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Asymptotic expansions in the problem of a solitary wave

Published online by Cambridge University Press:  26 April 2006

E. A. Karabut
E. A. Karabut
Affiliation:
Lavrentyev Institute of Hydrodynamics, Novosibirsk, 630090, Russia
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Abstract

There are a number of papers devoted to the construction of the exact solitary wave solution using a series. Power series in amplitude or Fourier series have usually been used. In the present paper we accomplish the exact summation of the Witting (1975) series and show that this series describes other flows, not solitary waves. One such flow is fluid suction under a curvilinear roof. The left half of it is similar to the left half of a maximal-amplitude solitary wave flow.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 319 , 25 July 1996 , pp. 109 - 123
Copyright
© 1996 Cambridge University Press

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References

Boussinesq, J. 1871 Theorie de l'intumescence liquide appelée on de solitaire on de translation se propageant dans un canal rectangulaire. Comptes Rendus 72, 755.Google Scholar
Byatt-Smith, J. G. B. & Longuet-Higgins, M. S. 1976 On the speed and profile of steep solitary waves. Proc. R. Soc. Lond. A 350, 175.Google Scholar
Craig, W. & Sternberg, P. 1988 Symmetry of solitary waves. Commun. Partial Diffl Equat. 13, 603.Google Scholar
Davies, T. V. 1951 The theory of symmetrical gravity waves of finite amplitude I. Proc. R. Soc. Lond. A 208, 475.Google Scholar
Fenton, J. 1972 A ninth-order solution for the solitary wave. J. Fluid Mech. 53, 257.Google Scholar
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. Maths 1, 81.Google Scholar
Friedrichs, K. O. & Hyers, D. H. 1954 The existence of solitary waves. Commun. Pure Appl. Maths 7, 517.Google Scholar
Goody, A. J. & Davies, T. V. 1957 The theory of symmetrical gravity waves of finite amplitude IV. Steady, symmetrical, periodic waves in a channel of finite depth. Q. J. Mech. Appl. Maths 10, 1.Google Scholar
Grimshaw, R. 1971 The solitary wave in water of variable depth. Part 2. J. Fluid Mech. 46, 611.Google Scholar
Gurevich, M. I. 1965 Theory of Jets in Ideal Fluids. Academic Press.
Hunter, J. K. & Vanden-Broeck, J.-M. 1983 Accurate computations for steep solitary waves. J. Fluid Mech. 136, 63.Google Scholar
Karabut, E. A. 1994 The numerical analysis of asymptotical representation of solitary waves. Prikl. Mekh. i Tekh. Fiz. (5), 44 (in Russian).Google Scholar
Keller, J. B. 1948 The solitary wave and periodic waves in shallow water. Commun. Pure Appl. Maths 1, 323.Google Scholar
Korteweg, D. J. & Vries, G. de 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. (5) 39, 422.Google Scholar
Laitone, E. V. 1960 The second approximation to solitary and cnoidal waves. J. Fluid Mech. 9, 430.Google Scholar
Lavrentyev, M. A. 1946 On the theory of long waves. Zb. Praz Inst. Matemat. Akad. Ukr. Nayk. 8, 13 (in Ukraine).Google Scholar
Longuet-Higgins, M. S. & Fenton, J. D. 1974 On the mass, momentum, energy and circulation of a solitary wave. II. Proc. R. Soc. Lond. A 340, 471.Google Scholar
Ovsyannikov, L. V. 1991 On an asymptotic representation of solitary waves. Dokl. Acad. Nauk SSSR (3) 318, 556 (in Russian).Google Scholar
Packham, B. A. 1952 The theory of symmetrical gravity waves of finite amplitude II. The solitary wave. Proc. R. Soc. Lond. A 213, 238.Google Scholar
Pennel, S. A. 1987 On a series expansion for the solitary wave. J. Fluid Mech. 179, 557.Google Scholar
Pennel, S. A. & Su, C. H. 1984 A seventeenth-order series expansion for the solitary wave. J. Fluid Mech. 149, 431.Google Scholar
Plotnikov, P. I. 1983 Stokes conjecture proof in the theory of surface water waves. Dokl. Acad. Nauk SSSR (1) 269, 80 (in Russian).Google Scholar
Plotnikov, P. I. 1991 Nonuniqueness of solitary water waves and bifurcation theorem for critical points of smooth functionals. Izv. AN SSSR, Matem. (2) 55, 339 (in Russian).Google Scholar
Rayleigh, Lord 1876 On waves. Phil. Mag. (5) 1, 257.Google Scholar
Richardson, A. R. 1920 Stationary waves in water. Phil. Mag. (6) 256, 97. N 235.Google Scholar
Russel, J. S. 1838 Report of the Committee on waves. Rep Brit. Assn Adv. Sci., 1837, p. 417.Google Scholar
Stokes, G. G. 1880 On the theory of oscillatory waves. Mathematical and Physical Papers, vol. 1, pp. 197, 314.Google Scholar
Toland, J. F. 1978 On the existence of a wave of greatest height and Stokes conjecture. Proc. R. Soc. Lond. A 363, 469.Google Scholar
Villat, H. 1915 Sur l’écoulement des fluides resant. Ann. Sci. École Norm. supér. 32.Google Scholar
Williams, J. M. 1981 Limiting gravity waves in water of finite depth. Phil. Trans. R. Soc. Lond. A 302, 139.Google Scholar
Witting, J. 1975 On the highest and other solitary waves. SIAM J. Appl. Maths. 28, 700.Google Scholar
Witting, J. 1981 High solitary waves in water: results of calculations. NRL Rep. 8505.Google Scholar