Multifrequency control of Faraday wave patterns

We show how pattern formation in Faraday waves may be manipulated by varying the harmonic content of the periodic forcing function. Our approach relies on the crucial influence of resonant triad interactions coupling pairs of critical standing wave modes with damped, spatiotemporally resonant modes. Under the assumption of weak damping and forcing, we perform a symmetry-based analysis that reveals the damped modes most relevant for pattern selection, and how the strength of the corresponding triad interactions depends on the forcing frequencies, amplitudes, and phases. In many cases, the further assumption of Hamiltonian structure in the inviscid limit determines whether the given triad interaction has an enhancing or suppressing effect on related patterns. Surprisingly, even for forcing functions with arbitrarily many frequency components, there are at most five frequencies that affect each of the important triad interactions at leading order. The relative phases of those forcing components play a key role, sometimes making the difference between an enhancing and suppressing effect. In numerical examples, we examine the validity of our results for larger values of the damping and forcing. Finally, we apply our findings to one-dimensional periodic patterns obtained with impulsive forcing and to two-dimensional superlattice patterns and quasipatterns obtained with multifrequency forcing.

Figure 1

Fourier space diagram of spatially resonant triads satisfying Eq. (2). The two neutrally stable modes have wave number $\mid {\mathbf{k}}_1\mid =\mid {\mathbf{k}}_2\mid =k_c$ and oscillate with dominant frequency $m∕2$. The damped mode has $\mid {\mathbf{k}}_3\mid =k_d$ and oscillates with dominant frequency $\Omega$. (a) $k_d<k_c$. (b) $k_c<k_d<2k_c$.

Figure 2

Three qualitatively different phase portraits corresponding to Eq. (9) with $a<0$, ${\Lambda }_1>0$. Top: $b<a$. Middle: $a<b<-a$. Bottom: $b>-a$.

Figure 3

Coupling coefficient $b\left(\theta \right)$ in Eq. (9) computed from Eq. (36) using three-frequency $(m,n,p)=(8,7,2)$ forcing, $\gamma =0.1$, ${\Gamma }_0=16$, $\mid f_n\mid ∕\mid f_m\mid =0.4$, $\mid f_p\mid ∕\mid f_m\mid =0.08$, ${\varphi }_m={\varphi }_n={\varphi }_p=0$. The angle $\theta$ is given in degrees.

Figure 4

Resonant contribution $b_{\mathrm{res}}$ as a function of the damping parameter $\gamma$. The dots correspond to a numerical computation using Eq. (36). The straight line of slope 1 confirms the $b_{\mathrm{res}}\propto \gamma$ scaling predicted by symmetry arguments. The capillarity and forcing parameters used are the same as those in Fig. 3.

Figure 5

Half-width $\Psi$ of the resonant “dip” as a function of the damping $\gamma$. The dots correspond to a numerical computation using Eq. (36). The straight line of slope 1 confirms the predicted $\Psi \propto \gamma$ scaling. The capillarity and forcing parameters used are the same as those in Fig. 3.

Figure 6

The values of $\Phi$ at which $b\left({\theta }_{\mathrm{res}}\right)$ takes on its minimum and maximum values as a function of the damping $\gamma$. The dots correspond to numerical data, while the lines at 90° and $-90°$ show the predicted minimum and maximum respectively. The capillarity number and forcing amplitudes used are the same as those in Fig. 3.

Figure 7

(a)–(c) Dependence of $b\left({\theta }_{\mathrm{res}}\right)$ on the phase $\Phi$ ($\Phi$ given in degrees). (a) With damping $\gamma =0.04$. As predicted by the symmetry arguments in Sec. 3, the phase dependence is sinusoidal with minimum and maximum near $\pm 90°$. (b) $\gamma =0.2$. The phase dependence is sinusoidal, but there is a phase shift of approximately 45°. (c) $\gamma =1$. The dependence is no longer sinusoidal. (d) Fourier transform of the data in (c). The zero component has been removed and the remaining data have been normalized so that the strongest component has magnitude 1. The dependence on higher harmonics, e.g., $2\Phi$, $3\Phi$, $4\Phi$ is apparent. For all plots, the capillarity number and forcing amplitudes used are the same as those in Fig. 3.

Figure 8

Effect of relative forcing phase on the first harmonic resonance, i.e., resonance with the $\Omega =m$ mode, for $(m,n)=(1,2)$ forcing. The relevant phase $\Phi$ is given in Table . (a) Cross-coupling coefficient $b\left(\theta \right)$ with $\Phi =90°$ and $\Phi =-90°$; the single frequency case (dashed line) is shown for reference. (b) Dip magnitude $b\left({\theta }_{\mathrm{res}}\right)$ versus $\Phi$. For these calculations, the parameters in Eq. (36) are $\gamma =0.008$ and ${\Gamma }_0=0.125$, and the forcing amplitude ratio is $\mid f_n\mid ∕\mid f_m\mid =0.396$. Both $\theta$ and $\Phi$ are given in degrees.

Figure 9

Schematic representation of the asymmetric $\delta$-function forcing specified by Eq. (45).

Figure 10

Schematic of the Fourier wave vectors corresponding to the 12 dominant waves which comprise an SL-I superlattice pattern. The vectors point to the vertices of two hexagons, one rotated by an angle ${\theta }_{\mathrm{h}}<30°$ with respect to the other.

Figure 11

(a) Coupling coefficient for computing superlattice-I pattern stability. We use three-frequency forcing with $(m,n,p)=(8,10,11)$. The two small “bumps” at $({\theta }_{n-m},{\theta }_{p-m})=(20.3°,39.9°)$ are due to resonance with the modes oscillating with the difference frequencies $\Omega =n-m$ and $\Omega =p-m$. No superlattice patterns are stable. (b) Like (a), but with additional forcing frequency components $(q,r)=(4,6)$ which parametrically force the difference frequency modes. The result from (a) is duplicated as a dotted line for comparison. The two bumps become two very large spikes, and the superlattice pattern with angle ${\theta }_{\mathrm{h}}\simeq 20.3°$ is stabilized. For (a) and (b), we have added vertical arrows to guide the eye to the effects at the resonant angles. The fluid parameters used are $\gamma =0.1$ and ${\Gamma }_0=5.26$. The forcing amplitude ratios and phases used are given in the text. The region around 60° corresponds to a hexagonal interaction not captured by our calculation, and thus has been removed.

Figure 12

Coupling coefficient $b\left(\theta \right)$ in Eq. (9) as computed from Eq. (36) for seven-frequency $(m,n,p,q,r,s,t)=(12,17,20,10,16,30)$ forcing. The forcing phases are all 0 for the dotted line, while the solid line corresponds to the optimal choice ${\varphi }_{r,s,t}=90°$. The three damped modes with frequencies $n-m$, $p-m$, and $q-m$ are parametrically forced by the $(r,s,t)$ components. These three difference frequencies lead to spikes in $b\left(\theta \right)$ at angles which may help stabilize a 14-fold quasipattern; the desired locations of these spikes (as determined by geometry) are indicated by vertical arrows. The small spike around 42° is due to another difference frequency resonance $(s-m)$ not of interest here. As in Fig. 11, the region around 60° has been removed.