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27
Superlattice patterns in surface waves
 Physica D
, 1998
"... We 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 ("supe ..."
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We 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 ("superlatticeI") consisting of two hexagonal lattices oriented at a relative angle of 22o 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 3, oriented at a relative angle of 30o. The time dependence of this "superlatticeII " wave pattern is unusual. The instantaneous patterns reveal a timeperiodic stripe modulation that breaks the 6fold 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 superlatticeII state to a 12fold quasicrystalline 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.
Pattern Formation in Weakly Damped Parametric Surface Waves Driven By Two Frequency Components
, 1997
"... this paper an extension of a weakly nonlinear model previously introduced by Zhang & Vi~nals (1996a) and (1996b), which is valid in the limit of low fluid viscosity, large aspect ratio and large depth, to study pattern selection near onset of Faraday waves when the driving force has two independ ..."
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Cited by 15 (1 self)
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this paper an extension of a weakly nonlinear model previously introduced by Zhang & Vi~nals (1996a) and (1996b), which is valid in the limit of low fluid viscosity, large aspect ratio and large depth, to study pattern selection near onset of Faraday waves when the driving force has two independent frequency components. The model developed by Zhang & Vi~nals (1996b) was based on a quasipotential approximation to the equations governing fluid motion. In it, the flow is considered to be potential in the bulk, subject to effective boundary conditions at the moving free surface that incorporate the effect of the rotational component of the flow within a thin boundary layer near the free surface. We further assumed without rigorous justification that for low viscosity fluids, only linear viscous terms need to be retained in the resulting equations (the socalled "linear damping quasipotential equations", or LDQPE's). A multiscale analysis of the resulting LDQPE's led to the prediction of standing wave patterns of square symmetry near onset for capillary waves, in agreement with experiments. For mixed capillarygravity waves, patterns of hexagonal symmetry or quasiperiodic patterns were predicted depending on the value of the damping coefficient. 2 WENBIN ZHANG AND JORGE VI ~ NALS
Twofrequency forced Faraday waves: Weakly damped modes and pattern selection
 Physica D
, 2000
"... Recent experiments [1] on two–frequency parametrically excited surface waves exhibit an intriguing “superlattice ” wave pattern near a codimension–two bifurcation point where both subharmonic and harmonic waves onset simultaneously, but with different spatial wavenumbers. The superlattice pattern is ..."
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Cited by 12 (7 self)
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Recent experiments [1] on two–frequency parametrically excited surface waves exhibit an intriguing “superlattice ” wave pattern near a codimension–two bifurcation point where both subharmonic and harmonic waves onset simultaneously, but with different spatial wavenumbers. The superlattice pattern is synchronous with the forcing, spatially periodic on a large hexagonal lattice, and exhibits small–scale triangular structure. Similar patterns have been shown to exist as primary solution branches of a generic 12–dimensional D6 ˙+T 2 –equivariant bifurcation problem, and may be stable if the nonlinear coefficients of the bifurcation problem satisfy certain inequalities [2]. Here we use the spatial and temporal symmetries of the problem to argue that weakly damped harmonic waves may be critical to understanding the stabilization of this pattern in the Faraday system. We illustrate this mechanism by considering the equations developed by Zhang and Viñals [3] for small amplitude, weakly damped surface waves on a semi–infinite fluid layer. We compute the relevant nonlinear coefficients in the bifurcation equations describing the onset of patterns for excitation frequency ratios of 2/3 and 6/7. For the 2/3 case, we show that there is a fundamental difference in the pattern selection problems for subharmonic and harmonic instabilities near the bicritical point. Also, we find that the 6/7 case is significantly different from the 2/3 case due to the presence of additional weakly damped harmonic modes. These additional harmonic modes can result in a stabilization of the superpatterns. 1 1
Parametrically Excited Surface Waves: TwoFrequency Forcing, Normal Form Symmetries, and Pattern Selection
 Phys. Rev. E
, 1999
"... Motivated by experimental observations of exotic free surface standing wave patterns in the twofrequency Faraday experiment, we investigate the role of normal form symmetries in the associated pattern selection problem. With forcing frequency components in ratio m=n, where m and n are coprime inte ..."
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Cited by 10 (5 self)
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Motivated by experimental observations of exotic free surface standing wave patterns in the twofrequency Faraday experiment, we investigate the role of normal form symmetries in the associated pattern selection problem. With forcing frequency components in ratio m=n, where m and n are coprime integers that are not both odd, there is the possibility that both harmonic waves and subharmonic waves may lose stability simultaneously, each with a different wave number. We focus on this situation and compare the case where the harmonic waves have a longer wavelength than the subharmonic waves with the case where the harmonic waves have a shorter wavelength. We show that in the former case a normal form transformation can be used to remove all quadratic terms from the amplitude equations governing the relevant resonant triad interactions. Thus the role of resonant triads in the pattern selection problem is greatly diminished in this situation. We verify our general bifurcation theoretic resu...
Spatial periodmultiplying instabilities of hexagonal Faraday waves
, 2000
"... A recent Faraday wave experiment with twofrequency forcing reports two types of `superlattice' patterns that display periodic spatial structures having two separate scales [1]. These patterns both arise as secondary states once the primary hexagonal pattern becomes unstable. In one of these ..."
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Cited by 5 (4 self)
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A recent Faraday wave experiment with twofrequency forcing reports two types of `superlattice' patterns that display periodic spatial structures having two separate scales [1]. These patterns both arise as secondary states once the primary hexagonal pattern becomes unstable. In one of these patterns (socalled `superlatticetwo') the original hexagonal symmetry is broken in a subharmonic instability to form a striped pattern with a spatial scale increased by a factor of 2 p 3 from the original scale of the hexagons. In contrast, the timeaveraged pattern is periodic on a hexagonal lattice with an intermediate spatial scale ( p 3 larger than the original scale) and apparently has 60 ffi rotation symmetry. We present a symmetrybased approach to the analysis of this bifurcation. Taking as our starting point only the observed instantaneous symmetry of the superlatticetwo pattern presented in [1] and the subharmonic nature of the secondary instability, we show (a) that a...
Design of parametrically forced patterns and quasipatterns
 SIAM J. Appl. Dyn. Syst
, 2009
"... Abstract. The Faraday wave experiment is a classic example of a system driven by parametric forcing, and it produces a wide range of complex patterns, including superlattice patterns and quasipatterns. Nonlinear threewave interactions between driven and weakly damped modes play a key role in determ ..."
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Abstract. The Faraday wave experiment is a classic example of a system driven by parametric forcing, and it produces a wide range of complex patterns, including superlattice patterns and quasipatterns. Nonlinear threewave interactions between driven and weakly damped modes play a key role in determining which patterns are favoured. We use this idea to design single and multifrequency forcing functions that produce examples of superlattice patterns and quasipatterns in a new model PDE with parametric forcing. We make quantitative comparisons between the predicted patterns and the solutions of the PDE. Unexpectedly, the agreement is good only for parameter values very close to onset. The reason that the range of validity is limited is that the theory requires strong damping of all modes apart from the driven patternforming modes. This is in conflict with the requirement for weak damping if threewave coupling is to influence pattern selection effectively. We distinguish the two different ways that threewave interactions can be used to stabilise quasipatterns, and present examples of 12, 14 and 20fold approximate quasipatterns. We identify which computational domains provide the most accurate approximations to 12fold quasipatterns, and systematically investigate the Fourier spectra of the most accurate approximations. Key words. Pattern formation, quasipatterns, superlattice patterns, mode interactions, Faraday waves. AMS subject classifications. 35B32, 37G40, 52C23, 70K28, 76B15 1. Introduction. The
Secondary Instabilities of Hexagons: a bifurcation analysis of experimentally observed Faraday wave patterns
, 2002
"... We examine three experimental observations of Faraday waves generated by twofrequency forcing, in which a primary hexagonal pattern becomes unstable to three dierent superlattice patterns. We analyse the bifurcations involved in creating the three new patterns using a symmetrybased approach. E ..."
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Cited by 2 (2 self)
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We examine three experimental observations of Faraday waves generated by twofrequency forcing, in which a primary hexagonal pattern becomes unstable to three dierent superlattice patterns. We analyse the bifurcations involved in creating the three new patterns using a symmetrybased approach. Each of the three examples reveals a dierent situation that can arise in the theoretical analysis.
Harmonic Resonance Theory: An Alternative to the "Neuron Doctrine" Paradigm of Neurocomputation to Address Gestalt properties of perception
, 2000
"... neurocomputation involves discrete signals communicated along fixed transmission lines between discrete computational elements. This concept is shown to be inadequate to account for invariance in recognition, as well as for the holistic global aspects of perception identified by Gestalt theory. A Ha ..."
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Cited by 1 (0 self)
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neurocomputation involves discrete signals communicated along fixed transmission lines between discrete computational elements. This concept is shown to be inadequate to account for invariance in recognition, as well as for the holistic global aspects of perception identified by Gestalt theory. A Harmonic Resonance theory is presented as an alternative paradigm of neurocomputation, that exhibits both the property of invariance, and the emergent Gestalt properties of perception, not as special mechanisms contrived to achieve those properties, but as natural properties of the resonance itself.