Transitions to chaotic cross-waves: An experimental investigation.

Abstract

Experiments were carried out to investigate the modulated nature of cross-waves in a long, deep, rectangular channel with a plane-flap type wavemaker at one end. Poincare sections, power spectra, Lyapunov exponents, correlation dimensions and winding numbers were generated from local wave height data to reveal the temporal behavior of modulated states. Information about the transverse spatial structure was obtained from digital image processing and multiprobe techniques. In general, three frequencies were observed: the subharmonic and two slow modulation frequencies. A one-mode modulation was associated with streamwise variation, and a sloshing modulation was associated with spanwise variation. The interaction of the oscillations produced a pattern of overlapping frequency-locked tongues in parameter space. In some tongues, all three frequencies were locked. The frequency-locked tongues had winding numbers in Farey tree sequences, and the overall picture was similar to the Arnold tongues of one-dimensional circle maps. Near the intersection of the neutral stability curves for two adjacent modes, a standing wave localized on one side of the tank was observed, in agreement with the coupled mode analysis of Ayanle (1989). At large forcing amplitudes or negative detunings, the transverse sloshing motion broke up into more than one soliton-like structure propagating in the spanwise direction and reflected by the sidewalls.