Hostname: page-component-7c8c6479df-24hb2 Total loading time: 0 Render date: 2024-03-29T08:11:45.072Z Has data issue: false hasContentIssue false

Eulerian and Lagrangian aspects of surface waves

Published online by Cambridge University Press:  21 April 2006

M. S. Longuet-Higgins
Affiliation:
Department of Applied Mathematics and Theoretical Physics, Silver Street, Cambridge CB3 9EW, UK andInstitute of Oceanographic Sciences, Wormley, Surrey GU8 5UB, UK

Abstract

Surface waves can be recorded in two kinds of ways, either with a fixed (Eulerian) probe or with a free-floating (Lagrangian) buoy. In steep waves, the differences between corresponding properties can be very marked.

By a simple physical model and by accurate calculation it is shown that the Lagrangian wave period may differ from the Eulerian wave period by as much as 38 %. The Lagrangian mean level is also higher than the Eulerian mean, leading to possible discrepancies in remote sensing of the ocean from satellites.

Surface accelerations are of interest in relation to the incidence of breaking waves, and for interactions between short (gravity or capillary) waves and longer gravity waves. Eulerian accelerations tend to be very non-sinusoidal, with large downwards peaks, sometimes exceeding - g in magnitude, near to sharp wave crests. Lagrangian accelerations are much smoother; for uniform gravity waves they lie between −0.388g and +0.315g. These values are verified by laboratory experiments. In wind-generated waves the limits are probably wider.

In progressive gravity waves in deep water the horizontal accelerations generally exceed the vertical accelerations. In steep waves, the subsurface accelerations can slightly exceed those at the free surface.

A novel application is made to the rolling motion of ships. In very steep, irrotational waves it is shown theoretically that the flow near the wave crest can lead to the rotation of the hull through angles up to 120° by a single wave, even if the wave is not breaking. This is confirmed by simple experiments. The efficiency of the keel appears to promote capsizing.

Type
Research Article
Copyright
© 1986 Cambridge University Press

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

Ewing, J. A., Longuet-Higgins, M. S. & Srokosz, M. A. 1986 Measurements of the vertical acceleration in wind waves. J. Phys. Oceanogr. (in press).Google Scholar
Himeno, Y. 1981 Prediction of ship roll damping - state of the art. University of Michigan, Dept of Naval Architecture Rep. No. 239, Sept. 1981, 65 pp.Google Scholar
Kirkman, K. L., Nagle, T. J. & Salsich, J. O. 1983 Sailing yacht capsizing. Proc. Chesapeake Sailing Yacht Symp., Annapolis Md. Jan. 1983. New York: Soc. Nav. Arch. Mar. Eng.
Lamb, H. 1932 Hydrodynamics, 6th edn. Cambridge University Press, 738 pp.
Longuet-Higgins, M. S. 1953 Mass transport in water waves. Phil. Trans. R. Soc. Lond. A 245, 535581.Google Scholar
Longuet-Higgins, M. S. 1963 The generation of capillary waves by steep gravity waves. J. Fluid Mech. 16, 138159.Google Scholar
Longuet-Higgins, M. S. 1974 Breaking waves - in deep or shallow water. Proc. 10th Symp. on Naval Hydrodynamics, Cambridge, Mass. June 1974, pp. 597608. Arlington, Va., Office of Nav. Res., 792 pp.
Longuet-Higgins, M. S. 1975 Integral properties of periodic gravity waves of finite amplitude. Proc. R. Soc. Lond. A 342, 157174.Google Scholar
Longuet-Higgins, M. S. 1979a Why is a water wave like a grandfather clock?. Phys. Fluids 22, 18281829.Google Scholar
Longuet-Higgins, M. S. 1979bThe trajectories of particles in steep, symmetric gravity waves. J. Fluid Mech. 94, 497517.Google Scholar
Longuet-Higgins, M. S. 1979c The almost-highest wave: a simple approximation. J. Fluid Mech. 94, 269273.Google Scholar
Longuet-Higgins, M. S. 1985 Accelerations in steep gravity waves. J. Phys. Oceanogr. 15, 15701579.Google Scholar
Longuet-Higgins, M. S. 1986 The propagation of short surface waves on longer gravity waves. Proc. R. Soc. Lond. A. (in press).Google Scholar
Longuet-Higgins, M. S. & Fox, M. J. H. 1977 Theory of the almost-highest wave: the inner solution. J. Fluid Mech. 80, 721740.Google Scholar
Longuet-Higgins, M. S. & Fox, M. J. H. 1978 Theory of the almost-highest wave. Part 2. Matching and analytic extension. J. Fluid Mech. 85, 769786.Google Scholar
Ochi, M. K. & Tsai, C.-H. 1983 Prediction of breaking waves in deep water. J. Phys. Oceanogr. 13, 20082019.Google Scholar
Phillips, O. M. 1958 The equilibrium range in the spectrum of wind generated waves. J. Fluid Mech. 4, 426434.Google Scholar
Srokosz, M. A. 1986 A note on the probability of wave breaking. J. Phys. Oceanogr. 16, 382385.Google Scholar
Srokosz, M. A. & Longuet-Higgins, M. S. 1986 On the skewness of sea-surface elevation. J. Fluid Mech. 164, 487497.Google Scholar
Taylor, G. I. 1921 Diffusion by continuous movements. Proc. Lond. Math. Soc. (ser. 2) 20, 196212.Google Scholar
Taylor, G. I. 1953 An experimental study of standing waves. Proc. R. Soc. Lond. A 218, 4459.Google Scholar
Williams, J. M. 1985 Near-limiting waves in water of finite depth. Phil. Trans. R. Soc. Lond. A 314, 353377.Google Scholar