Superlattice patterns in surface waves

Abstract

International audienceWe report novel superlattice wave patterns at the interface of a fluid layer driven vertically. These patterns are described most naturally in terms of two interacting hexagonal sublattices. Two frequency forcing at very large aspect ratio is utilized in this work. A superlattice pattern (“superlattice-I”) consisting of two hexagonal lattices oriented at a relative angle of 22° is obtained with a 6:7 ratio of forcing frequencies. Several theoretical approaches that may be useful in understanding this pattern have been proposed. In another example, the waves are fully described by two superimposed hexagonal lattices with a wavelength ratio of \sqrt{3}, oriented at a relative angle of 30°. The time dependence of this “superlattice-II” wave pattern is unusual. The instantaneous patterns reveal a time-periodic stripe modulation that breaks the sixfold symmetry at any instant, but the stripes are absent in the time average. The instantaneous patterns are not simply amplitude modulations of the primary standing wave. A transition from the superlattice-II state to a 12-fold quasi-crystalline pattern is observed by changing the relative phase of the two forcing frequencies. Phase diagrams of the observed patterns (including superlattices, quasicrystalline patterns, ordinary hexagons, and squares) are obtained as a function of the amplitudes and relative phases of the driving accelerations

Superlattice, Rhombus, Square, and Hexagonal Standing Waves in Magnetically Driven Ferrofluid Surface

“…The observed superlattice pattern is similar but different from the ones previously observed by others [3][4][5][6]. In particular, the snapshot of superlattice [ Fig.…”

“…In subsequent experiments on a similar experimental setup, Kudrolli et al observed two different types of superlattice pattern ("type-I" and "type-II") in addition to the 12-fold symmetric quasicrystalline pattern [3]. More recently, Arbell and Fineberg report yet another kinds of superlattice pattern ("SSS" and "2MS") [4].…”

Diffusive Instabilities and Chemical Reactions

“…The resulting pattern is composed of two discrete but interacting sublattices and thus presents a spatial order on two different length scales. Such dynamic superlattices have also been obtained in nonlinear optical devices [Pampaloni et al, 1997;Musslimani & Pismen, 2000] and in two-frequencies Faraday experiments [Kudrolli et al, 1998;Arbell & Fineberg, 1998]. Finally, for specific relations between the angles of the triads of wave vectors of active modes, quasiperiodic patterns may also be generated.…”

Dynamics of nearly inviscid Faraday waves in almost circular containers

“…In large domains recent work has focused on two-frequency forcing [1], in which the two frequency components select distinct wavenumbers. These experiments, generally carried out in quite viscous liquids, have revealed various superlattice patterns, as well as nonperiodic states resembling quasipatterns [12,27,28,1]. In these experiments, the frequencies used are in (low order) resonance, so that the forcing acceleration remains periodic in time, with the frequency ratio, their relative phase, and their relative contribution to the forcing providing parameters to be tuned in the experiment.…”