Third-harmonic resonance in the interaction of capillary and gravity waves

Abstract

The method of multiple scales is used to determine the temporal and spatial variation of the amplitudes and phases of capillary-gravity waves in a deep liquid at or near the third-harmonic resonant wave-number. This case corresponds to a wavelength of 2·99 cm in deep water. The temporal variation shows that the motion is always bounded, and the general motion is an aperiodic travelling wave. The analysis shows that pure amplitude-modulated waves are not possible in this case contrary to the second-harmonic resonant case. Moreover, pure phase-modulated waves are periodic even near resonance because the non-linearity adjusts the phases to yield perfect resonance. These periodic waves are found to be unstable, in the sense that any disturbance would change them into aperiodic waves.