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The second approximation to cnoidal and solitary waves

Published online by Cambridge University Press:  28 March 2006

E. V. Laitone
E. V. Laitone
Affiliation:
University of California, Berkeley
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Abstract

The expansion method introduced by Friedrichs (1948) for the systematic development of shallow-water theory for water waves of large wavelength was used by Keller (1948) to obtain the first approximation for the finite-amplitude solitary wave of Boussinesq (1872) and Rayleigh (1876), as well as for periodic waves of permanent type, corresponding to the cnoidal waves of Korteweg & de Vries (1895).

The present investigation extends Friedrich's method so as to include terms up to the fourth order from shallow-water theory for a flat horizontal bottom, and thereby obtains the complete second approximations to both cnoidal and solitary waves. These second approximations show that, unlike the first approximation, the vertical motions cannot be considered as negligible, and that the pressure variation is no longer hydrostatic.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 9 , Issue 3 , November 1960 , pp. 430 - 444
Copyright
© 1960 Cambridge University Press

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References

Boussinesq, J. 1872 Théorie des ondes et des remous qui se propagent de long d'un canal rectangulaire horizontal, en commuiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond. J. Math. Pures appl. (Series 2), 17, 55–108.Google Scholar
Daily, J. W. & Stephan, S. C. Jr., 1952 The solitary wave. M.I.T., Hydrodynamics Laboratory, Technical Report, no. 8. Also Proc. 3rd Conf. Coastal Engineering, Cambridge, Mass. (Oct. 1952), pp. 13–30.
Friedrichs, K. O. 1948 On the derivation of the shallow water theory. Appendix to The formation of breakers and bores by J. J. Stoker. Commun. Pure Appl. Math., 1, 81–5.Google Scholar
Friedrichs, K. O. & Hyers, D. H. 1954 The existence-of solitary waves. Commun. Pure Appl. Math. 7, 517–50.Google Scholar
Hunt, J. N. 1955 On the solitary wave of finite amplitude. Houille blanche, 10, 197–203.Google Scholar
Ippen, A. T. & Kulin, G. 1955 Shoaling and breaking characteristics of the solitary wave. M.I.T., Hydrodynamics Laboratory, Technical Report, no. 15 (April).Google Scholar
Keller, J. B. 1948 The solitary wave and periodic waves in shallow water. Commun. Pure Appl. Math., 1, 323–39.Google Scholar
Korteweg, D. J. & De Vries, G. 1895 On the change of form of long waves advancing in a rectangular canal and on a new type of long stationary waves. Phil. Mag. (Ser. 5), 39, 422–43. (London, Dublin and Edinburgh.)Google Scholar
Lamb, H. 1932 Hydrodynamics, 6th ed. Cambridge University Press.
Littman, Walter. 1957 On the existence of periodic waves near critical speed. Commun. Pure Appl. Math. 10, no 2, 241–69.Google Scholar
Long Robert, R., 1956 Solitary waves in the one- and two-fluid systems. Tellus, 8, no 4, 460–71.Google Scholar
McCowan, J. 1894 On the highest wave of permanent type. Phil. Mag. (Ser. 5), 38, 351–8. (London, Dublin and Edinburgh.)Google Scholar
Rayleigh, Lord. 1876 On waves. Phil. Mag. (Ser. 5), 1, 257–79. (London, Dublin and Edinburgh.) Or (1880) Scientific Papers, 1, 252–71.Google Scholar
Stoker, J. J. 1957 Water Waves. New York: Interscience Publishers.
Stokes, G. G. 1880 Mathematical and Physical Papers, 1, 197–229, 314–26. Cambridge University Press.
Truesdell, C. 1956 Leonhardi Euleri Commentationes Mechanicae ad Theoriam Corporum Fluidorum Pertinentes (Euleri Opera Omnia, Ser. ii, Vol. 13). Zürich: Orell Füssli.
Weinstein, A. 1926 Sur la vitesse de propagation de l'onde solitaire. Proc. 2nd Int. Congr. Appl. Mech. Zurich (Sept.), pp., 445–8.
Yamada, Hikoji. 1957 On the highest solitary wave. Reports of Research Institute for Applied Mechanics, Kyushu University, vol v, no. 18, pp. 53–67.