Pattern formation in two-frequency forced parametric waves

Abstract

We present an experimental investigation of superlattice patterns generated on the surface of a fluid via parametric forcing with two commensurate frequencies. The spatiotemporal behavior of four qualitatively different types of superlattice patterns is described in detail. These states are generated via a number of different three-wave resonant interactions. They occur either as symmetry-breaking bifurcations of hexagonal patterns composed of a single unstable mode or via nonlinear interactions between the two primary unstable modes generated by the two forcing frequencies. A coherent picture of these states together with the phase space in which they appear is presented. In addition, we describe a number of new superlattice states generated by four-wave interactions that arise when symmetry constraints rule out three-wave resonances.

Three-Wave Interactions and Spatiotemporal Chaos

“…In some systems, modes with a second wavelength can play an important role in pattern formation if these modes are either unstable or only weakly damped. The interaction between two waves of one wavelength with a third wave of the other wavelength is known both experimentally and theoretically to play a key role in producing a rich variety of interesting phenomena such as quasipatterns, superlattice patterns and spatio-temporal chaos (STC) [1][2][3][4][5][6][7][8].In this paper, we focus on three-wave interactions (3WIs) involving two comparable wavelengths and develop a criterion for when such interactions are likely to lead to STC, as opposed to steady patterns and quasipatterns. The mechanism we describe is generic, and will apply to any system in which such 3WIs can occur, such as the Faraday wave experiment [1-3], coupled Turing systems [4] and some optical systems [5].…”

“…Using a multi-component forcing f (t) enables the excitation of waves with comparable wavelengths. Experimentally, the phases and amplitudes of the different components of f (t) have been shown to determine whether simple pattterns, superlattice patterns, quasipatterns or STC are seen [1][2][3]. A theoretical understanding of the stabilization of some superlattice patterns has been developed using a single 3WI [2,6,7], but up until this point there has been no explanation for the presence of STC.…”

“…In some systems, modes with a second wavelength can play an important role in pattern formation if these modes are either unstable or only weakly damped. The interaction between two waves of one wavelength with a third wave of the other wavelength is known both experimentally and theoretically to play a key role in producing a rich variety of interesting phenomena such as quasipatterns, superlattice patterns and spatio-temporal chaos (STC) [1][2][3][4][5][6][7][8].…”

Dynamic Mechanism of Photochemical Induction of Turing Superlattices in the Chlorine Dioxide−Iodine−Malonic Acid Reaction−Diffusion System

“…8,9 In nonequilibrium systems, superlattice patterns are formed by Faraday waves. [10][11][12] They have been found in other hydrodynamic, 13 magnetohydrodynamic, 14 and optical 15 systems. Superlattice patterns are seen on leopard and jaguar skins.…”

Pattern formation in large domains

“…A notable example of this is quasi-patterns, which are most readily found in Faraday wave experiments in which a tray of liquid is subjected to vertical vibrations with two commensurate forcing frequencies (Edwards & Fauve 1994). A recent survey of experimental results can be found in Arbell & Fineberg (2002), and one experimental example of a quasi-pattern is shown in¯gure 2a. This pattern is quasi-periodic in any horizontal direction, that is, the amplitude of the pattern (taken along any direction in the plane) can be regarded as a sum of modes with incommensurate spatial frequencies.…”