Waves of finite amplitude trapped by oscillating gates

Arrays of oscillating gates located at the inlet channels of Venice Lagoon have been recently proposed as a means to protect Venice from the phenomenon of high water associated with the propagation of high tides. It has sometimes been observed in laboratory investigations that the gates subjected to the action of incident monochromatic waves propagating along the inlet channel respond with oscillations which are out of phase among each other and not synchronous with the incident wave. The latter observations have been recently explained by Blondeaux et al. (Rend. Matem. Acc. Lincei 59 (1993), 291-298; 299-305) as a Mathieu type phenomenon whereby the incident wave excites free transverse modes of oscillation of the gate system. In the present investigation the above instability process is followed in the weakly nonlinear regime in the interesting case when the response is subharmonic. A bifurcation is shown to occur which may be either supercritical or subcritical in different regions of the parameter space. The equilibrium configurations are shown to correspond to attractive foci of the cubic amplitude equation describing the bifurcation process. This result bears some analogy with those found in a different context by Hall & Seminara (J. Fluid Mech. 101 (1980), 423-444) though differences originate from the `dissipative' effect exhibited by the gate system which is able to radiate energy through the generation of travelling waves.