Sideband instability and modulations of Faraday waves

Three-mode interaction equations are derived for Faraday waves in a long rectangular container. Two water depths are studied, revealing very different behaviour. Our equations include conservative and non-conservative cubic nonlinear terms, and quintic conservative nonlinear terms. Instability of a single standing wave to neighbouring modes is examined. Resultant three-mode interactions display rich structure, with fast and slow timescales. For the smaller depth, but not the greater, recurrent nearly calm intervals are separated by strong wave activity. Our results agree quite well with experimental findings of Craik and Armitage [Faraday excitation, hysteresis and wave instability in a narrow rectangular wave tank, Fluid Dyn. Res. 15 (1995) 129-143] and with unpublished observations by Armitage, Craik and Sterratt, here briefly described. (C) 1999 Elsevier Science B.V. All rights reserved.