FARADAY RESONANCE IN WATER-WAVES AT NEARLY CRITICAL DEPTHS

For the Faraday resonance in a rectangular basin, the dependences of w ave amplitude on excitation frequency for a given wave harmonic are in vestigated both theoretically and experimentally in the case that the fluid depth is equal or close to the critical depth. The third-order n onlinear correction to the wave frequency predicted by the linear theo ry is known to vanish at the critical depth. We give a comprehensive d escription 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 d epth close to the critical value, Physics-Doklady 39 (2) (1994) 126-12 7, translated from Dokl. Acad. Nauk. 334(6) 710-711]. We use the Lagra ngian formulation to write out the exact nonlinear equations and the d ynamic 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 res onance curve consists of two separate branches; one branch has a soft- spring character and the other branch, a hard-spring character. This r esults in the hysteresis effect, which has an unusual form for the par ametric resonance. We also present experimental data, which justify th e predicted properties of the parametrically excited water waves. (C) 1998 Elsevier Science Inc. All rights reserved.