Abstract
A simple and efficient way to calculate steep gravity waves is described, which avoids the use of power series expansions or integral equations. The method exploits certain relations between the coefficients in Stokes’s expansion which were discovered by the author in 1978.
The method yields naturally the critical wave steepnesses for bifurcation of regular waves into non-uniform steady waves. Moreover, truncation of the series after only two terms yields a simple model for Class 2 bifurcation.
The analysis can be used to discuss the stability of steep gravity waves and to derive new integral relations. Particularly relevant to breaking waves are some new relations for the angular momentum. The level of action ya for a limiting wave can also be expressed in terms of the Fourier coefficients.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Chen, B. and Saffman, P.G. 1980 Numerical evidence for the existence of new types of gravity waves of permanent form on deep water. Studies in Appl. Math. 62, 1–21.
Lamb, H. 1932 Hydrodynamics, 6th ed. Cambridge Univ. Press 738 pp.
Longuet-Higgins, M.S. 1975 Integral properties of periodic gravity waves of finite amplitude. Prof. R. Soc. Lond. A. 342, 157–174.
Longuet-Higgins, M.S. 1978a Some new relations between Stokes’s coefficients in the theory of gravity waves. J. Inst. Maths. Applics. 22, 261–273.
Longuet-Higgins, M.S. 1978b The instabilities of gravity waves of finite amplitude in deep water. I. Superharmonics. Proc. R. Soc. Lond. A. 360, 471–488.
Longuet-Higgins, M.S. 1980 Spin and angular momentum in gravity waves. J. Fluid Mech. 97, 1–25.
Longuet-Higgins, M.S. 1984a (Paper I) New integral relations for gravity waves of finite amplitude. J. Fluid Mech 149, 205–215.
Longuet-Higgins, M.S. 1984b (Paper III) On the stability of steep gravity waves. Proc. R. Soc. Lond. A 396, 269–280.
Longuet-Higgins, M.S. 1985 (Paper II) Bifurcation in gravity waves. J. Fluid Mech. (in press).
Longuet-Higgins, M.S. and Fox, M.J.H. 1978 Theory of the almost-highest wave. Part 2. Matching and analytic extension. J. Fluid Mech. 85, 769–786.
McLean, J.W. 1982 Instabilities of finite-amplitude water waves. J. Fluid Mech. 114, 315–330.
Schwartz, L.W. 1974 Computer extension and analytic continuation of Stokes’ expansion for gravity waves. J. Fluid Mech. 62, 553–578.
Stokes, G.G. 1880 Supplement to a paper on the theory of oscillatory waves. Mathematical and Physical Papers 1, 225–228, Cambridge Univ. Press.
Tanaka, M. 1983 The stability of steep gravity waves. J. Phys. Soc. Japan 52, 3047–3055.
Williams, J.M. 1981 Limiting gravity waves in water of finite depth. Phil.Trans. R. Soc. Lond. A 302, 139–188.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1985 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Longuet-Higgins, M.S. (1985). A New Way to Calculate Steep Gravity Waves. In: Toba, Y., Mitsuyasu, H. (eds) The Ocean Surface. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-7717-5_1
Download citation
DOI: https://doi.org/10.1007/978-94-015-7717-5_1
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-8415-6
Online ISBN: 978-94-015-7717-5
eBook Packages: Springer Book Archive