Abstract
Marine hydrodynamics is characterised by both weak nonlinearities, as seen for example in drift forces, and strong nonlinearities, as seen for example in wave breaking. In many cases their relative importance is still a controversial matter. The phenomenon of particle escape, seen in linear theory, appears to offer a guide to when strongly nonlinear effects will start to become important, and what will happen when they do.
In the case of the “ringing” of vertical cylinders in steep waves, particle escape is shown to correspond approximately to local wave breaking, which leads to the cavitation responsible for “ringing”. Another example is rogue waves, where recent results from weakly nonlinear theory are disappointing, and also fail to explain the rogue waves seen in relatively shallow water, as in the data from the Draupner and Gorm platforms. Recent laboratory experiments, too, show wave crests continuing to grow in height after all frequency components have come into phase, which is inconsistent with weakly nonlinear theory. Particle escape, which is more frequent in shallow water, offers a simple alternative explanation for these observations, as well as for the violent motion at the wave crests, which often confuses rogue-wave data. Extreme wave crests have long been known to be strongly nonlinear, so it appears possible that rogue waves are primarily a strongly nonlinear phenomenon.
Fully nonlinear computations of two interacting regular waves are presented, to explore further the connection between particle escape and wave breaking. They are combined with Monte-Carlo simulations of particle escape in hurricane conditions, and the very few measurements of large breaking waves during hurricanes. It is concluded that large breaking waves will have occurred about once per hour, and once per 100 h, respectively, in the recent hurricanes LILI and IVAN. These findings call into question the use of non-breaking wave models in the design codes for fixed steel offshore structures.
Similar content being viewed by others
References
Grue J (ed) (2005) Proc. 20th Int. Workshop on Water Waves and Floating Bodies. R. Inst. Naval Architects, London (see www.rina.org.uk)
Stokes GG (1880) On the theory of oscillatory waves. Trans. Camb. Phil. Soc. (1847). In: Mathematical & physical papers by G.G. Stokes, vol 1, Cambridge University Press, pp 197–219
Schwartz LW (1974) Computer extension and analytic continuation of Stokes’ expansion for gravity waves. J Fluid Mech 62:553–578
Malenica S, Molin B (1995) Third-harmonic wave diffraction by a vertical cylinder. J Fluid Mech 302:203–229
Faltinsen OM, Newman JN, Vinje T (1995) Nonlinear wave loads on a slender vertical cylinder. J Fluid Mech 289:179–198
Grue J, Bjorshol G, Strand Ø (1995) Higher harmonic wave exciting forces on a vertical cylinder. Institute of Mathematics, University of Oslo Preprint. No. 2. ISBN 82-553-0862-8
Chaplin JR, Rainey RCT, Yemm RW (1997) Ringing of a vertical cylinder in waves. J Fluid Mech 350:119–147
Rainey RCT, Chaplin JR (2003) Wave breaking and cavitation around a vertical cylinder: experiments and linear theory In: Clement AH, Ferrant P (eds), Proc. 18th Int. Workshop on Water Waves and Floating Bodies. R. Inst. Naval Architects, London (see www.rina.org.uk)
Sarpkaya T, Isaacson M (1981) Mechanics of Wave Forces on Offshore Structures. Van Nostrand Reinhold, New York
Benjamin TB, Feir JE (1967) The disintegration of wavetrains in deep water, Part 1, Theory. J Fluid Mech 27:417–430
Mei CC (1989) The applied dynamics of ocean surface waves. World Scientific, Singapore
Peregrine DH (1983) Water waves, nonlinear Schrödinger equations and their solutions. J Aust Math Soc B25 Pt 1:16–43
Janssen PAEM (2003) Nonlinear four-wave interactions and freak waves. J Phys Oceanogr 33:863–884
Socquet-Juglard H, Dysthe K, Trulsen K, Krogstad HE, Liu J (2005) Probability distributions of surface gravity waves during spectral change. J Fluid Mech 542:195–216
Walker DAG, Taylor PH, Eatock Taylor R (2005) The shape of large surface waves on the open sea and the Draupner new year wave. Appl Ocean Res 26:73–83
Janssen PAEM, Onorato M (2005) The shallow water limit of the Zakharov equation and consequences for (freak) wave prediction. Tech. Memorandum No. 464, European Centre for Medium-Range Weather Forecasts
Skourup J, Hansen N-EO, Andreasen KK (1997) Non-Gaussian extreme waves in the central North Sea. Trans ASME 119:146–150
Ramamonjiarisoa A, Coantic M (1976) Loi experimentale de dispersion des vagues produites par le vent sur une faible longueur d’action. Cr hebd Séanc Acad Sci Paris B282:111–113
Chaplin JR (1996) On frequency-focusing unidirectional waves. Int J Offshore Polar Engng 6(2):131–137
Longuet-Higgins MS, Phillips OM (1962) Phase velocity effects in tertiary wave interactions. J Fluid Mech 12:333-336
Rice JA (1995) Mathematical statistics and data analysis 2nd edn. Wadsworth Publishing Co, Belmont CA
Banner ML, Peregrine DH (1993) Wave breaking in deep water. Annu Rev Fluid Mech 25:373–397
Waseda T (2006) Nonlinear wave generation in a tank (in Japanese) KANRIN J. Japan Soc. Naval Architects and Ocean Engineers, vol. 4
Sterndorff MJ, Gronbech J (2000) Short term probability distributions for extreme crest heights. Proc. 19th Int. Conf. on Offshore Mechanics and Arctic Engng. Trans ASME
Forristall GZ (2000) Wave crest distributions: observations and second-order theory. J Phys Oceanogr 30:1931–1943
Stansell P (2005) Distributions of extreme wave, crest and trough heights measured in the North Sea. Ocean Engng 32:1015–1036
Ochi MK (1998) Ocean waves—the stochastic approach. Cambridge University Press
Buckley WH (2005) Extreme waves for ship and offshore platform design: an overview. Proc. 3rd Int. Conf. on Design and Operation for Abnormal Conditions. R. Inst. Naval Architects, London (see www.rina.org.uk)
Earle MD (1975) Extreme conditions during hurricane Camille. J Geophys Res 80(3):377–379
Longuet-Higgins MS (1969) On wave breaking and the equilibrium spectrum of wind-generated waves. Proc R Soc London A310:151–159
Longuet-Higgins MS, Fox MJH (1977) Theory of the almost-highest wave: the inner solution. J Fluid Mech 80:721–741
Rainey RCT (2002) “Escape” of particle trajectories in linear irregular waves: a new explanation for wave breaking and model of breaking waves. In: Rainey RCT, Lee SF (eds) Proc. 17th Intl Workshop on Water Waves and Floating Bodies. R. Inst. Naval Architects, London, pp 155–158 (see www.rina.org.uk)
Longuet-Higgins MS, Tanaka M (1997) On the crest instabilities of steep surface waves. J Fluid Mech 336:51–68
Longuet-Higgins MS, Dommermuth DG (1997) Crest instabilities of gravity waves. Part 3. Nonlinear development and breaking. J Fluid Mech 336:33–50
Longuet-Higgins MS (1981) On the overturning of gravity waves. Proc R Soc London A376:377–400
Dold JW, Peregrine DH (1986) An efficient boundary-integral method for steep unsteady water waves. In: Morton KW, Baines MJ (eds) Numerical Methods for Fluid Dynamics II. Oxford University Press, pp 671–679
Dold JW (1992) An efficient surface-integral algorithm applied to unsteady gravity waves. J Comp Phys 103(1):90–115
Longuet-Higgins MS, Cokelet ED (1976) The deformation of steep surface waves on water. I. A numerical method of computation. Proc R Soc London A358:1–26
Longuet-Higgins MS, Stewart RW (1960) Changes in the form of short gravity waves on long waves and tidal currents. J Fluid Mech 8:565–583
Longuet-Higgins MS, Cleaver RP (1994) Crest instabilities of gravity waves. Part 1. The almost-highest wave. J Fluid Mech 258:115–119
Tuck EO (1965) The effect of nonlinearity at the free surface on flow past a submerged cylinder. J Fluid Mech 22:401–404
Wehausen JV (1973) The wave resistance of ships. Adv Appl Mech 13:93–245
Lamb H (1932) Hydrodynamics, 6th edn. Cambridge University Press
Biesel F (1952) Study of wave propagation in water of gradually varying depth. Gravity Waves, U.S. National Bureau of Standards, Washington, Circular no. 521, pp. 243–253
Gerstner F (1802) Theorie der Wellen Abh. d. k. Böhm. Ges. d. Wiss. (1802) [Gilbert’s Annalen d. Physik, xxxii (1809)]
Gjøsund SH, Moe G, Arntsenø A (2001) Kinematics in broad-banded irregular ocean waves by a Lagrangian formulation. Proc. 20th Int. Conf. on Offshore Mech. and Arctic Engng. Conf. pp 681–688. Trans. ASME
Wheeler JD (1970) Method of calculating forces produced by irregular waves, Proc. Offshore Technology Conference (OTC, Houston) vol 1, pp 71–82
Unna PJH (1941) “White horses”. Nature, London. 148:226–227
Phillips OM (1981) The dispersion of short wavelets in the presence of a dominant long wave. J Fluid Mech 107:465–485
Longuet-Higgins MS (1987) The propagation of short surface waves on longer gravity waves. J Fluid Mech 177:293–306
Zhang J, Hong K, Yue DKP (1993) Effects of wavelength ratio on wave modelling. J Fluid Mech 248:107–127
American Petroleum Institute (2000) Recommended Practice for Planning, Designing and Constructing Fixed Offshore Platforms (RP 2A-WSD) 21st edn.
New AL, McIver P, Peregrine DH (1985) Computations of overturning waves. J Fluid Mech 150:233–251
Rainey RCT (1989) A new equation for calculating wave loads on offshore structures. J Fluid Mech 204:295–324
Rainey RCT (1995) Slender-body expressions for the wave load on offshore structures. Proc R Soc London A450:391–416
Rainey RCT (1989) Breaking wave loads on immersed members of offshore structures. HSE Rep. OTH 89 311 HSE Books, London
Heideman JC, Weaver TO (1992) Static wave force procedure for platform design. Proc. 5th Int. Conf. on Civil Engineering in the Oceans. College Station, Texas. American Society of Civil Engineers
Creamer DB, Henyey F, Schult R, Wright J (1989) Improved linear representation of ocean surface waves. J Fluid Mech 205:135–161
Longuet-Higgins MS (1973) On the form of the highest progressive and standing waves in deep water. Proc R Soc London A331:445-456
Rainey RCT, Longuet-Higgins MS (2006) A close one-term approximation to the highest Stokes wave on deep water. Ocean Engng 33:2012–2024
Bebernes J, Eberly D (1989) Mathematical problems from combustion theory. Springer Verlag
Sulem C, Sulem P-L (1999) The nonlinear schrodinger equations: self-focusing and wave collapse. Springer Verlag, New York
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Rainey, R.C.T. Weak or strong nonlinearity: the vital issue. J Eng Math 58, 229–249 (2007). https://doi.org/10.1007/s10665-006-9126-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10665-006-9126-2