Figure 1

Above: instantaneous realization of hexagaonal Faraday waves in a domain of size $L_x\times L_y\times (h_1+h_2)$. The density and viscosity are ${\rho }_2$, ${\nu }_2$ above the free surface at $z=\zeta (x,y,t)$; below, they are ${\rho }_1$, ${\nu }_1$. The displacement amplitude of $\zeta$ is comparable to $h_1$; at its lowest points, $\zeta$ almost touches the lower boundary. Below: projection of the horizontal doubly periodic rectangular domain of dimensions $L_x\times L_y=2{\lambda }_c/\sqrt{3}\times 2{\lambda }_c$ shows that it is compatible with a hexagonal lattice. The minima of $\zeta$ (above) are located at the vertices and centers of the hexagons (below). The right corner represents a portion of the rectangular computational grid; the actual grid is twice as fine in each direction.Reuse & Permissions

Figure 2

Maximum interface heights ${max}_{x,y}\zeta (x,y,t)$ (rapidly oscillating curve) and ${max}_{x,y,[t,t+T]}\zeta (x,y,t)$ (smooth envelope). Surrounding visualizations show instantaneous contour plots of $\zeta (x,y,t)$ at times indicated by black dots on rapidly oscillating curve and along abscissa. The range of contour levels is scaled independently for each plot. Over the large white areas, the interface is very close to the bottom and almost flat. Visualizations shown at $t_i+jT/4$ for $j=0$, 1, 2, 3 (i.e., over one subharmonic period) for hexagons at $t_1$, symmetric and nonsymmetric beaded stripes at $t_2$ and $t_4$, and quasihexagons at $t_3$ and $t_5$.Reuse & Permissions

Figure 3

Time-filtered spatial Fourier spectra ${\zeta }_{mn}(t_i)$: hexagons at $t_1$, beaded stripes at $t_2$, quasihexagons at $t_3$ and $t_5$, nonsymmetric beaded stripes at $t_4$ and $t_6$.Reuse & Permissions

Figure 4

(a) Spatial Fourier grid of the domain. Hexagonal symbols: sets of six ${\mathbf{k}}_{mn}$ which share the same length $k$. This portion of the grid contains three hexagons, with $k/k_c=1$, $\sqrt{3}$, and 2. The hexagonal modes are located at the vertices (and not the sides) of the large hexagons indicated by solid and dashed lines. Square symbols: sets of four ${\mathbf{k}}_{mn}$ which share the same $k$. Circles: other modes. Colors differentiate between ${\mathbf{k}}_{mn}$ and its images under rotations of $l\pi /3$ for hexagonal modes or under reflections across the $n$ axis for the others. (b) Temporal evolution of ${\zeta }_{mn}(t)$, grouped by length $k$. The color code is that used in the spatial Fourier grid. Dashed blue curves: hexagonal modes (0, 2), (1, 3), (0, 4) and nonhexagonal modes ($-1$, 2), ($-2$, 1). Solid black curves: hexagonal modes (1, 1), (2, 0), (2, 2) and nonhexagonal modes (1, 0), (1, 2), (2, 1). Dash-dotted orange curves: hexagonal modes ($-1$, 1), ($-1$, 3), ($-2$, 2). (c) Three-dimensional phase portrait of temporal evolution of modes for an extended simulation. Points are equally spaced in time; their color evolves in time from dark to light. Times $t_1,\dots ,t_6$ marked by crosses and by vertical lines in (b).Reuse & Permissions