Faraday wave lattice as an elastic metamaterial

Metamaterials enable the emergence of novel physical properties due to the existence of an underlying subwavelength structure. Here, we use the Faraday instability to shape the fluid-air interface with a regular pattern. This pattern undergoes an oscillating secondary instability and exhibits spontaneous vibrations that are analogous to transverse elastic waves. By locally forcing these waves, we fully characterize their dispersion relation and show that a Faraday pattern presents an effective shear elasticity. We propose a physical mechanism combining surface tension with the Faraday structured interface that quantitatively predicts the elastic wave phase speed, revealing that the liquid interface behaves as an elastic metamaterial.

Figure 1

(a) Side view of the standing Faraday instability wave pattern obtained for $\varepsilon >0$ at $f_0=72$ Hz. The length of the white segment represents the Faraday wavelength (here ${\lambda }_F=5.1$ mm). (b) Sketch of the experimental setup. (c) Top view of the stable square pattern. The white segment has a length equal to ${\lambda }_F=5.1$ mm. (d) Top view of the oscillating Faraday pattern. The white segment has a length equal to $4{\lambda }_F=20.4$ mm, which is the wavelength of the spontaneous oscillations.

Figure 2

The forcing frequency is $f_0=72$ Hz for this figure. (a) Typical Fourier spectrum ${\tilde{x}}_{mn}\left(\omega \right)$ of the peak $(18,18)$ in the center of the pattern. (b) Modulus of spatial Fourier 2D spectrum $|\widehat{y}(k_x,k_y,f)|$ for $\varepsilon =0.976$, averaged for $f=1.52\pm 0.1$ Hz. Open diamonds show the position of the Fourier peaks for a stable pattern [Fig. 1]. (c) Amplitude of the spontaneous vibration as a function of the normalized control parameter, averaged over all the antinodes of the lattice. The up (red) and down (blue) triangles correspond to $|\tilde{y}|$ and $|\tilde{x}|$, respectively. Dashed line is a square root fit. Gray shades denote (from left to right) stable pattern, spontaneous vibrations of the lattice, and chaotic behavior.

Figure 3

(a) Device used to force the vibrations of the Faraday pattern. The black arrow shows the vertical motion of the whole vessel, and the dotted white arrow shows the direction of the comb vibration. The width of the vessel is 12 cm. (b) Map of the real part of the fast Fourier transform peak $\text{Re}\left(\tilde{y}\right)$ for a forcing frequency of $3.7\mathrm{Hz}$ and a vertical parametric forcing at $f_0=120$ Hz. The forcing device is on the left; each pixel represents a bright point of our images. (c) Dispersion relation $f\left(k_T\right)$ and $f\left(k_L\right)$. Blue down triangles: transversal waves in the $y$ direction. Red up triangles: transversal waves in the $x$ direction. Open circles: longitudinal waves in the $y$ direction. Solid black line: linear fit. Dashed line: gravity-capillary wave dispersion relation. Dotted line and gray background: prediction from Eq. (4) and its associated uncertainty.

Figure 4

Open circles: evolution of $S\left(\gamma \right)$ with $\gamma$ computed numerically for ${\lambda }_F=3.5\mathrm{mm}$ and $A_0/{\lambda }_F=13.5\%$. Dark line: theoretical prediction from Eq. (3). Inset (left): reference surface. Inset (right): sheared surface, with $tan\theta =\gamma$.