The role of the meniscus in determining the temporal response of parametrically-excited surface waves

Recent experiments by Shao et al. 2021, JFM, have revealed complex surface wave dynamics in a mechanically-vibrated fluid bath related to the geometry of the meniscus formed at the container sidewall, including the observation of i) harmonic axisymmetric waves, ii) subharmonic asymmetric waves, and iii) the mixing of harmonic axisymmetric waves and subharmonic asymmetric waves at a fixed driving frequency. We develop a corresponding theoretical model for this system including meniscus effects through detuning the driving acceleration of the container, which results in an inhomogeneous Mathieu equation that governs the wave dynamics whose spatial structure is defined by the mode number pair (n, m), with n and m the radial and azimuthal mode numbers, respectively. A perturbation method (Poincare-Lindstedt) is used to compute the instability tongues for the axisymmetric m=0 modes, while the asymmetric m≠ 0 modes are unaffected by the meniscus and satisfy a homogeneous Mathieu equation from which we compute the instability tongues using Floquet theory. Our model results explicitly show how the shape of the meniscus and spatial structure of the wave determine the temporal response, as it depends upon the Galilei number (Ga), Bond number (Bo), and meniscus wave amplification factor (f*). Our model predictions are in excellent agreement with prior experimental observations for both pure modes and mixed modes. Notably, the meniscus only affects the shape of the harmonic instability tongues for the axisymmetric m=0 modes, which can have a lower threshold acceleration and larger bandwidth than the subharmonic instability tongues, consistent with the experiment and suggesting our model recovers the essential physics associated with these complex surface wave dynamics.