Pattern Control via Multifrequency Parametric Forcing

We use symmetry considerations to investigate control of a class of resonant three-wave interactions relevant to pattern formation in weakly damped, parametrically forced systems near onset. We classify and tabulate the most important damped, resonant modes and determine how the corresponding resonant triad interactions depend on the forcing parameters. The relative phase of the forcing terms may be used to enhance or suppress the nonlinear interactions. We compare our symmetry-based predictions with numerical and experimental results for Faraday waves. Our results suggest how to design multifrequency forcing functions that favor chosen patterns in the lab.

Figure 1

Fourier space diagram of resonant triads composed of two critical modes ($\left|{\mathbf{k}}_1\right|=\left|{\mathbf{k}}_2\right|=k_1$) and a damped mode (${\mathbf{k}}_3={\mathbf{k}}_1+{\mathbf{k}}_2$, $\left|{\mathbf{k}}_3\right|=k_3$) with (a) $k_3<k_1$, (b) $k_1<k_3<2k_1$.

Figure 2

Effect of $\Phi$ on the stabilizing $\Omega =n-m$ resonance that selects superlattice patterns [14] with $(m,n,p)=(6,7,2)$ forcing. (a) $b\left(\theta \right)$ with the optimal phase $\Phi =90°$; the singular region heralding hexagons at $\theta =60°$ is removed. The large dip near $\theta =67°$ is due to the strongly suppressing $\Omega =m$ resonance in the first section of Table . (b) Close-up of $b\left(\theta \right)$ near $\theta =22°$ with $\Phi =90°$ and $\Phi =-90°$; the two-frequency result (dotted line) with $f_p=0$ is also shown. (c) Spike magnitude versus $\Phi$ [see Eq. (7)]. For these calculations $\left|f_n\right|/\left|f_m\right|=0.75$, $\left|f_p\right|/\left|f_m\right|=0.1$. The rescaled fluid parameters (see 10) in the Zhang-Viñals equations are $\gamma =0.08$, $G_0=1.5$.