with spherical wavenumbers $\ell=2$, 3, 4, 5, 6 originate at $\omega/\omega_c =2\sqrt{\ell(\ell-1)(\ell+2)}/n$. []{data-label="fig:BLUE_tongues"}](fig6a.eps "fig:"){width="110.00000%"}\
![Instability tongues resulting from oscillatory forcing with amplitude $a$ and angular frequency $\omega$ for an inviscid drop with parameters given in table \[tab:forcing\_response\]. Diamond shapes designate the parameter values of our simulations. Solid curves bound subharmonic tongues and dashed curves bound harmonic tongues. (a) Tongues corresponding to gravitational instability with spherical wavenumbers $\ell=1$, 2, 3, 4 originate at $\omega/\omega_g =
  2\sqrt{\ell}/n$, with $n=1$ for subharmonic tongues and $n=2$ for harmonic tongues. (b) Tongues corresponding to capillary instability with spherical wavenumbers $\ell=2$, 3, 4, 5, 6 originate at $\omega/\omega_c =2\sqrt{\ell(\ell-1)(\ell+2)}/n$. []{data-label="fig:BLUE_tongues"}](fig6b.eps "fig:"){width="110.00000%"}\

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  (a)
  \[8cm\] (b)
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These formulas for functions with a given symmetry are given by [@Busse1975] and [@Riahi1984] for patterns with a single $\ell$, aligned along the $z$ axis. When the patterns are rotated to a different orientation, the value of $\ell$ is conserved, but the combinations of $m$ values change. Patterns with other symmetries are also possible, but are not used or not achieved here. For $\ell=1$, the initial condition for the interface is a sphere perturbed only by its representation on a triangular mesh. In all cases, the initial velocity is 