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Some exact solutions to the nonlinear shallow-water wave equations

  • William Carlisle Thacker (a1)
Abstract

These exact solutions correspond to time-dependent motions in parabolic basins. A characteristic feature is that the shoreline is not fixed. It is free to move and must be determined as part of the solution. In general, the motion is oscillatory and has the appropriate small-amplitude limit. For the case in which the parabolic basin reduces to a flat plane, there is a solution for a flood wave. These solutions provide a valuable test for numerical models of inundating storm tides.

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References
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Carrier, G. F. & Greenspan, H. P. 1958 Water waves of finite amplitude on a sloping beach. J. Fluid Mech. 4, 97109.
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Lamb, H. 1945 Hydrodynamics, 6th edn. Dover.
Longuet-Higgins, M. S. 1972 A class of exact, time-dependent, free-surface flows. J. Fluid Mech. 55, 529543.
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Rankine, W. M. J. 1863 On the exact form of waves near the surface of deep water. Phil. Trans. Roy. Soc. A 153, 127138.
Sachdev, P. L. 1980 Exact, self-similar, time-dependent free surface flows under gravity. J. Fluid Mech. 96, 797802.
Thacker, W. C. 1977 Irregular grid finite-difference techniques: simulations of oscillations in shallow circular basins. J. Phys. Oceanogr. 7, 284292.
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Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
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