Hostname: page-component-7d684dbfc8-zgpz2 Total loading time: 0 Render date: 2023-09-23T23:35:24.176Z Has data issue: false Feature Flags: { "corePageComponentGetUserInfoFromSharedSession": true, "coreDisableEcommerce": false, "coreDisableSocialShare": false, "coreDisableEcommerceForArticlePurchase": false, "coreDisableEcommerceForBookPurchase": false, "coreDisableEcommerceForElementPurchase": false, "coreUseNewShare": true, "useRatesEcommerce": true } hasContentIssue false

Theory of the almost-highest wave: the inner solution

Published online by Cambridge University Press:  11 April 2006

M. S. Longuet-Higgins
Department of Applied Mathematics and Theoretical Physics, Silver Street, Cambridge and Institute of Oceanographic Sciences, Wormley, Surrey
M. J. H. Fox
Department of Applied Mathematics and Theoretical Physics, Silver Street, Cambridge and Institute of Oceanographic Sciences, Wormley, Surrey


This paper investigates the flow near the summit of steep, progressive gravity wave when the crest is still rounded but the flow is approaching Stokes's corner flow. The natural length scale in the neighbourhood of the summit is seen to be l =q2/2g, where g denotes gravity and q is the particle speed at the crest in a reference frame moving with the wave speed. We show that a class of self-similar smooth local flows exists which satisfy the free-surface condition and which tend to Stokes's corner flow when the radial distance r becomes large compared withl. The behaviour of the solution at large values of r/l is shown to depend on the roots of the transcendental equation \[ K \tan h K = \pi/2\surd{3}. \] The two real roots correspond to a damped oscillation of the free surface decaying like (l/r)½. The positive imaginary roots correspond to perturbations vanishing like higher negative powers of r.

The complete flow is calculated by transforming the domain onto the interior of a circle in the complex plane and expanding the potential at the surface in a Fourier series. The computation is checked by an independent method, based on approximating the flow by a sequence of dipoles. The profile of the surface is found to intersect its asymptote at large values of r/l. This implies that the maximum slope slightly exceeds 30°. The computed value 30·37° is in close agreement with that obtained by extrapolating the maximum slopes of steep gravity waves, as calculated by previous authors. The vertical acceleration of a particle at the crest is 0·388g. In the far field, however, the acceleration tends to the value ½g corresponding to the Stokes corner flow.

Research Article
© 1977 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.)


Byatt-Smith, J. G. B. & Longuet-Higgins, M. S. 1976 On the speed and profile of steep solitary waves. Proc. Roy. Soc. A 350, 175189.Google Scholar
Cokelet, E. D. 1976 Steady and unsteady nonlinear water waves. Ph.D. thesis, Cambridge University.
Grant, M. A. 1973 The singularity at the crest of a finite amplitude progressive Stokes wave. J. Fluid Mech. 59, 257262.Google Scholar
Havelock, T. H. 1918 Periodic irrotational waves of finite height. Proc. Roy. Soc. A 95, 3851.Google Scholar
Keady, G. & Pritchard, W. G. 1974 Bounds for surface solitary waves. Proc. Camb. Phil. Soc. 76, 345358.Google Scholar
Krasovskii, Yu. P. 1961 On the theory of steady-state waves of finite amplitude. Zl. Vych. Mat. Mat. Fiz. 1, 836.Google Scholar
Lenau, C. W. 1966 The solitary wave of maximum amplitude. J. Fluid Mech. 26, 309320.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. 1973 On the form of the highest progressive and standing waves in deep water. Proc. Roy. Soc. A 331, 445456.Google Scholar
Longuet-Higgins, M. S. 1974 On the mass, momentum, energy and circulation of a solitary wave. Proc. Roy. Soc. A 337, 113.Google Scholar
Longuet-Higgins, M. S. 1975 Integral properties of periodic gravity waves of finite amplitude. Proc. Roy. Soc. A 342, 157174.Google Scholar
Longuet-Higgins, M. S. & Fenton, J. D. 1974 On the mass, momentum energy and circulation of a solitary wave. II. Proc. Roy. Soc. A 340, 471493.Google Scholar
Michell, J. H. 1893 The highest waves in water. Phil. Mag. 36, 430437.Google Scholar
Sasaki, K. & Murakami, T. 1973 Irrotational, progressive surface gravity waves near the limiting height. J. Oceanogr. Soc. Japan, 29, 94105.Google Scholar
Schwartz, L. W. 1974 Computer extension and analytic continuation of Stokes's expansion for gravity waves. J. Fluid Mech. 62, 553578.Google Scholar
Schwitters, D. J. 1966 The exact profile of the solitary wave. Ph.D. dissertation, University of Arizona.
Stokes, G. G. 1880 On the theory of oscillatory waves. Appendix B: Considerations relative to the greatest height of oscillatory irrotational waves which can be propagated without change of form. Math. Phys. Papers 1, 225228.Google Scholar
Yamada, H. 1957a Highest waves of permanent type on the surface of deep water. Rep. Res. Inst. Appl. Mech. Kyushu Univ. 5, 3752.Google Scholar
Yamada, H. 1957b On the highest solitary wave. Rep. Res. Inst. Mech. Kyushu Univ. 5, 5367.Google Scholar