Faraday resonance in water waves at nearly critical depths

For the Faraday resonance in a rectangular basin, the dependences of wave amplitude on excitation frequency for a given wave harmonic are investigated both theoretically and experimentally in the case that the fluid depth is equal or close to the critical depth. The third-order nonlinear correction to the wave frequency predicted by the linear theory is known to vanish at the critical depth. We give a comprehensive description of the fifth-order theory proposed and briefly described by Bordakov et al. [G.A. Bordakov, I.I. Karpov, S.Ya. Sekerh-Zen'kovich, I.K. Shingareva, Parametric excitation of surface waves for a fluid depth close to the critical value, Physics-Doklady 39 (2) (1994) 126-127, translated from Dokl. Acad. Nauk. 334(6) 710-711]. We use the Lagrangian formulation to write out the exact nonlinear equations and the dynamic and the kinematic boundary conditions and develop an asymptotic procedure based on the Krylov-Bogolyubov averaging method. The theory predicts the following properties of the resonance curves: (i) if the fluid depth is equal to or greater than the critical depth, then the resonance curve bears a soft-spring character; (ii) otherwise, the resonance curve consists of two separate branches; one branch has a soft-spring character and the other branch, a hard-spring character. This results in the hysteresis effect, which has an unsual form for the parametric resonance. We also present experimental data, which justify the predicted properties of the parametrically excited water waves.