Abstract
Euler’s equations with standard boundary conditions for the problem of potential surface waves of an arbitrary amplitude in a homogeneous liquid layer with a flat bottom are converted into the new system, including integral and differential equations for the of the potential and its time derivative near the surface. The basic formula of the theory of infinitesimal waves, paired Korteweg-de Vries (KdV) and Kadomtsev− Petviashvili (KP) equations, the envelope Zakharov−Shabat soliton follows from the system in limiting case. The resulting generalized equation, unlike traditional KdFand KP-equations is suitable for the description of waves on the surface of the initially quiescent fluid. A new exact solutions for gravity waves in a deep water, expressed in terms of complex Lambert’s functions are constructed.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Gradshtein, I.S. and Ryzhik, I.M., Tables of Integrals, Sums, Series and Products, Moscow: GIFML, 1962.
Kadomtsev, B.B. and Petviashvili, V.I., On the stability of solitary waves in weakly dispersing media, Sov. Phys. Doklady, 1970, vol. 15, pp. 539–541.
Kistovich, A.V. and Chashechkin, Yu.D., Integral model of the propagation of stationary potential waves in fluids, Dokl. Phys., 2008, vol. 53, iss. 7, pp. 395–400.
Nekrasov, A.I., On waves of steady species, Math. Ivanovo-Voznesensky Polytechnic. Inst., 1921, no. 3, pp. 52–65.
Ovsyannikov, L.V., Makarenko, N.I., Nalimov, V.I., Liapidevskii, V.Y., Plotnikov, P.I., Sturova, I.V., Bukreev, V.I., and Vladimirov, V.A., Nonlinear Problems of the Theory of Surface and Internal Waves, Novosibirsk: Sci., 1985 [in Russian].
Zeytounian, P.D., Nonlinear long waves on water surface and solitons, Physics-Uspekhi, 1995, vol. 165, no. 12 pp. 1403–1456.
Ablowitz, M.J., Fokas, A.S., and Musslimani, Z.H., On a new non-local theory of water waves, J. Fluid Mech., 2006, vol. 562, pp. 313–343.
Babanin, A., Breaking and Dissipation of Ocean Surface Waves, Cambridge: CUP, 2011.
Byatt-Smith, J.G.B., An integral equation for unsteady surface waves and a comment on the Boussinesq equation, J. Fluid Mech., 1971, vol. 49, Pt. 4, pp. 625–633.
Chen, B. and Saffman, P.G., Numerical evidence for the existence of new types of gravity waves of permanent form on deep water, Stud. Appl. Math., 1980, vol. 62, pp. 1–21.
Clamond, D., Note on the velocity and related fields of steady irrotational two-dimensional surface gravity waves, Phil. Trans. R. Soc. A, 2012, vol. 370, pp. 1572–1586, DOI: 10.1098/rsta.2011.0470.
Clamond, D., On the Lagrangian description of steady surface gravity waves, J. Fluid Mech., 2007, vol. 589, pp. 433–454, DOI: http://dxdoiorg/10.1017/ S0022112007007811.
Clamond, D. and Constantin, A., Recovery of steady periodic wave profiles from pressure measurements at the bed, J. Fluid Mech., 2013, vol. 714, pp. 463–475, DOI: 10.1017/jfm.2012.490.
Dutykh, D. and Clamond, D., Efficient computation of steady solitary gravity waves, Wave Motion, 2014, vol. 51, pp. 86–99.
Henry, D., Steady periodic flow induced by the Korteweg-de Vries equation, Wave Motion, 2009, vol. 46, pp. 403–411.
Ionescu-Kruse, D., On the particle paths and the stagnation points in small-amplitude deep-water waves, J. Math. Fluid Mechanics, 2013, vol. 15, pp. 41–54, DOI: 10.1007/s00021 -012-0102-5.
Korteweg, D.J. and de Vries, G., On the change form of long waves advancing in a rectangular canal, and on a new type of long stationary waves, Phyl. Mag., 1895, vol. 5, pp. 422–443.
Lamb, H., Hydrodynamics, 6th ed., Front Cover: CUP, 1932.
Landau, L.D. and Lifshitz, E.M., Fluid Mechanics (Volume 6 of A Course of Theoretical Physics), 2nd English ed., Oxford: Pergamon Press, 1987.
Mei, C.C., Stiassnie, M., and Yue, D.K.-P., Theory and Applications of Ocean Surface Waves: Nonlinear aspects, World Scientific, 2005.
Nayfeh, A.H., Introduction to Perturbation Techniques, N.Y.: John Wiley &, Sons, 1981, XIV.
Ochi, M.K. and Tsai, C.-H., Prediction of breaking waves in deep water, J. Phys. Oceanogr., 1983, vol. 13, pp. 2008–2019.
Rankine, W.J.M., On the exact form of waves near the surface of deep water, Philos. Trans. R. Soc., 1863, pp. 127–138.
Shemer, L. and Liberzon, D., Lagrangian kinematics of steep waves up to the inception of a spilling breaker, Physics of Fluids, 2014, vol. 26, 016601, DOI: 10.1063/1.4860235.
Stokes, G.G., On the theory of oscillatory waves, Trans. Camb. Philos. Soc., 1847, vol. 8, pp. 441–455.
Stokes, G.G., Supplement to a paper on the Theory of oscillatory waves, Math. and Phys. Papers, Cambridge: CUP, 1880, vol. 1, pp. 314–326.
Yuen, H.C. and Lake, B.M., Nonlinear deep water waves: Theory and experiment, The Phys. of Fluids, 1975, vol. 18, no. 8 pp. 956–960.
Zufiria, J.A., Non-symmetric gravity waves on water of infinite depth, J. Fluid Mech., 1987, vol. 181, pp. 17–39, DOI: 10.1017/S002211208700199X.
A “rogue wave” is large, unexpected, and dangerous, http://oceanservicenoaagov/facts/roguewaveshtml
Author information
Authors and Affiliations
Corresponding author
Additional information
This work was partially supported by the Russian Academy of Sciences (Program P23) and RFBR (project 15-01-09235).
The article is published in the original.
Rights and permissions
About this article
Cite this article
Kistovich, A.V., Chashechkin, Y.D. Analytical models of stationary nonlinear gravitational waves. Water Resour 43, 86–94 (2016). https://doi.org/10.1134/S0097807816120083
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0097807816120083