Surface waves in a square container due to its resonant horizontal elliptic motion

Surface waves in a square container due to its resonant horizontal elliptic or linear motion are investigated theoretically. The motion of the container is characterized by the ratio, expressed as tan phi, of the length of the minor axis to the length of the major axis of its elliptic orbit, and by the angle theta between the directions of the major axis and one of its sidewalls. Using the reductive perturbation method, non-linear time evolution equations for the complex amplitudes of two degenerate modes excited by this motion are derived with the inclusion of linear damping. When tan phi is small, for any theta these equations have two kinds of stable stationary solutions corresponding to regular co-rotating waves whose direction of rotation is the same as that of the container and regular counter-rotating waves of the opposite direction of rotation. As tan phi increases to one, the region of forcing frequency in which stable regular counter-rotating waves are observed shrinks and then disappears for any theta. Solutions with chaotic or periodic slow variations in amplitude and phase of excited surface waves are also obtained for forcing frequencies where no stable stationary solutions exist. Non-stationary solutions are either uni-directionally or bidirectionally rotating waves. For theta = 0 degrees, chaotic waves and bidirectionally rotating waves are observed more frequently for smaller tan phi. For theta = phi = 0 degrees, for sufficiently small fluid depth, regular non-rotating waves are expected to occur for any forcing frequency. Moreover, stable stationary and non-stationary solutions obtained for phi = 0 degrees are found to agree fairly well with the experimental results in a preceding study.