Weak quadratic interactions of two-dimensional waves

Abstract

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.