Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-27T12:44:09.308Z Has data issue: false hasContentIssue false
  • English
  • Français

Reflection of a shallow-water soliton. Part 1. Edge layer for shallow-water waves

Published online by Cambridge University Press:  20 April 2006

N. Sugimoto  and
T. Kakutani
N. Sugimoto
Affiliation:
Department of Mechanical Engineering, Faculty of Engineering Science. Osaka University, Toyonaka, Osaka 560, Japan
T. Kakutani
Affiliation:
Department of Mechanical Engineering, Faculty of Engineering Science. Osaka University, Toyonaka, Osaka 560, Japan
Get access Rights & Permissions [Opens in a new window]

Abstract

To investigate reflection of a shallow-water soliton at a sloping beach, the edge-layer theory is developed to obtain a ‘reduced’ boundary condition relevant to the simplified shallow-water equation describing the weakly dispersive waves of small but finite amplitude. An edge layer is introduced to take account of the essentially two-dimensional motion that appears in the narrow region adjacent to the beach. By using the matched-asymptotic-expansion method, the edge-layer theory is formulated to cope with the shallow-water theory in the offshore region and the boundary condition at the beach. The ‘reduced’ boundary condition is derived as a result of the matching condition between the two regions. An explicit edge-layer solution is obtained on assuming a plane beach.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 146 , September 1984 , pp. 369 - 382
Copyright
© 1984 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

Cole, J. D. 1968 Perturbation Methods in Applied Mathematics. Blaisdell.
Funakoshi, M. & Oikawa, M. 1982 A numerical study on the reflection of a solitary wave in shallow water. J. Phys. Soc. Japan 51, 1018–1023.Google Scholar
Hibberd, S. & Peregrine, D. H. 1979 Surf and run-up on a beach: a uniform bore. J. Fluid Mech. 95, 323–345.Google Scholar
Johnson, R. S. 1973 On the development of a solitary wave moving over an uneven bottom. Proc. Camb. Phil. Soc. 73, 183–203.Google Scholar
Kakutani, T. 1971 Effects of an uneven bottom on gravity waves. J. Phys. Soc. Japan 30, 272–276 (and errata: 30, 593).Google Scholar
Keller, H. B., Levine, D. A. & Whitham, G. B. 1960 Motion of a bore over a sloping beach. J. Fluid Mech. 7, 302–316.Google Scholar
Kim, S. K., Liu P. L.-F. & Liggett, J. A. 1983 Boundary integral equation solutions for solitary wave generation, propagation and run-up. Coastal Engng 7, 299–317.Google Scholar
Mei, C. C. & Le Méhauté, B. 1966 Note on the equations of long waves over an uneven bottom. J. Geophys. Res. 71, 393–400.Google Scholar
Meyer, R. E. & Taylor, A. D. 1972 Run-up on beaches. In Waves on Beaches (ed. R. D. Meyer), Academic.
Miles, J. W. 1977a Obliquely interacting solitary waves. J. Fluid Mech. 79, 157–169.Google Scholar
Miles, J. W. 1977b Resonantly interacting solitary waves. J. Fluid Mech. 79, 171–179.Google Scholar
Miles, J. W. 1980 Solitary waves. Ann. Rev. Fluid Mech. 12, 11–43.Google Scholar
Milne-Thomson, L. M. 1962 Theoretical Hydrodynamics. Macmillan.
Oikawa, M. & Yajima, N. 1973 Interactions of solitary waves-a perturbation approach to nonlinear systems. J. Phys. Soc. Japan 34, 1093–1099.Google Scholar
Pedersen, G. & Gjevik, B. 1983 Run-up of solitary waves. J. Fluid Mech. 135, 283–299.Google Scholar
Peregrine, D. H. 1967 Long waves on a beach. J. Fluid Mech. 27, 815–827.Google Scholar
Sugimoto, N. 1981a Nonlinear theory for flexural motions of thin elastic plate, part 1: higher order theory. Trans. ASME E: J. Appl. Mech. 48, 377–382.Google Scholar
Sugimoto, N. 1981b Nonlinear theory for flexural motions of thin elastic plate, part 2: boundary layer theory near the edge. Trans. ASME E: J. Appl. Mech. 48, 383–390.Google Scholar
Whitham, G. B. 1974 Linear and Nonlinear Waves. Wiley-Interscience.
Whitham, G. B. 1979 Lectures on Wave Propagation. Springer.