The generation, evolution, and work–energy conversion in Faraday waves

The triggering of Faraday waves in a tank can be considered the accumulation of the liquid mechanical energy due to the work done by vertical external excitations. To investigate the mechanism of resonant Faraday waves, the work–energy conversion in a noninertial reference frame is first established, which can convert the complicated calculation of the work done by surface forces into that by body forces, i.e., the inertial forces. Subsequently, the subharmonic resonant Faraday waves and nonresonant response are simulated by using an in-house numerical model. It is found that for subharmonic resonance, the frequency of vertical momentum change can match that of the inertial force, so positive work done by the inertial force can be accumulated in build-up stage, leading to the resonance. However, for the nonresonant response, the positive and negative work done by the inertial force are almost comparable in magnitude, so no energy can be accumulated and no resonance can be generated. Furthermore, the short-time Fourier transform method is utilized to analyze the nonlinearity of Faraday waves. The modes of vertical momentum with the same frequency as the external excitation provide the main contribution to the instability. Then, the liquid energy increases and decreases when the phase difference between this mode and the external excitation lies within the intervals 0, 0.5π∪1.5π, 2π and (0.5π, 1.5π), respectively. When the nonlinearity becomes strong, the phase of response can increase and shift; under the resonant excitation, when the nonlinearity is weak, the response yields to the external excitation and the phase can approach that of the external excitation.