Skip to main content
×
Home
    • Aa
    • Aa

Surface waves of large amplitude beneath an elastic sheet. Part 1. High-order series solution

  • Lawrence K. Forbes (a1) (a2)
Abstract

Two-dimensional periodic waves beneath an elastic sheet resting on the surface of an infinitely deep fluid are investigated using a high-order series-expansion technique. The solution is found to have certain features in common with capillary-gravity waves; specifically, there is a countably infinite set of values of the flexural rigidity of the sheet at which the series solution fails, and these values are conjectured to be bifurcation points of the solution. Limiting waves of maximum height are found at each value of the flexural rigidity investigated. These are characterized by a cusp singularity in the elastic bending moment at the wave crest, and infinite fluid pressure there. For at least one value of the flexural rigidity, the series solution shows that the wave of maximum height also travels with infinite speed.

Copyright
References
Hide All
Cokelet, E. D. 1977 Steep gravity waves in water of arbitrary uniform depth. Phil. Trans. R. Soc. Lond. A 286, 183230.
Domb, C. & Sykes, M. F. 1957 On the susceptibility of a ferromagnetic above the Curie point. Proc. R. Soc. Lond. A 240, 214228.
Donnell, L. H. 1976 Beams, Plates and Shells. McGraw-Hill.
Evans, D. V. & Davies, T. V. 1968 Wave-ice interaction. Davidson Laboratory Report 1313, Stevens Institute of Technology, Hoboken, New Jersey.
Hogan, S. J. 1980 Some effects of surface tension on steep water waves. Part 2. J. Fluid Mech. 96, 417445.
Hogan, S. J. 1981 Some effects of surface tension on steep water waves. Part 3. J. Fluid Mech. 110, 381410.
Longuet-Higgins, M. S. 1975 Integral properties of periodic gravity waves of finite amplitude. Proc. R. Soc. Lond. A 342, 157174.
Müller, A. & Ettema, R. 1984a Dynamic response of an ice-breaker hull to ice breaking. Iowa Institute of Hydraulic Research Report IIHR273, The University of Iowa, Iowa City.
Müller, A. & Ettema, R. 1984b Dynamic response of an icebreaker hull to ice breaking. In Proc. IAHR Ice Symp., Hamburg, vol. II, pp. 287296.
Schwartz, L. W. 1974 Computer extension and analytic continuation of Stokes’ expansion for gravity waves. J. Fluid Mech. 62, 553578.
Schwartz, L. W. & Fenton, J. D. 1982 Strongly nonlinear waves. Ann. Rev. Fluid Mech. 14, 3960.
Schwartz, L. W. & Vanden Broeck, J.-M. 1979 Numerical solution of the exact equations for capillary-gravity waves. J. Fluid Mech. 95, 119139.
Shanks, D. 1955 Non-linear transformation of divergent and slowly convergent sequences. J. Maths & Phys. 34, 142.
Squire, V. A. 1984 On the critical angle for ocean waves entering shore fast ice. Cold Reg. Sci. Tech. 10, 5968.
Stoker, J. J. 1957 Water Waves. Wiley-Interscience.
Stokes, G. G. 1849 On the theory of oscillatory waves. Trans. Camb. Phil. Soc. 8, 441455.
Stokes, G. G. 1880 In Mathematical and Physical Papers, vol. 1. Cambridge University Press.
Tuck, E. O. 1982 An inviscid theory for sliding flexible sheets. J. Austral. Math. Soc. B 23, 403415.
Wilton, J. R. 1915 On ripples. Phil. Mag. 29, 688700.
Wynn, P. 1966 On the convergence and stability of the epsilon algorithm. SIAM J. Numer. Anal. 3, 91121.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 21 *
Loading metrics...

Abstract views

Total abstract views: 118 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 19th October 2017. This data will be updated every 24 hours.