Surface waves due to resonant horizontal oscillation

Abstract

Experiments on surface waves were made using a cylindrical container oscillated horizontally with the period T close to that associated with the two known degenerate modes. Outside a certain region in the (T,x0)-plane, where x0 is the amplitude of the forcing displacement, surface waves exhibit either of the two kinds of regular motions whose amplitudes are constant. Within this region, however, the wave amplitude slowly changes, expressing the irregular or periodic motion of surface waves. In order to analyse these motions in detail, the slow evolution of four variables associated with the amplitudes and phases of the two modes is computed from the free-surface displacement at two measuring points. It is shown that the most common attractor corresponding to the irregular wave motion is the strange attractor with a positive maximum Liapunov exponent and a correlation dimension of 2.1–2.4. Furthermore, another kind of chaotic attractor and a few periodic orbits are found in a small parametric region. The route to chaos associated with period-doubling bifurcation is also observed. The above experimental results are compared with the solutions to weakly nonlinear evolution equations derived by Miles. We find that the equations can explain well many of the experimental results on regular and irregular wave motions. In particular, the most common chaotic attractors both in the experiments and in the theory have similar shapes in a phase space, and also yield similar values for maximum Liapunov exponents and correlation dimensions.