Surface Waves in Closed Basins under Parametric And Internal Resonances

The method of multiple scales is used to analyze the nonlinear response of the free surface of a liquid in a cylindrical container to a harmonic vertical oscillation in the presence of a two‐to‐one internal (autoparametric) resonance. Four first‐order ordinary‐differential equations are derived for the modulation of the amplitudes and phases of the two modes involved in the internal resonance with the lower mode is excited by a principal parametric resonance. In the presence of small damping, the long‐time response may be any of (a) a trivial motion, (b) a limit cycle involving both modes, (c) an amplitude‐ and phase‐modulated sinusoid (motion on a two torus), and (d) a chaotically modulated sinusoid.