Surface Waves in a Square Container Due to Resonant Horizontal Oscillations

Abstract

Resonantly forced water waves in a square container due to its horizontal oscillations are examined. The excited waves are assumed to be gravity waves for infinite depth. Using the reductive perturbation method and including the effect of a linear damping, we derive an evolution equation for the complex amplitudes of two degenerate resonant modes. When the angle θ between the direction of the oscillations and that along one of the sidewalls of the container is 0° or 45°, we obtain planar stationary solutions without the rotation of wave pattern as well as a pair of non-planar ones associated with the clockwise or anti-clockwise rotation. If 0° < θ< 45° , however, no planar stationary solution exists, and the symmetry between these non-planar solutions for θ=0° or 45° is broken. We find the pitchfork bifurcations of the stationary solution for θ=0° and 45°, and also the Hopf and saddle-node bifurcations of this solution for 0° ≤θ≤45°. Furthermore, periodic or chaotic solutions exist within the parameter re...