Observations on the evolution of wave modulation

Abstract

A series of laboratory experiments on the long-time evolution of nonlinear wave trains in deep water was carried out in a super wave flume (300×5.0×5.2 m) at Tainan Hydraulics Laboratory of National Cheng Kung University. Two typical wave trains, namely uniform wave and imposed sideband wave, were generated by a piston-type wavemaker. Detailed discussions on the evolution of modulated wave trains, such as transient wavefront, fastest growth mode and initial wave steepness effect, are given and the results are compared with existing experimental data and theoretical predictions. Present results on the evolution of initial uniform wave trains cover a wide range of initial wave steepness ( ) and thus, greatly extend earlier studies that are confined only to the larger initial wave steepness region ( ). The amplitudes of the fastest growth sidebands exhibit a symmetric exponential growth until the onset of wave breaking. At a further stage, the amplitude of lower sideband becomes larger than the carrier wave and upper sideband after wave breaking, which is known as the frequency downshift. The investigations on the evolution of initial imposed sideband wave trains for fixed initial wave steepness but different sideband space indicate that the most unstable mode of initial wave train will manifest itself during evolution through a multiple downshift of wave spectrum for the wave train with the smaller sideband space. It reveals that the spectrum energy tends to shift to a lower frequency as the wave train propagates downstream due to the sideband instability. Experiments on initial imposed sideband wave trains with varied initial wave steepness illustrate that the evolution of the wave train is a periodic modulation and demodulation at post-breaking stages, in which most of the energy of the wave train is transferred cyclically between the carrier wave and two imposed sidebands. Meanwhile, the wave spectra show both temporal and permanent frequency downshift for different initial wave steepness, suggesting that the permanent frequency downshift induced by wave breaking observed by earlier researchers is not permanent. Additionally, the local wave steepness and the ratio of horizontal particle velocity to linear phase velocity at wave breaking in modulated wave group are very different from those of Stokes theory.