Spatiotemporal characterization of interfacial Faraday waves by means of a light absorption technique

We present measurements of the complete spatiotemporal Fourier spectrum of Faraday waves. The Faraday waves are generated at the interface of two immiscible index matched liquids of different density. By use of a light absorption technique we are able to determine the bifurcation scenario from the flat surface to the patterned state for each complex spatial and temporal Fourier component separately. The surface spectra at onset are found to be in good agreement with the predictions from the linear stability analysis. For the nonlinear state our measurements show in a direct manner how energy is transferred from lower to higher harmonics and we quantify the nonlinear coupling coefficients. Furthermore we find that the nonlinear coupling generates static components in the temporal Fourier spectrum leading thus to a contribution of a nonoscillating permanent sinusoidal deformed surface state. A comparison of hexagonal and rectangular patterns reveals that spatial resonance can give rise to a spectrum that violates the temporal resonance conditions given by the weakly nonlinear theory.

Figure 1

Experimental setup: L—halogen lamp, Ln—lens, C—container filled by two liquids: SOIL and WSS with the same refractive indices, DS—diffusive screen, IF—interference filter, CCD—high speed CCD camera.

Figure 2

Snapshots of the square Faraday pattern [(a) and (b)] and their power spectra [(c) and (d)] at $\Omega ∕2\pi =12\mathrm{Hz}$ and $\epsilon =0.17$ $(a_0=30.0\mathrm{m}∕{\mathrm{s}}^2)$ for two different temporal phases corresponding to maximum [(a)] and minimum [(b)] surface elevation as indicated by arrows ($t_1$ and $t_2$) in Fig. 4.

Figure 3

Critical acceleration $a_c$ [(a)] and critical wave number $k_c$ [(b)] for different driving frequencies $\Omega$. The symbols are the experimental data, the lines are theoretical calculations based on the linear stability analysis. The size of the symbols coincides with the size of the error bars.

Figure 4

Temporal behavior of amplitudes $A$ corresponding to spatial modes $k\left(10\right)$ [(a)], $k\left(20\right)$ [(b)], and $k\left(11\right)$ [(c)] of the square Faraday pattern at $\Omega ∕2\pi =12\mathrm{Hz}$ and $\epsilon =0.17$ $(a_0=30.0\mathrm{m}∕{\mathrm{s}}^2)$. (d) shows the driving signal $a\left(t\right)$. Please note, that in contrast to Ref. 2, not the square root of the power spectra but the amplitude $A$ of the surface elevation $h(\mathbf{r},t)=A\cos [\mathbf{k}\left(ij\right)\bullet \mathbf{r}]$ is shown.

Figure 5

Temporal Fourier spectra $(\Omega ∕2\pi =12\mathrm{Hz})$ of the amplitudes $A\left[\mathbf{k}(ij,\omega )\right]$ corresponding to several spatial modes of the oscillating square Faraday pattern [(a), (b), and (c)] $(\epsilon =0.18)$ and hexagonal Faraday pattern [(d), (e), and (f)] $(\epsilon =0.38)$.

Figure 6

Amplitudes $A(n\Omega ∕2)$ of the $\mathbf{k}\left(10\right)$ [(a)] and $\mathbf{k}\left(11\right)$ [(b)] modes at $\Omega ∕2\pi =12\mathrm{Hz}$ as a function of the driving strength $\epsilon$. (c) The square of the sum of the subharmonic components of the (10)-mode versus $\epsilon$. As would be expected for a forward bifurcation the data can be linearly fitted, at least up to driving strength of $\epsilon \approx 0.1$

Figure 7

Ratio of the amplitudes $A(3\Omega ∕2)∕A(\Omega ∕2)$ of the $\mathbf{k}\left(10\right)$ mode at $\epsilon =0$ for different driving frequencies. The symbols mark experimental, the lines data from the linear theory. The experimental data are extrapolated from measurements at $\epsilon >0$ shown in the inset: The amplitude ratios at $\Omega ∕2\pi =12$ and 29 Hz as a function of the driving strength $\epsilon$.

Figure 8

Temporal phases $\psi (\Omega ∕2)$ [(a)] and $\psi (3\Omega ∕2)$ [(b)] of the $\mathbf{k}\left(10\right)$ mode for two driving frequencies versus driving strength $\epsilon$. The symbols are the experimental data, the lines are the theoretical calculations. Squares and broken line: $\Omega ∕2\pi =57\mathrm{Hz}$; circles and dotted line: $\Omega ∕2\pi =12\mathrm{Hz}$. In the range $0.2<\epsilon <0.28$ a transition from squares to hexagons takes place leading thus to a disordered state. For this reason an extraction of the phases is not possible there.

Figure 9

Amplitudes $A\left[\mathbf{k}\left(20\right)\right]$ and $A\left[\mathbf{k}\left(11\right)\right]$ versus the square of the amplitude of the fundamental mode $A\left[\mathbf{k}\left(10\right)\right]$ $(\Omega ∕2\pi =12\mathrm{Hz})$. Coefficient $\beta$ is the slope of the linear fit and characterizes the efficiency of the nonlinear interaction.

Figure 10

Snapshots of the hexagonal Faraday pattern [(a), (b), and (c)] and corresponding power spectra [(d), (e), and (f), respectively] at $\Omega ∕2\pi =12\mathrm{Hz}$ and $\epsilon =0.37$ $(a_0=39.3\mathrm{m}∕{\mathrm{s}}^2)$ for three different temporal phases. (a) Down hexagons; (b) pattern near the minimal surface elevation, (c) up hexagons. See Refs. 2, 13 for further explanations on the switch from up to down hexagons in the Faraday Experiment.

Figure 11

Vector diagram of the interacting modes for the hexagonal surface state.

Figure 12

Snapshots of the Faraday pattern [(a),(c), and (e)] and the power spectra [(b), (d), and (f)] at $\Omega ∕2\pi =20\mathrm{Hz}$ and $\epsilon =0.6$ $(a_0=54.4\mathrm{m}∕{\mathrm{s}}^2)$ [(a),(b)]; $\Omega ∕2\pi =20\mathrm{Hz}$ and $\epsilon =0.08$ $(a_0=36.7\mathrm{m}∕{\mathrm{s}}^2)$ [(c),(d)]; $\Omega ∕2\pi =57\mathrm{Hz}$ and $\epsilon =0.11$ $(a_0=116.3\mathrm{m}∕{\mathrm{s}}^2)$ [(e),(f)].

Figure 13

Amplitudes $A(\Omega ∕2)$ of the $\mathbf{k}\left(10\right)$ and $\mathbf{k}\left(01\right)$ modes at $\Omega ∕2\pi =29\mathrm{Hz}$ [(a)], and $\mathbf{k}\left(1\right)$ mode at $\Omega ∕2\pi =57\mathrm{Hz}$ [(b)] as a function of the driving strength $\epsilon$. The marked strip in (a) indicates the region of the crossed rolls state.

Figure 14

The amplitudes of spatial modes $A\left(k\right)$ [(a)] and $A\left(2k\right)$ [(b)] in the line state at $\Omega ∕2\pi =57\mathrm{Hz}$ and $\epsilon =0.11$ $(a_0=116.3\mathrm{m}∕{\mathrm{s}}^2)$. The temporal constant offset in (b) is indicated by $A(2k,0)$. (c) shows the dependence of $A(2k,0)$ versus $A^2(k,\Omega ∕2)$.