Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-25T14:27:33.788Z Has data issue: false hasContentIssue false
  • English
  • Français

Bounds for surface solitary waves

Published online by Cambridge University Press:  24 October 2008

G. Keady  and
W. G. Pritchard
G. Keady
Affiliation:
Fluid Mechanics Research Institute, University of Essex
W. G. Pritchard
Affiliation:
Fluid Mechanics Research Institute, University of Essex
Get access Rights & Permissions [Opens in a new window]

Extract

In these notes we give proofs of some properties of surface solitary waves. Assuming the existence of solitary-wave solutions to the nonlinear boundary-value problem (P) defined below, it is shown (i) that the wave is a wave of elevation alone, and (ii) that at large distances it is asymptotic to a uniform supercritical stream (i.e. the Froude number , where c is the speed of the stream, h is its depth, and g is the gravity constant).

We also deduce a number of inequalities relating F2 to a/h, where a is the maximum displacement of the free surface from its value at infinity. In particular, it is shown for the wave of greatest height that 1·480 < F2 < 2.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 76 , Issue 1 , July 1974 , pp. 345 - 358
Copyright
Copyright © Cambridge Philosophical Society 1974

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

REFERENCES

(1)Abramowitz, M. & Stegun, I. A.Handbook of mathematical functions (Dover, 1965).Google Scholar
(2)Camfield, F. E. & Street, R. L.Shoaling of solitary waves on small slopes. Proc. ASCE, Jnl of Waterways & Harbours Div. 91 (1969), 1.Google Scholar
(3)Fenton, J.A ninth-order solution for the solitary wave. J. Fluid Mech. 53 (1972), 257.CrossRefGoogle Scholar
(4)Gilbarg, D. Jets and cavities, . In Handbuch der Physik, vol. 9 (ed. Flugge, , Springer, 1960).Google Scholar
(5)Gradshteyn, I. S. & Ryzhik, J. M. 1965 Tables of integrals, series and products (Academic, 1966).Google Scholar
(6)Krasovskii, Yu. P.Existence of aperiodic flows with free boundaries. Dokl. Akad. Nauk 133 (1960), 768.Google Scholar
(7)Lavrentiev, M. A.Variational methods (Noordhoff, 1964).Google Scholar
(8)Lewy, H.A note on harmonic functions and a hydrodynamical application. Proc. Amer. Math. Soc. 3 (1952), 111.CrossRefGoogle Scholar
(9)Long, R. R.Solitary waves in one and two-fluid systems. Telles 8 (1956), 460.CrossRefGoogle Scholar
(10)Scott, Russell J. Report on waves. Rep. 14th meeting of the British Association, p. 311 (London; John Murray, 1845).Google Scholar
(11)Serrin, J. B.Existence theorems for some hydrodynamical free boundary problems. J. Rat. Mech. Anal. 1 (1952), 1.Google Scholar
(12)Starr, J. B.Two hydrodynamic comparison theorems. J. Rat. Mech. Anal. 1 (1952), 563.Google Scholar
(13)Starr, V. P.Momentum and energy integrals for gravity waves of finite height. J. Marine Res. 6 (1947), 176.Google Scholar
(14)Wehausen, J. V. & Laitone, E. V. Surface waves. In Handbuch der Physik, vol. 9 (ed. Flugge, , Springer, 1960).Google Scholar