The work–energy conversion and nonlinearity in harmonic resonant Faraday waves

Following the previous study by Liu et al. [“The generation, evolution, and work–energy conversion in Faraday waves,” Phys. Fluids 37, 102101 (2025)], in which a work–energy calculation system was developed in a non-inertial reference frame for analyzing subharmonic Faraday waves, the present work employs the same system to explore the instability mechanism of harmonic Faraday waves. The generation of harmonic resonance in a tank can be regarded as the accumulation of liquid mechanical energy induced by vertical excitation, which occurs when the excitation frequency equals the natural frequency of the liquid. Consequently, the harmonic resonant Faraday waves are simulated by employing an in-house numerical model. It is found that the direction of inertial force consistently aligns with that of the liquid vertical velocity on one side. Therefore, the positive work continuously acts on that side of the liquid, leading to energy accumulation and amplification of the free surface response. Furthermore, the nonlinearity of harmonic Faraday waves and the characteristics of each mode are analyzed by utilizing the short-time Fourier transform (STFT) method. It is found that the mode of vertical momentum with the same frequency as the external excitation provides the main contribution to the instability. When the nonlinearity becomes strong, the phase of this mode can shift away from that of the external excitation, resulting in the weakening of the response; when the nonlinearity is weak, the response yields the excitation and the phase can approach that of the external excitation, thus exciting the response once more.