Model of Faraday waves in a cylindrical container with force detuning

Recent experiments by Shao et al. [“Surface wave pattern formation in a cylindrical container,” J. Fluid Mech. 915, A19 (2021)] have revealed complex wave dynamics on the surface of a liquid bath in a vertically vibrated cylindrical container that are related to the presence of a meniscus on the container sidewall. We develop a corresponding theoretical model for this system by 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. Asymmetric m≠0 modes are unaffected by the detuning parameter, which is related to the meniscus shape and satisfy a homogeneous Mathieu equation with the shape of the instability tongues computed by the Floquet theory. The Poincaré–Lindstedt method is used to compute the instability tongues for the axisymmetric m=0 modes, which have a lower threshold acceleration and larger bandwidth that depend upon the detuning parameter. Our model results explicitly show how the shape of the meniscus and spatial structure of the wave determine the temporal response and are in good agreement with prior experimental observations for both pure modes and mixed modes.