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Reflection of a shallow-water soliton. Part 2. Numerical evaluation.

Published online by Cambridge University Press:  21 April 2006

N. Sugimoto ,
Y. Kusaka  and
T. Kakutani
N. Sugimoto
Affiliation:
Department of Mechanical Engineering, Faculty of Engineering Science, Osaka University, Toyonaka, Osaka 560, Japan
Y. Kusaka
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
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Abstract

Reflection of a shallow-water soliton at a plane beach is investigated numerically based on the edge-layer theory developed in Part 1 of this series. The offshore behaviour in the shallow-water region is first obtained by solving the Boussinesq equation under the ‘reduced’ boundary condition. The spatial and temporal variations of the surface elevation are displayed for two typical values of the inclination angle of the beach. Using these solutions, the nearshore behaviour is then evaluated to obtain the surface elevation and the velocity distribution in the edge layer. Both offshore and nearshore behaviours furnish a full knowledge of the reflection problem of a shallow-water soliton. To check the applicability of the edge-layer theory, a ‘computational experiment’ is carried out based on the boundary-element method, in which the Laplace equation is solved numerically under the full nonlinear boundary conditions at the free surface without introducing the edge-layer concept. Both results show a fairly good agreement for the overall reflection behaviour of a shallow-water soliton except for the surging movement at the shoreline.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 178 , May 1987 , pp. 99 - 117
Copyright
© 1987 Cambridge University Press

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References

Brebbia C. A. (ed.) 1984 Topics in Boundary Element Research, vol. 1. Springer.
Carrier, G. F. & Greenspan, H. P. 1958 Water waves of finite amplitude on a sloping beach. J. Fluid Mech. 4, 97–109.Google Scholar
Gjevik, B. & Pedersen, G. 1981 Run-up of long waves on an inclined plane. Preprint Series Institute of Mathematics, University of Oslo.
Hall, J. V. & Watts, G. M. 1953 Laboratory investigation of the vertical rise of solitary waves on impermeable slopes. US Army, Corps of Engrs, Beach Erosion Board, Tech. Memo no. 33. (cited in R. L. Wiegel 1964 Oceanographical Engineering. Prentice-Hall).
Ippen, A. T. & Kulin, G. 1955 The shoaling and breaking of the solitary wave. In Proc. 5th Conf. Coastal Engng, pp. 27–47.
Jaswon, M. A. & Symm, G. T. 1977 Integral Equation Methods in Potential Theory and Elastostatics. Academic.
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
Laitone, E. V. 1960 The second approximation to cnoidal and solitary waves. J. Fluid Mech. 9, 430–444.Google Scholar
Miles, J. W. 1977 Obliquely interacting solitary waves. J. Fluid Mech. 79, 157–169.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
Street, R. L. & Camfield, F. E. 1966 Observations and experiments on solitary wave deformation. In Proc. 10th Conf. Coastal Engng, pp. 284–301.
Sugimoto, N. & Kakutani, T. 1984 Reflection of a shallow-water soliton. Part 1. Edge layer for shallow-water waves. J. Fluid Mech. 146, 369–382.Google Scholar
Ursell, F. 1953 The long-wave paradox in the theory of gravity waves. Proc. Camb. Phil. Soc. 49, 685–694.Google Scholar
Whitham, G. B. 1974 Linear and Nonlinear Waves. Wiley-Interscience.
Wu, T. Y. 1981 Long waves in ocean and coastal waters. J. Engng Mech. Div. ASCE 107, 501–522.Google Scholar