Observation of alternately localized Faraday waves in a narrow tank

There are many experimental works and analyses of gravity water surface waves in vibrating high-aspect-ratio rectangular tanks. In most cases, the waves are symmetric or antisymmetric in the direction along the short sides. Here we report an unusual alternately localized Faraday wave (ALFW) in this system which is neither symmetric nor antisymmetric along the short side direction. The peculiar feature is that close to the boundary there are a series of large oscillating regions and flat regions; i.e., the surface barely moves during the experiment. The large oscillating regions and the flat regions appear alternately not only in the direction along the long side of the tank, but also along the short side. The large surface deformation implies strong nonlinearities of the phenomenon. The spectrum of the discrete cosine transformation of the surface profile shows clearly that there are only two dominating modes. However, further analyses reveal that it is not simply a two-mode excitation through external driving, but a one mode excitation, then pumping the other mode excited through strong internal mode interactions in a leading-passive way. We use the phenomenological nonlinear mode competition model, which is a set of coupled nonlinear Mathieu equations, to reproduce the ALFW pattern. Theoretical analyses and numerical simulations indicate that both nonlinear internal mode interactions and nonlinear bounding effects account for this phenomenon. Phase locking and amplitude bounding can be reproduced quantitatively by the model. The instability boundary in the parameter space obtained by numerical simulations fits the one obtained by experiments very well.