Localized Faraday patterns under heterogeneous parametric excitation

Abstract

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. PACS numbers: 05.45.Yv, 05.45.-a, 89.75.Kd

Spatial asymmetries of resonant oscillations in periodically forced heterogeneous media

“…However, in some cases spatial heterogeneity is an inherent feature of the system and is paramount to the emergence of distinct type of localized oscillations. Examples are the cochlea of the inner ear [29,30,31], alligator water dance [32], and Faraday waves under heterogeneous parametric excitation [33]. In other cases, heterogeneities can be exploited to tune the performance of particular engineered outputs in a range of potential applications, including mechanical resonators [34,35,36], catalytic surface reactions [26] and plasmonic nanoparticles [13].…”

Drift instabilities in localised Faraday patterns

“…The first two terms in the right-hand side of Eq. ( 4) are Weber-like terms that give the known linear limit of the heterogeneous system [13]. At the modulational instability, the amplitude of the unstable Gauss-Hermite modes is saturated by the quintic nonlinearity given by the last term in Eq.…”

“…The effect of a heterogeneous pump in this system has been recently studied [13] by introducing the space-dependent Gaussian profile γ(x) = γ o exp (−x 2 /2σ 2 i ), where γ o is the pump strength and σ i its extension. The linear stability analysis under a slightly non-uniform Gaussian drive, i.e.…”

“…The linear stability analysis under a slightly non-uniform Gaussian drive, i.e. σ i √ α, shows that the envelope C 0 of the pattern has a discrete response given by the Gauss-Hermite polynomials [13]. Urra et al [13] showed that the threshold of the instability of the m-th Gauss-Hermite mode is given by…”

Heterogeneity-stabilized homogeneous states in driven media

“…We consider how Faraday instabilities are impacted by symmetry-broken geometries, defined by heterogeneous (i.e., non-flat) substrates. Past studies have considered the possibility of localization resulting from small corrugations in the container [24][25][26] or through localized driving forces 27 , but we stress that the potential for HSHS has not been considered in this literature. Here, we show how heterogeneity from more general substrate geometries can impact the onset of Faraday wave instabilities, which allows us to a b experimentally demonstrate the existence of HSHS for both sinusoidal and random substrates with suitably large heterogeneity.…”