Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-17T20:05:55.474Z Has data issue: false hasContentIssue false
  • English
  • Français

Infragravity waves induced by short-wave groups

Published online by Cambridge University Press:  26 April 2006

Hemming A. Schäffer
Hemming A. Schäffer
Affiliation:
Danish Hydraulic Institute, Agern Allé 5, DK-2970 Hørsholm, Denmark
Get access Rights & Permissions [Opens in a new window]

Abstract

A theoretical model for infragravity waves generated by incident short-wave groups is developed. Both normal and oblique short-wave incidence is considered. The depth-integrated conservation equations for mass and momentum averaged over a short-wave period are equivalent to the nonlinear shallow-water equations with a forcing term. In linearized form these equations combine to a second-order long-wave equation including forcing, and this is the equation we solve. The forcing term is expressed in terms of the short-wave radiation stress, and the modelling of these short waves in regard to their breaking and dynamic surf zone behaviour is essential. The model takes into account the time-varying position of the initial break point as well as a (partial) transmission of grouping into the surf zone. The former produces a dynamic set-up, while the latter is equivalent to the short-wave forcing that takes place outside the surf zone. These two effects have a mutual dependence which is modelled by a parameter K, and their relative strength is estimated. Before the waves break, the standard assumption of energy conservation leads to a variation of the radiation stress, which causes a bound, long wave, and the shoaling bottom results in a modification of the solution known for constant depth. The respective effects of this incident bound, long wave and of oscillations of the break-point position are shown to be of the same order of magnitude, and they oppose each other to some extent. The transfer of energy from the short waves to waves at infragravity frequencies is analysed using the depth-integrated conservation equation of energy. For the case of normally incident groups a semi-analytical steady-state solution for the infragravity wave motion is given for a plane beach and small primary-wave modulations. Examples of the resulting surface elevation as well as the corresponding particle velocity and mean infragravity-wave energy flux are presented. Also the sensitivity to the variation of input parameters is analysed. The model results are compared with laboratory experiments from the literature. The qualitative agreement is good, but quantitatively the model overestimates the infragravity wave activity. This can, in part, be attributed to the neglect of frictional effects.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 247 , February 1993 , pp. 551 - 588
Copyright
© 1993 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

Abramowitz, M. & Stegun, I. A. 1972 Handbook of Mathematical Functions. Dover, New York.
Bowers, E. C. 1977 Harbour resonance due to set-down beneath wave groups. J. Fluid Mech. 79, 71– 93.Google Scholar
Carrier, C. F. & Greenspan, H. P. 1958 Water waves of finite amplitude on a sloping beach. J. Fluid Mech. 4, 97– 109.Google Scholar
Chu, V. H. & Mei, C. C. 1970 On slowly varying Stokes waves. J. Fluid Mech. 41, 873– 87.Google Scholar
Goda, Y. 1970 A synthesis on breaker indices. Trans. Japan Soc. Civ. Engrs 2, (2), 227– 230.Google Scholar
Guza, R. T. & Thornton, E. B. 1982 Swash oscillations on a natural beach. J. Geophys. Res. 87, C1, 483– 491.Google Scholar
Guza, R. T. & Thornton, E. B. 1985 Observations of surf beat. J. Geophys. Res. 90, C2, 3161– 3171.Google Scholar
Hansen, J. B. 1990 Periodic waves in the surf zone: analysis of experimental data. Coastal Engng 14, 19– 41.Google Scholar
Kostense, J. K. 1984 Measurements of surf beat and set-down beneath wave groups. Proc. 19th Intl Conf. Coastal Engng, Houston, Texas, 1984, vol. 1, pp. 724– 40. ASCE, New York, 1985.
LIU, P. L.-F. 1989 A note on long waves induced by short-wave groups over a shelf. J. Fluid Mech. 105, 163– 170.Google Scholar
List, J. H. 1991 Wave groupiness variations in the nearshore. Coastal Engng 15, 475– 496.Google Scholar
Lo, J.-M. 1988 Dynamic wave setup. Proc. 21st Intl Conf. Coastal Engng, Malaga, Spain, 1988, vol. 2, pp. 999– 1010. ASCE, New York, 1989.
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 stresses in water waves: a physical discussion with applications. Deep Sea Res. 11, 529– 562.Google Scholar
Mei, C. C. 1983 The Applied Dynamics of Ocean Surface Waves. Wiley.
Mei, C. C. & Agnon, Y. 1989 Long period oscillations in harbours induced by incident short waves. J. Fluid Mech. 208, 595– 608.Google Scholar
Mei, C. C. & Benmoussa, C. 1984 Long waves induced by short-wave groups over an uneven bottom. J. Fluid Mech. 139, 219– 235.Google Scholar
Meyer, R. E. & Taylor, A. D. 1972 Run-up on beaches, In Waves on Beaches. (ed. R. E. Meyer), pp. 357– 412. Academic.
Molin, F. 1982 On the generation of long-period second-order free-waves due to changes in the bottom profile. Ship. Res. Inst. Papers, vol. 68. Tokyo.Google Scholar
Munk, W. H. 1949 Surf beats. Trans. Am. Geophys. Union 30, 849– 854.Google Scholar
Nakaza, E. & Hino, M. 1991 Bore-like surf beat in a reef caused by wave groups of incident short period waves. Fluid dyn. Res. 7 (Jap. Soc. of Fluid Mech.), 89– 100.Google Scholar
Ottesen Hansen, N.-E. 1978 Long period waves in natural wave trains. Prog. Rep. 46, 13– 24. Inst. Hydrodyn. Hydr. Engng (ISVA), Technical University of Denmark.Google Scholar
Phillips, O. M. 1977 The Dynamics of the Upper Ocean, 2nd edn. Cambridge University Press.
Roelvink, J. A. 1991 Modelling of cross-shore flow and morphology. Proc. Coastal Sediments '91, Seattle, USA, vol. 1, pp. 603– 617. ASCE, New York 1991.
Sand, S. E. 1982 Long wave problems in laboratory models. J. Waterway, Port, Coastal and Ocean Div., Proc. ASCE, 108, 492– 503.Google Scholar
Schäffer, H. A. 1990 Infragravity water waves induced by short-wave groups. PhD thesis. Series paper 50, Inst. Hydrodyn. Hydr. Eng. (ISVA), Technical University of Denmark, 168 pp.
Schäffer, H. A. 1993 Laboratory wave generation correct to second order. Proc. Intl Conf. Wave Kinematics and Environmental Forces, London, England, 1993. Soc. Underwater Tech.
Schäffer, H. A. & Jonsson, I. G. 1990 Theory versus experiments in two-dimensional surf beats. Proc. 22nd Intl Conf. Coastal Engng, Delft, The Netherlands 1990, vol. 2, pp. 1131– 1143. ASCE, New York, 1991.
Schäffer, H. A., Jonsson, I. G. & Svendsen, I. A. 1990 Free and forced cross-shore long waves. In Water Wave Kinematics (ed. A. Tørum & O. T. Gudmestad), pp. 367– 385. Kluwer Academic Publishers, Dordrecht.
Schäffer, H. A. & Svendsen, I. A. 1988 Surf beat generation on a mild slope beach. Proc. 21st Intl Conf. Coastal Engng, Malaga, Spain, 1988, vol. 2, pp. 1058– 1072. ASCE, New York, 1989.
Svendsen, I. A. 1984 Wave heights and set-up in a surf zone. Coastal Engng 8, 303– 329.Google Scholar
Symonds, G. & Bowen, A. J. 1984 Interactions of nearshore bars with incoming wave groups. J. Geophys. Res. 89, C2, 1953– 1959.Google Scholar
Symonds, G., Huntley, G. A. & Bowen, A. J. 1982 Two dimensional surf beat: long wave generation by a time-varying break point. J. Geophys. Res. 87, Cl, 492– 498.Google Scholar
Tucker, M. J. 1950 Surf beats: sea waves of 1 to 5 min. period. Proc. R. Soc. Lond. A 202, 565– 573.Google Scholar
Van Leeuwen, P. J. & Battjes, J. A 1990 A model for surf beat. Proc. 22nd Intl Conf. Coastal Engng, Delft, The Netherlands 1990, vol. 1, pp. 32– 40. ASCE, New York, 1991.
Wright, L. D., Guza, R. T. & Short, A. D. 1982 Surf zone dynamics on a high energy dissipative beach. Mar. Geol. 45, 41– 62.Google Scholar
Wu, J.-K. & Liu, P. L.-F. 1990 Harbour excitations by incident wave groups. J. Fluid Mech. 217, 595– 613.Google Scholar