Skip to main content
×
Home
    • Aa
    • Aa

Highly nonlinear short-crested water waves

  • A. J. Roberts (a1) (a2)
Abstract

The properties of a fully three-dimensional surface gravity wave, the short-crested wave, are examined. Linearly, a short-crested wave is formed by two wavetrains of equal amplitudes and wavelengths propagating at an angle to each other. Resonant interactions between the fundamental and its harmonics are a major feature of short-crested waves and a major complication to the use at finite wave steepness of the derived perturbation expansion. Nonetheless, estimates are made of the maximum steepnesses, and wave properties are calculated over the range of steepnesses. Although results for values of the parameter θ near 20° remain uncertain, we find that short-crested waves can be up to 60% steeper than the two-dimensional progressive wave. At limits of the parameter range the results compare well with those for known two-dimensional progressive and standing water waves.

Copyright
References
Hide All
Ablowitz, M. J. 1971 Applications of slowly varying nonlinear dispersive wave theories Stud. Appl. Maths 50, 329344.
Ablowitz, M. J. 1972 Approximate methods for obtaining multi-phase modes in nonlinear dispersive wave problems Stud. Appl. Maths 51, 1755.
Ablowitz, M. J. 1975 A note on resonance and nonlinear dispersive waves Stud. Appl. Maths 54, 6170.
Ablowitz, M. J. & Benney, D. J. 1970 The evolution of multi-phase modes for nonlinear dispersive waves Stud. Appl. Maths 44, 225238.
Bender, C. M. & Orszag, S. A. 1978 Advanced Mathematical Methods for Scientists and Engineers. McGraw-Hill.
Bryant, P. J. 1982 Two-dimensional periodic permanent waves in shallow water J. Fluid Mech. 115, 525532.
Chappelear, J. E. 1961 On the description of short-crested waves. Beach Erosion Board, U.S. Army, Corps Engrs, Tech. Memo no. 125.
Chen, B. & Saffman, P. G. 1979 Steady gravity-capillary waves on deep water — 1. Weakly nonlinear waves Stud. Appl. Maths 60, 183210.
Cokelet, E. D. 1977 Steep gravity waves in water of arbitrary uniform depth. Phil. Trans. R. Soc. Lond A 286, 183230.
Concus, P. 1964 Standing capillary gravity waves of finite amplitude: corrigendum. J. Fluid Mech. 19, 264266.
Fenton, J. D. 1983 Short-crested waves and the wave forces on a wall. Submitted to J. Waterway, Port, Coastal and Ocean Div. ASCE.
Fuchs, R. A. 1952 On the theory of short-crested oscillatory waves. Gravity Waves, U.S. Nat. Bur. Stand. Circular 521, pp. 187200.
Holyer, J. Y. 1980 Large amplitude progressive interfacial waves J. Fluid Mech. 93, 433448.
Hsu, J. R. C., Silvester, R. & Tsuchiya, Y. 1980 Boundary-layer velocities and mass transport in short-crested waves J. Fluid Mech. 99, 321342.
Hsu, J. R. C., Tsuchiya, Y. & Silvester, R. 1979 Third-order approximation to short-crested waves J. Fluid Mech. 90, 179196.
Longuet-Higgins, M. S. & Fox, M. J. H. 1977 Theory of the almost-highest wave: the inner solution. J. Fluid Mech. 80, 721742.
Mcgoldrick, L. F. 1970 On Wilton's ripples: a special case of resonant interactions. J. Fluid Mech. 42, 193200.
Mcgoldrick, L. F. 1972 On the rippling of small waves: a harmonic nonlinear nearly resonant interaction. J. Fluid Mech. 52, 725751.
Meiron, D. I., Saffman, P. G. & Yuen, H. C. 1982 Calculation of steady three-dimensional deep-water waves J. Fluid Mech. 124, 109121.
Mollo-Christensen, E. 1981 Modulational stability of short-crested free surface waves Phys. Fluids 24, 775776.
Nayfeh, A. H. 1971 Third harmonic resonance in the interaction of capillary and gravity waves J. Fluid Mech. 48, 385395.
Penney, W. G. & Price, A. T. 1952 Some gravity wave problems in the motion of perfect liquids. Part 2: Finite periodic stationary gravity waves. Phil. Trans. R. Soc. Lond A 244, 251284.
Roberts, A. J. 1981 The behaviour of harmonic resonant steady solutions to a model differential equation Q. J. Mech. Appl. Maths 34, 287310.
Roberts, A. J. 1982 Nonlinear buoyancy effects in fluids. Ph.D. thesis, University of Cambridge.
Roberts, A. J. & Peregrine, D. H. 1983 Notes on long-crested water waves J. Fluid Mech. 135, 323335.
Roberts A. J. & Schwartz, L. W. 1983 The calculation of nonlinear short-crested gravity waves. Submitted to Phys. Fluids.
Rottman, J. W. 1982 Steep standing waves at a fluid interface J. Fluid Mech. 124, 283306.
Schwartz, L. W. 1974 Computer extension and analytic continuation of Stokes’ expansion for gravity waves J. Fluid Mech. 62, 553578.
Schwartz, L. W. & Whitney, A. K. 1981 A semi-analytic solution for nonlinear standing waves in deep water J. Fluid Mech. 107, 147171.
Stokes, G. G. 1880 Considerations relating to the greatest height of oscillatory waves which can be propagated without change of form. In Mathematical and Physical Papers, vol. 1, pp. 225228. Cambridge University Press.
Tadjbakhsh, I. & Keller, J. B. 1960 Standing surface waves of finite amplitude J. Fluid Mech. 8, 442451.
Whitham, G. B. 1974 Linear and Nonlinear Waves. Wiley.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 21 *
Loading metrics...

Abstract views

Total abstract views: 59 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 22nd October 2017. This data will be updated every 24 hours.