Hostname: page-component-848d4c4894-x24gv Total loading time: 0 Render date: 2024-06-02T04:24:37.339Z Has data issue: false hasContentIssue false
  • English
  • Français

Cylindrical solitary waves

Published online by Cambridge University Press:  21 April 2006

P. D. Weidman  and
R. Zakhem
P. D. Weidman
Affiliation:
Department of Mechanical Engineering, University of Colorado, Boulder, CO 80309, USA
R. Zakhem
Affiliation:
Department of Mechanical Engineering, University of Colorado, Boulder, CO 80309, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

Experiments on the radial propagation of axisymmetric free-surface solitary waves are reported and compared with theoretical and numerical solutions of the cylinderical Korteweg–de Vries (CKdV) equation. A new experimental technique to obtain a continuous amplitude signature on photographic paper is reported. These measurements show that an isolated disturbance evolves into a slowly varying solitary wave with amplitude decaying as $r^{-\frac{2}{3}}$, where r is the radius measured from the centre of the disturbance. A numerical study of the CKdV equation is made to interpret the transient development of these waves into the nonlinear asymptotic regime. It is further pointed out that the CKdV equation also describes weakly nonlinear axisymmetric internal waves, and a comparison of theory for this case with internal-wave trajectory measurements reported by Maxworthy (1980) exhibit good agreement.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 191 , June 1988 , pp. 557 - 573
Copyright
© 1988 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

Abramovitz, M. & Stegun, I. A. 1975 Handbook of Mathematical Functions. Dover.
Benjamin, T. B. 1967 Internal waves of permanent form in fluids of great depth. J. Fluid Mech. 29, 559.Google Scholar
Benney, D. J. 1966 Long nonlinear waves in fluid flows. J. Math. Phys. 45, 52.Google Scholar
Calogero, F. & Degasperis, A. 1978 Solution by the spectral-transform method of a nonlinear evolution equation including as a special case the cylindrical KdV equation. Lett. Nuovo Cim. 23, 150.Google Scholar
Chang, P., Melville, W. K. & Miles, J. W. 1979 On the evolution of a solitary wave in a gradually varying channel. J. Fluid Mech. 95, 401.Google Scholar
Chwang, A. T. & Wu, T. Y. 1976 Cylindrical solitary waves. Proc. IUTAM Symp. on Water Waves in Water of Varying Depth, Canberra, Australia.
Cumberbatch, E. 1978 Spike solution for radially symmetric solitary waves. Phys. Fluids 21, 374.Google Scholar
Davis, R. E. & Acrivos, A. 1967 Solitary internal waves in deep water. J. Fluid Mech. 29, 593.Google Scholar
Grimshaw, R. 1979 Slowly varying solitary waves. I. Korteweg—de Vries equation. Proc. R. Soc. Lond. A 368, 359.Google Scholar
Hammack, J. L. & Segur, H. 1974 The Korteweg—de Vries equation and water waves. Part 2. Comparison with experiments. J. Fluid Mech. 65, 625.Google Scholar
Ippen, A. T., Kulin, G. & Raza, M. A. 1955 Damping characteristics of the solitary wave. Hydrodyn. Lab., M.I.T. Tech. Rep. 16.
Johnson, R. S. 1973 On an asymptotic solution of the Korteweg—de Vries equation with slowly varying coefficients. J. Fluid Mech. 60, 813.Google Scholar
Kakutani, T. 1971 Effect of an uneven bottom on gravity waves. J. Phys. Soc. Japan 30, 272.Google Scholar
Kaup, D. J. & Newell, A. C. 1978 Solitons as particles, oscillators, and in slowly changing media: a singular perturbation theory. Proc. Roy. Soc. Lond. A 361, 413.Google Scholar
Ko, K. 1978 Korteweg—de Vries soliton in a slowly varying medium. PhD thesis, University of Southern California.
Ko, K. & Kuehl, H. H. 1978 Korteweg—de Vries soliton in a slowly varying medium. Phys. Rev. Lett. 40, 233.Google Scholar
Ko, K. & Kuehl, H. H. 1979 Cylindrical and spherical Korteweg—de Vries solitary waves. Phys. Fluids 22, 1343.Google Scholar
Leibovich, S. & Randall, J. D. 1973 Amplification and decay of long nonlinear waves. J. Fluid Mech. 53, 481.Google Scholar
Maxon, S. & Viecelli, J. 1974 Cylindrical solitons. Phys. Fluids 17, 1614.Google Scholar
Maxworthy, T. 1980 On the formation of nonlinear internal waves from the gravitation collapse of mixed regions in two and three dimensions. J. Fluid Mech. 96, 47.Google Scholar
Miles, J. W. 1977 Note on a solitary wave in a slowly varying channel. J. Fluid Mech. 80, 149.Google Scholar
Miles, J. W. 1978 An axisymmetric Boussinesq wave. J. Fluid Mech. 84, 181.Google Scholar
Ono, H. 1975 Algebraic solitary waves in stratified fluids. J. Phys. Soc. Japan 39, 1082.Google Scholar
Ostrovskiy, L. A. & Pelinovskiy, E. N. 1975 Refraction of nonlinear sea waves in a coastal zone. Izv. Acad. Nauk SSSR, Atmos. Ocean. Phys. 11, 37.Google Scholar
Perroud, P. H. 1957 The solitary reflection along a straight vertical wall at oblique incidence. PhD thesis, University of California, Berkeley.
Shuto, N. 1974 Nonlinear waves in a channel of variable section. Coastal Engng Japan 17, 1.Google Scholar
Su, C. H. & Mirie, R. M. 1980 On head-on collisions between two solitary waves. J. Fluid Mech. 98, 509.Google Scholar
Vliegenthart, A. C. 1971 On finite difference methods for the KdV equation. J. Engng Maths 5, 2.Google Scholar
Weidman, P. D. & Johnson, M. 1982 Experiments on leapfrogging internal solitary waves. J. Fluid Mech. 122, 195.Google Scholar
Weidman, P. D. & Maxworthy, T. 1978 Experiments on strong interactions between solitary waves. J. Fluid Mech. 85, 417.Google Scholar