Gravity–capillary multi-component wave patterns generated by a single-frequency wave-maker oscillation and subsequent resonances

Abstract

Cross-waves are standing waves with crests perpendicular to a wave-maker; they are subharmonic waves excited by parametric instability. The modulational and chaotic behaviours of nonlinear cross-waves have been studied widely since the 1970s. Most of the previous work has focused on gravity waves where surface tension can be neglected. In this work we study cross-waves that are highly dependent on surface tension as well as gravity. By oscillating a planar wave-maker either vertically or horizontally with frequencies of 25 Hz through 40 Hz at one end of a rectangular basin, two-dimensional multi-component surface patterns are realized. Using the free-surface synthetic Schlieren technique to measure the surface elevations, multi-dimensional Fourier transforms are utilized to track the evolutionary spectrum of the water surface in both the temporal and spatial domains. Wavelet transforms are implemented to show the development of the various frequency components. Three-wave resonances with and without first subharmonics are observed for small nonlinearity. Three-dimensional oblique propagating cross-waves are generated at higher nonlinearity; unlike most previous cross-wave experiments, this staggered pattern propagates far downstream. Experimental evidence shows that two oblique propagating waves form a two-dimensional short-crested pattern, and that the lateral component of the waves develops into parametric sloshing modes corresponding to the width of the tank. Two regimes of nonlinear wave patterns, resonant triads and oblique propagating cross-waves, are delineated.