Spatial evolution of an initially narrow-banded wave train

Abstract

Nonlinear evolution of narrow-banded unidirectional gravity wave trains along wave flume is studied both experimentally and numerically. The spatial version of the Zakharov equation serves as the theoretical model. The frequency domains of spatially linearly unstable disturbances to a monochromatic wave, as well as the frequencies of the most linearly unstable modes are determined theoretically as functions of the carrier wave steepness; these results serve as a reference for the following study. The effect of the spectral width on the evolution along the tank is considered for bi-modal initial spectra, as well as for spectra consisting of a carrier wave and two sidebands with small but finite amplitude. Good agreement between the experimental results and numerical simulations is obtained. The variation of the frequency spectra along the tank resulting from nonlinearity, as well as of the maximum envelope and crest height is investigated as a function of the initial conditions. Fermi–Pasta–Ulam recurrence is obtained for frequency spacing between the initial spectral harmonics approximately corresponding to most unstable disturbance. For narrower spectra, the evolution pattern becomes irregular, numerous additional harmonics are generated by nonlinearity; in this process very steep waves can be generated. The relevance of those findings to appearance of rogue waves is discussed.