Localized Faraday patterns under heterogeneous parametric excitation

Faraday waves are a classic example of a system in which an extended pattern emerges under spatially uniform forcing. Motivated by systems in which uniform excitation is not plausible, we study both experimentally and theoretically the effect of heterogeneous forcing on Faraday waves. Our experiments show that vibrations restricted to finite regions lead to the formation of localized subharmonic wave patterns and change the onset of the instability. The prototype model used for the theoretical calculations is the parametrically driven and damped nonlinear Schrödinger equation, which is known to describe well Faraday-instability regimes. For an energy injection with a Gaussian spatial profile, we show that the evolution of the envelope of the wave pattern can be reduced to a Weber-equation eigenvalue problem. Our theoretical results provide very good predictions of our experimental observations provided that the decay length scale of the Gaussian profile is much larger than the pattern wavelength.

Figure 1

Experimental setup used to generate a localized injection of energy in a Faraday-wave configuration. The soft bottom is attached to a set of pistons, each linked to a rotary cam system attached to a common shaft and a brushless motor. Amplitude and frequency can be independently programed.

Figure 2

Snapshots of the observed structures under localized forcing for different excitation regions: from top to bottom ${\sigma }_i=96,176,208\mathrm{mm}$. Green continuous lines depict the detected free-surface deformations at the wall, while blue dashed curves, the profiles of the effective acceleration at the surface.

Figure 3

Spatiotemporal evolution of free surface (Faraday waves) at the wall $\eta (x,t)$ for (a) $n=6$ pistons and (b) $n=13$ pistons.

Figure 4

Injected forcing variance ${\sigma }_i$ as a function of pattern variance ${\sigma }_w$ for $\nu =0.07$ and $\mu =0.0152$. Theoretical (solid black line), experimental (blue squares), and numerical (green circles) data are depicted.

Figure 5

Localized Faraday-wave amplitude ${\eta }_{max}$ as a function of the forcing amplitude $\mathrm{\Gamma }$ for $n=13$ pistons. The wave displays four nodes at frequency $f=14.61\mathrm{Hz}$. The two series of experimental data show increasing (blue $▴$) and decreasing (green $▾$) ramps in $\mathrm{\Gamma }$. The data is compared with a $1/4\mathrm{th}$ power law, $\mathrm{\Lambda }{(\mathrm{\Gamma }-{\mathrm{\Gamma }}_c)}^{1/4}$ (dashed line) with parameters $\mathrm{\Lambda }=0.604$ and ${\mathrm{\Gamma }}_0=0.265$ derived from theory (no-fitted parameters).

Figure 6

Numerical simulation of the onset of a localized Faraday wave from the homogeneous zero solution triggered by small-amplitude noise. (a) A wave pattern obtained with ${\gamma }_0=0.22,\nu =0.2,\mu =0.14$, and $\sigma =8.0$ (${\gamma }_0^{\left(1\right)}<{\gamma }_0<{\gamma }_0^{\left(1\right)}$). (b) Temporal evolution of the pattern-envelope amplitude, showing the nonlinear saturation. The envelope of the wave at the end of the simulation has a nearly Gaussian-like profile, as shown in (c), suggesting the instability of the fundamental Gauss-Hermite mode. Here, the green solid line represents $\text{Re}\left(\varphi \right)$, while the black dashed line represents the envelope of $\text{Re}\left(\varphi \right)$ obtained using the Hilbert transform. The localized injection is also depicted as a reference (dash-dotted blue line). Time is dimensionless (measured in fundamental period units, $T$).

Figure 7

Spectra obtained numerically for $\nu =1.0,\mu =0.45$, and ${\sigma }_i=16.0$. (a) Spectrum for ${\gamma }_0^{\left(0\right)}<{\gamma }_0<{\gamma }_0^{\left(1\right)}$. The inset shows a zoom-out of the spectrum, showing a continuum set of eigenvalues with a nonvanishing imaginary part. (b) Spectrum for ${\gamma }_0^{\left(1\right)}<{\gamma }_0<{\gamma }_0^{\left(2\right)}$. The real part of the eigenfunctions of the first two critical modes are indicated in each case.

Figure 8

Real part of the eigenvalues of the Gauss-Hermite modes as a function of ${\gamma }_0$, obtained from several numerical simulations for $\nu =1.0,\mu =0.45$, and ${\sigma }_i=16.0$. The modes turn unstable at the values predicted by the linear stability analysis.