$R/\Delta x=26.7$ rather than 32. We have also carried out simulations with grids of $256^3$ (to confirm our results) and of $64^3$ (for visualisation of our patterns). Parallelization is achieved through domain decomposition, in which each subdomain is assigned to its own process thread. Here, we use $512$ subdomains or processes, each with $16^3$ or $32^3$ gridpoints for the global resolutions of $128^3$ and $256^3$, respectively.

The time step $\Delta t$ is chosen at each iteration in order to satisfy a criterion based on $$\label{eq:steps}
\left\{\Delta t_{\rm CFL}, \Delta t_{\rm int}, \Delta t_{\rm vis}, \Delta t_{\rm cap}\right\}$$ which ensures stability of the calculations. These bounds are defined by: $$\label{eq:Time_steps}
\begin{array}{ll}
\displaystyle{\Delta t_{\rm CFL} \equiv \min_j \left( \min_{\rm domain} \left(\frac{\Delta x_j}{u_j}\right) \right)} &
\displaystyle{\Delta t_{\rm int} \equiv \min_j \left( \min_{\Gamma(t)} \left( \frac{\Delta x_j}{\|{\bf V}\|}\right) \right)}\\
\displaystyle{\Delta t_{\rm vis} \equiv \min\left(\frac{\rho_{2}}{\mu_{2}},\frac{\rho_{1}}{\mu_{1}}\right)\frac{\Delta x_{\min}^2}{6}}&
\displaystyle{\Delta t_{\rm cap} \equiv \frac{1}{2}\bigg(\frac{(\rho_{1} + \rho_{2}) \Delta x_{\min}^3}{\pi \sigma} \bigg)^{1/2}}
\end{array}
$$ where $\Delta x_{\min} = \min_j \left(\Delta x_j \right)$. In our simulations, this minimum is realized by $\Delta t_{\rm CFL}\approx\Delta t_{\rm int}$ for our gravitational waves and by $\Delta t_{\rm cap}\approx\Delta t_{\rm vis}$ for our capillary waves.

Spherical harmonic transform {#sec:sht}
----------------------------

In order to analyze the shape 