Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-28T06:19:19.038Z Has data issue: false hasContentIssue false
  • English
  • Français

Wave action and set-down for waves on a shear current

Published online by Cambridge University Press:  12 April 2006

I. G. Jonsson ,
O. Brink-Kjaer  and
G. P. Thomas
I. G. Jonsson
Affiliation:
Institute of Hydrodynamics and Hydraulic Engineering, Technical University of Denmark, DK-2800 Lyngby
O. Brink-Kjaer
Affiliation:
Institute of Hydrodynamics and Hydraulic Engineering, Technical University of Denmark, DK-2800 Lyngby Present address: Computational Hydraulics Centre, Danish Hydraulic Institute, DK-2970 Hørsholm.
G. P. Thomas
Affiliation:
School of Mathematics, University of Bristol, England Present address: Department of Civil Engineering, University of Bristol, England.
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper considers steady, slowly varying water waves propagating over a gently sloping bed on a steady current. The current varies linearly with depth, and so has constant vorticity ω. The analysis is two-dimensional and dissipation is neglected. Definitions, and expressions correct to second order in the amplitude, are given for the radiation stress, wave energy density E and total energy flux. An average La-grangian [Lscr ], obtained by heuristic arguments from Clebsch potentials, leads to the result that for this particular problem E equals the wave action [Lscr ]ω times the angular frequency ωrm relative to a frame of reference moving with the average-over-depth current velocity Um. This determines the variation of the amplitude with distance explicitly. An analytical expression for the height of the mean water surface is found by a heuristic argument which compares the conservation equations for total energy and wave action. All the results have been checked directly by substitution back into the basic equations. Graphs illustrate the effect of the vorticity ω on the wavelength, amplitude and set-down.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 87 , Issue 3 , 15 August 1978 , pp. 401 - 416
Copyright
© 1978 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

Abdullah, A. J. 1949 Wave motion at the surface of a current which has an exponential distribution of vorticity. Ann. N.Y. Acad. Sci. 51, 425–441.Google Scholar
Biesel, F. 1950 Étude théorique de la houle eneau courante. Houille Blanche 5, 279–285.Google Scholar
Bretherton, F. P. 1968 Propagation in slowly varying wave guides. Proc. Roy. Soc. A 302, 555–576.Google Scholar
Bretherton, F. P. 1970 A note on Hamilton's principle for perfect fluids. J. Fluid Mech. 44, 19–31.Google Scholar
Bretherton, F. P. 1978 Conservation of wave action and angular momentum in a spherical atmosphere. J. Fluid Mech. (to appear).Google Scholar
Bretherton, F. P. & Garrett, C. J. R. 1968 Wave-trains in inhomogeneous moving media. Proc. Roy. Soc. A 302, 529–554.Google Scholar
Brevik, I. 1976 The stopping of linear gravity waves in currents of uniform vorticity. Physica Norvegica 8, 157–162.Google Scholar
Brink-Kjáer, O. 1976 Gravity waves on a current: the influence of vorticity, a sloping bed, and dissipation. Inst. Hydrodyn. Hydraul. Engng (ISVA), Tech. Univ. Denmark, Series Paper no. 12.
Brink-Kjáer, O. & Jonsson, I. G. 1975 Radiation stress and energy flux in water waves on a shear current. Inst. Hydrodyn. Hydraul. Engng (ISVA), Tech. Univ. Denmark, Prog. Rep. no. 36, pp. 27–32.Google Scholar
Burns, J. C. 1953 Long waves in running water. Proc. Camb. Phil. Soc. 49, 695–705.Google Scholar
Dalrymple, R. A. 1973 Water wave models and wave forces with shear currents. Coastal Ocean. Engng Lab., Univ. Florida Tech. Rep. no. 20.Google Scholar
Dalrymple, R. A. & Cox, J. C. 1976 Symmetric finite-amplitude rotational water waves. J. Phys. Ocean. 6, 847–852.Google Scholar
Fenton, J. D. 1973 Some results for surface gravity waves on shear flows. J. Inst. Math. Appl. 12, 1–20.Google Scholar
Fredsøe, J. 1974 Rotational channel flow over small three-dimensional bottom irregularities. J. Fluid Mech. 66, 49–66.Google Scholar
Hayes, W. D. 1970 Conservation of action and modal wave action. Proc. Roy. Soc. A 320, 187–208.Google Scholar
Hunt, J. N. 1955 Gravity waves in flowing water. Proc. Roy. Soc. A 231, 496–504.Google Scholar
Jonsson, I. G. 1971 A new proof of an energy equation for surface gravity waves propagating on a current. Coastal Engng Lab. & Hydraul. Lab., Tech. Univ. Denmark, Prog. Rep. no. 23, pp. 11–19.Google Scholar
Jonsson, I. G. 1977 The dynamics of waves on currents over a weakly varying bed. In Waves on Water of Variable Depth (ed. D. G. Provis & R. Radok). Lecture Notes in Physics, vol. 64, pp. 133–144. Springer.
Jonsson, I. G. 1977 Energy flux and wave action in gravity waves propagating on a current. J. Hydraul. Res. (to appear).Google Scholar
Jonsson, I. G., Skougaard, C. & Wang, J. D. 1971 Interaction between waves and currents. Proc. 12th Coastal Engng Conf., Sept. 1970, Wash. vol. 1, pp. 489–507. New York: A.S.C.E.
Landau, L. D. & Lifshitz, E. M. 1975 The Classical Theory of Fields, 4th English edn. Pergamon.
Longuet-Higgins, M. S. & Stewart, R. W. 1960 Changes in the form of short gravity waves on long waves and tidal currents. J. Fluid Mech. 8, 565–583.Google Scholar
Longuet-Higgins, M. S. & Stewart, R. W. 1961 The changes in amplitude of short gravity waves on steady non-uniform currents. J. Fluid Mech. 10, 529–549.Google Scholar
Longuet-Higgins, M. S. & Stewart, R. W. 1962 Radiation stress and mass transport in gravity waves, with application to ‘surf beats’. J. Fluid Mech. 13, 481–504.Google Scholar
Longuet-Higgins, M. S. & Stewart, R. W. 1964 Radiation stress in water waves; a physical discussion with applications. Deep-Sea Res. 11, 529–562.Google Scholar
Luke, J. C. 1967 A variational principle for a fluid with a free surface. J. Fluid Mech. 27, 395–397.Google Scholar
Lundgren, H. 1963 Wave thrust and wave energy level. Int. Assoc. Hydraul. Res., Proc. 10th Congr., London, vol. 1, pp. 147–151.Google Scholar
Mcintyre, M. E. 1977 Wave transport in stratified, rotating fluids. 38th Int. Astron. Un. Coll. Problems of Stellar Convection. Lecture Notes in Physics, vol. 71, pp. 290–314. Springer.
Peregrine, D. H. 1976 Interaction of water waves and currents. Adv. Appl. Mech. 16, 9–117.Google Scholar
Phillips, O. M. 1966 The Dynamics of the Upper Ocean. Cambridge University Press.
Sarpkaya, T. 1955 Oscillatory gravity waves in flowing water. Proc. A.S.C.E., Engng Mech. Div. 81, 815-1–815-33.Google Scholar
Sarpkaya, T. 1957 Oscillatory gravity waves in flowing water. Trans. A.S.C.E. 122, 564–586.Google Scholar
SUN’ TSAO 1959 Behaviour of surface waves on a linearly varying flow. Moskow Fiz.-Tech. Inst. Issled. Mekh. Prikl. Mat. 3, 66–84.Google Scholar
Taylor, G. I. 1955 The action of a surface current used as a breakwater. Proc. Roy. Soc. A 231, 466–478.Google Scholar
Thompson, P. D. 1949 The propagation of small surface disturbances through rotational flow. Ann. N.Y. Acad. Sci. 51, 463–474.Google Scholar
Whitham, G. B. 1974 Linear and Nonlinear Waves. Wiley.
Yu, Y.-Y. 1952 Breaking of waves by opposing currents. Trans. Am. Geophys. Un. 33, 39–41.Google Scholar