammack, J. L. Solitary Models of Long Internal Waves [J]. J. Fluid Mech., 1982, 118: 285–304
Ono H. Algebraic Solitary Waves in Stratified Fluid [J], J. Phys. Soc. Japan, 1975, 39(4): 1082–1091
Grimshaw, R. Theory of solitary waves in shallow fluids [M], Encyclopedia of Fluid Mechanics, vol. 2, Gulf Publishing Co. 1986: 1–25
Maslowe, S.A., Redekopp, L.G., Long Nonlinear Waves in Stratified Shear Flows [J], J. Fluid Mech., 101: 321–348, 1980
Grimshaw, R. Evolution Equations for Long Nonlinear Internal Waves in Stratified Shear Flows [J], Studies Appl. Mech., 1981, 65: 159–1
Author information
Authors and Affiliations
Additional information
Project supported by the National Natural Science Foundation of China (Grant No: 10272044).
Biography: Cheng You-liang (1963-), Male, PhD, Professor.
Rights and permissions
About this article
Cite this article
Cheng, Y., Fan, Z. The evolution equation for the second-order internal solitary waves in stratified fluids of finite depth. J Hydrodyn 18 (Suppl 1), 300–305 (2006). https://doi.org/10.1007/BF03400464
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF03400464