Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-29T12:42:12.836Z Has data issue: false hasContentIssue false
  • English
  • Français

The solitary wave on a stream with an arbitrary distribution of vorticity

Published online by Cambridge University Press:  28 March 2006

T. Brooke Benjamin
T. Brooke Benjamin
Affiliation:
Department of Engineering, University of Cambridge
Get access Rights & Permissions [Opens in a new window]

Abstract

The theoretical work reported herein makes a departure from the many previous analyses of the solitary wave which have treated the wave as an example of irrotational fluid motion. The present analysis is of more general scope in that it covers the whole category of examples in which the wave may propagate in either direction on a horizontal stream whose primary velocity distribution U(y) is an arbitrary function (i.e. there is no restriction on the extent of the variations of U(y)). An approximate form of the wave profile is found in general to be a sech2 {(x − ct)/b}, as it is according to previous theories applicable to the wave on a uniform stream, but the relationships amongst the wave amplitude a, the length scale b, and the two propagation velocities c (positive downstream and negative upstream) depend in complicated fashion on the form of U(y).

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 12 , Issue 1 , January 1962 , pp. 97 - 116
Copyright
© 1962 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Benjamin, T. B. & Lighthill, M. J. 1954 On cnoidal waves and bores. Proc. Roy. Soc. A, 224, 448.Google Scholar
Biésel, F. 1950 Etude théorique de la houle en eau courante. Houille blanche, 5, 279.Google Scholar
Boussinesq, J. 1871 Théorie de l'intumescence liquide appeleé onde solitaire ou de translation se propageant dans un canal rectangulaire. Comptes rendus 73, 755.Google Scholar
Boussinesq, J. 1877 Essai sur la théorie des eaux courantes. Mem. pres. Acad. Sci., Paris, Vol. 23.
Burns, J. C. 1953 Long waves in running water. Proc. Camb. Phil. Soc. 48, 695.Google Scholar
Friedrichs, K. O. & Hyers, D. H. 1954 The existence of solitary waves. Comm. Pure Appl. Math. 7, 517.Google Scholar
Keulegan, G. H. & Patterson, G. W. 1940 Mathematical theory of irrotational translation waves. J. Res. nat. Bur. Stand, 24, 47.Google Scholar
Keller, J. B. 1948 The solitary wave and periodic waves in shallow water. Comm. Pure Appl. Math. 1, 323.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. (5), 39, 422.Google Scholar
Lamb, H. 1932 Hydrodynamics, 6th ed. Cambridge University Press.
Long, R. R. 1953 Some aspects of the flow of stratified fluids. Tellus 3, 42.Google Scholar
Long, R. R. 1956 Solitary waves in one- and two-fluid systems. Tellus 8, 460.Google Scholar
McCowan, J. 1891 On the solitary wave. Phil. Mag. (5), 32, 45.Google Scholar
Packham, B. A. 1952 The theory of symmetrical gravity waves of finite amplitude. II. The solitary wave. Proc. Roy. Soc. A, 213, 238.Google Scholar
Peters, A. S. & Stoker, J. J. 1960 Solitary waves in liquids having non-constant density. Comm. Pure Appl. Math. 8, 115.Google Scholar
Rayleigh, Lord 1876 On waves. Phil. Mag. (5), 1, 257. (Collected Scientific Papers, Vol. 1, p. 251.)Google Scholar
Stoker, J. J. 1957 Water Waves. New York: Interscience.
Thompson, P. D. 1949 The propagation of small surface disturbances through rotational flow. Ann. N. Y. Acad. Sci. 51, 463.Google Scholar
Ursell, F. 1953 The long-wave paradox in the theory of gravity waves. Proc. Camb. Phil. Soc. 49, 685.Google Scholar
Weinstein, A. 1926 Sur la vitesse de propagation de l'onde solitaire. Rend. Accad. Lincei (6), 3, 463.Google Scholar