Resonantly excited regular and chaotic motions in a rectangular wave tank

The resonant excitation of surface waves inside a rectangular wave tank of arbitrary water depth with a flap-type wavemaker on one side is considered. Depending on the length and width of the tank relative to the sinusoidal forcing frequency of the wave paddle, three classes of resonant mechanisms can be identified. The first two are the well-known synchronous, resonantly forced longitudinal standing waves and the subharmonic, parametrically excited transverse (cross) waves. These have been studied by a number of investigators, notably in deep water. The governing equations are re-derived and show good comparisons with the experimental data of Lin & Howard (1960). The third class is new and involves the simultaneous resonance of the synchronous longitudinal and subharmonic cross-waves and their internal interactions. In this case, temporal chaotic motions are found for a broad range of parameter values and initial conditions. These are studied by local bifurcation and stability analysis, direct numerical simulations, estimations of the Lyapunov exponents and power spectra and examination of Poincare surfaces. To obtain a global criterion for widespread chaos, the method of resonance overlap (Chirikov 1979) is adopted and found to be remarkably effective.