Hostname: page-component-fc487d775-274l9 Total loading time: 0 Render date: 2023-09-20T10:36:16.766Z Has data issue: false Feature Flags: { "corePageComponentGetUserInfoFromSharedSession": true, "coreDisableEcommerce": false, "coreDisableSocialShare": false, "coreDisableEcommerceForArticlePurchase": false, "coreDisableEcommerceForBookPurchase": false, "coreDisableEcommerceForElementPurchase": false, "coreUseNewShare": true, "useRatesEcommerce": true } hasContentIssue false

Steep, steady surface waves on water of finite depth with constant vorticity

Published online by Cambridge University Press:  21 April 2006

A. F. Teles Da Silva
School of Mathematics, Bristol University, University Walk, Bristol BS8 1TW, UK
D. H. Peregrine
School of Mathematics, Bristol University, University Walk, Bristol BS8 1TW, UK


Two-dimensional steady surface waves on a shearing flow are computed for the special case where the flow has uniform vorticity, i.e. in the absence of waves the velocity varies linearly with height. A boundary-integral method is used in the computation which is similar to that of Simmen & Saffman (1985) who describe such waves on deep water. Particular attention is given to the effects of finite depth with descriptions of waves of limiting steepness, waves with eddies beneath their crests and extremely high waves on high-speed flows.

Many qualitative features of these waves are relevant to steep waves propagating in shallow water, or on a strong wind-induced drift current. An important practical point in the interpretation of wave measurements of wind driven waves is that mean kinetic energy and potential energy densities are unequal even for infinitesimal waves. This may mean that wave energy spectra deduced from surface-elevation measurements in the conventional way may sometimes be misleading.

Research Article
© 1988 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.)


Banner, M. L. & Phillips O. M. 1974 On the incipient breaking of small scale waves. J. Fluid Mech. 65, 647656.Google Scholar
Benjamin T. B. 1962 The solitary wave on a stream with an arbitrary distribution of vorticity J. Fluid Mech. 12, 97116.Google Scholar
Caulliez G. 1987 Structure des mouvements associés aux vagues de capilarité-gravité générées par le vent. PhD thesis, l'Université de d'Aix-Marseille II.
Dalrymple R. A. 1974 A finite amplitude wave on a linear shear current. J. Geophys. Res. 79, 44984504.Google Scholar
Delachenal M. B. 1973 Existence d'écoulement permanent de type coin pour un fluide parfait a surface libre C. R. Acad. Sci. Paris A 276, 10211024.Google Scholar
Dold, J. W. & Peregrine D. H. 1986 An efficient boundary-integral method for steep unsteady water waves. In Numerical methods for Fluid Dynamics vol. 2 (ed. K. W. Morton & M. J. Baines), pp. 671679. Clarendon Press.
Hunter, J. K. & Vanden-Broeck J. M. 1983 Accurate computations for steep solitary waves. J. Fluid Mech. 136, 6371.Google Scholar
Peregrine D. H. 1974 Surface shear waves. J. Hydraul. Div. ASCE 100, 12151227. [Discussion in 101, 1032–1034 (1975).]Google Scholar
Peregrine D. H. 1976 Interaction of water waves and currents. Adv. Appl. Mech. 16, 9117.Google Scholar
Perry A. & Fairlie, B. D. 1975 A study of turbulent boundary-layer separation and reattachment. J. Fluid Mech. 69, 657672.Google Scholar
Phillips, O. M. & Banner M. L. 1974 Wave breaking in the presence of wind drift and swell. J. Fluid Mech. 66, 625640.Google Scholar
Pullin, D. I. & Grimshaw R. H. 1985 Interfacial progressive gravity waves in a two layer shear flow. Phys. Fluids 26, 17311739.Google Scholar
Pullin, D. I. & Grimshaw R. H. 1988 Finite amplitude solitary waves at the interface between two homogeneous fluids, Submitted for publication.
Simmen, J. A. & Saffman P. G. 1985 Steady deep water waves on a linear shear current. Stud. Appl. Maths 73, 3557.Google Scholar
Stokes G. G. 1847 On the theory of oscillatory waves. Trans. Camb. Phil. Soc. 8, 441455.Google Scholar
Tanaka M. 1983 The stability of steep gravity waves. J. Phys. Soc. Japan 52, 30473055.Google Scholar
Tanaka M. 1985 The stability of steep gravity waves. Part 2. J. Fluid Mech. 156, 281289.Google Scholar
Tsao S. 1959 Behaviour of surface waves on a linearly varying current. Tr. Mosk. Fiz.-Tekh. Inst. Issled. Mekh.Google Scholar