Finite amplitude Faraday waves induced by a random forcing

Abstract

In the present contribution we study the waves arising on the free surface of a liquid in a rectangular container undergoing vertical oscillations. Our aim in the work is to investigate the role of a random forcing characterized by a narrow-band spectrum on the wave amplitude close to subharmonic resonant conditions. The analysis is carried out theoretically by means of a weakly nonlinear analysis, assuming the ratio a0 between the acceleration of the tank and the gravitational acceleration to be small. We consider irrotational flow and take into account viscous effects by adding a linear dissipative term to the amplitude equation following Miles [“Nonlinear Faraday resonance,” J. Fluid Mech. 146, 285 (1984)]. Comparing the results with those obtained in the case of a monochromatic forcing, it appears that the range of unstable frequencies significantly widens. This finding agrees with the theoretical results found in the linear context by Zhang, Casademunt, and Viñals [“Study of the parametric oscillator driven by a narrow band noise to model the response of a fluid surface to time-dependent accelerations,” Phys. Fluids A 5, 3147 (1993)]. The maximum equilibrium amplitude of the free surface waves for the random forcing case turns out to be smaller than that of the monochromatic forcing and it decreases as the spectrum width is increased.