Skip to main content Accessibility help

Weak quadratic interactions of two-dimensional waves

  • Young Yuel Kim (a1) and Thomas J. Hanratty (a1)


This paper reports on weak quadratic interactions which can occur with two-dimensional waves on shallow water layers and in the capillary-gravity range on deep water layers. It supplies experimental support of theoretical predictions for resonant interactions, but, perhaps of more significance, it explores in detail interactions which occur under conditions near resonance.

Waves of approximately sinusoidal form are introduced on the surface of water in a long rectangular tank. For deep water a rapid distortion in the sinusoidal wave and sometimes additional crests are observed because of energy exchange among the first, second and third harmonics at frequencies where both surface tension and gravity are important (7·5–13 c/s). An even greater exchange of energy can be observed on shallow water layers at low frequencies. For example, a wave train with seven secondary crests can be observed when the wave maker is operated at 3·04 c/s in a water layer of 0·65 cm.

Measured amplitudes and phase angles of the Fourier components of the wave train are described by a system of equations using only quadratic interactions among participating harmonics. The exchange of energy among Fourier components under certain conditions is explained in terms of the rate of change of relative phase angles of the different harmonics.



Hide All
Benjamin, T. B. & Feir, J. E. 1967 J. Fluid Mech. 27, 417-430.
Benney, D. J. 1962 J. Fluid Mech. 14, 577-584.
Bretherton, F. P. 1964 J. Fluid Mech. 20, 457-479.
De, S. C. 1955 Proc. Camb. Phil. Soc. 51, 713-736.
Kim, Y. Y. 1970 Ph.D. thesis in Chemical Engineering, University of Illinois.
Longuet-Higgins, M. S. & Smith, N. D. 1966 J. Fluid Mech. 25, 417-436.
Mcgoldrick, L. F. 1965 J. Fluid Mech. 21, 305-331.
Mcgoldrick, L. F. 1970 J. Fluid Mech. 40, 251-271.
Mcgoldrick, L. F., Phillips, O. M., Huang, N. E. & Hodgson, T. H. 1966 J. Fluid Mech. 25, 437-456.
Phillips, O. M. 1960 J. Fluid Mech. 9, 193-217.
Pierson, W. J. & Fife, P. J. 1961 Geophys. Res. 66, 163179.
Simmons, W. F. 1969 Proc. Roy. Soc. A 309, 551575.
Wilton, J. R. 1915 Phil. Mag. 29, 688600.
MathJax is a JavaScript display engine for mathematics. For more information see

Weak quadratic interactions of two-dimensional waves

  • Young Yuel Kim (a1) and Thomas J. Hanratty (a1)


Full text views

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

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.