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Verification of the Benjamin–Lighthill conjecture about steady water waves

Published online by Cambridge University Press:  26 April 2006

T. Brooke Benjamin
T. Brooke Benjamin
Affiliation:
Mathematical Institute, 24–29 St. Giles, Oxford OX1 3LB, UK
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Abstract

The exact problem of steady periodic waves (Stokes waves) on the surface of an ideal liquid above a horizontal bottom is reconsidered in order to confirm a general property conjectured by Benjamin & Lighthill (1954). Specifically, in terms of parameters r and s proportional respectively to the total-head and flow-force constants for steady flows, such waves are proved to realize points (r, s) inside the region of the (r, s)-plane that is bounded by the cusped curve representing all possible uniform streams. A corresponding attribute of steady periodic waves on the surface of an infinitely deep ideal liquid will also be demonstrated. The concluding discussion refers to steady water waves that are not periodic Stokes waves, and comments with reference to Appendix B on the significance of the flow-force invariant s in Hamiltonian representations of the steady-wave problem.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 295 , 25 July 1995 , pp. 337 - 356
Copyright
© 1995 Cambridge University Press

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