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23
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 11 (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 9 (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 patt ..."
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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...
Resonant Triad Dynamics in Weakly Damped Faraday Waves With . . .
, 2003
"... Many of the interesting patterns seen in recent multifrequency Faraday experiments can be understood on the basis of threewave interactions (resonant triads). In this paper we consider twofrequency forcing and focus on a resonant triad that occurs near the bicritical point where two patternformi ..."
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Cited by 5 (2 self)
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Many of the interesting patterns seen in recent multifrequency Faraday experiments can be understood on the basis of threewave interactions (resonant triads). In this paper we consider twofrequency forcing and focus on a resonant triad that occurs near the bicritical point where two patternforming modes with distinct wavenumbers emerge simultaneously. This triad has been observed directly (in the form of rhomboids) and has also been implicated in the formation of quasipatterns and superlattices. We show how the symmetries of the undamped unforced problem (time translation, time reversal, and Hamiltonian structure) can be used, when the damping is weak, to obtain general scaling laws and additional qualitative properties of the normal form coefficients governing the pattern selection process near onset; such features help to explain why this particular triad is seen only for certain "low" forcing ratios, and predict the existence of drifting solutions and heteroclinic cycles. We confirm the anticipated parameter dependence of the coefficients and investigate its dynamical consequences using coecients derived numerically from a quasipotential formulation of the Faraday problem due to Zhang and Viñals.
PATTERN SELECTION FOR FARADAY WAVES IN AN INCOMPRESSIBLE VISCOUS FLUID ∗
"... Abstract. When a layer of fluid is oscillated up and down with a sufficiently large amplitude, patterns form on the surface, a phenomenon first observed by Faraday. A wide variety of such patterns have been observed from regular squares and hexagons to superlattice and quasipatterns and more exotic ..."
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Cited by 4 (0 self)
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Abstract. When a layer of fluid is oscillated up and down with a sufficiently large amplitude, patterns form on the surface, a phenomenon first observed by Faraday. A wide variety of such patterns have been observed from regular squares and hexagons to superlattice and quasipatterns and more exotic patterns such as oscillons. Previous work has investigated the mechanisms of pattern selection using the tools of symmetry and bifurcation theory. The hypotheses produced by these generic arguments have been tested against an equation derived by Zhang and Viñals in the weakly viscous and large depth limit. However, in contrast, many of the experiments use shallow viscous layers of fluid to counteract the presence of high frequency weakly damped modes that can make patterns hard to observe. Here we develop a weakly nonlinear analysis of the full Navier–Stokes equations for the twofrequency excitation Faraday experiment. The problem is formulated for general depth, although results are presented only for the infinite depth limit. We focus on a few particular cases where detailed experimental results exist and compare our analytical results with the experimental observations. Good agreement with the experimental results is found. Key words. Faraday waves, superlattice patterns, weakly nonlinear analysis AMS subject classification. 37N10 DOI. 10.1137/050639223 1. Introduction. Waves
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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Cited by 4 (0 self)
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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
Resonances and Superlattice Pattern Stabilization In TwoFrequency . . .
, 2002
"... We investigate the role weakly damped modes play in the selection of Faraday wave patterns forced with rationallyrelated frequency components mω and nω. We use symmetry considerations to argue for the special importance of the weakly damped modes oscillating with twice the frequency of the critical ..."
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Cited by 3 (1 self)
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We investigate the role weakly damped modes play in the selection of Faraday wave patterns forced with rationallyrelated frequency components mω and nω. We use symmetry considerations to argue for the special importance of the weakly damped modes oscillating with twice the frequency of the critical mode, and those oscillating primarily with the "difference frequency" n  mω and the "sum frequency" (n + m)ω. We then perform a weakly nonlinear analysis using equations of Zhang and Vinals [1] which apply to smallamplitude waves on weakly inviscid, deep fluid layers. For weak damping and forcing and onedimensional waves, we perform a perturbation expansion through fourth order which yields analytical expressions for onset parameters and the cubic bifurcation coefficient that determines wave amplitude as a function of forcing. For stronger damping and forcing we numerically compute these same parameters, as well as the cubic crosscoupling coefficient for competing standing waves oriented at an angle # relative to each other. The resonance effects predicted by symmetry are borne out in the perturbation results for one spatial dimension, and are supported by the numerical results in two dimensions. The difference frequency resonance plays a key role in stabilizing superlattice patterns of the SLI type observed by Kudrolli, Pier and Gollub [2].
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.
NearResonant Steady Mode Interaction: Periodic, Quasiperiodic, and Localized Patterns ∗
"... Abstract. Motivated by the rich variety of complex periodic and quasiperiodic patterns found in systems such as twofrequency forced Faraday waves, we study the interaction of two spatially periodic modes that are nearly resonant. Within the framework of two coupled onedimensional Ginzburg–Landau ..."
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Cited by 1 (0 self)
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Abstract. Motivated by the rich variety of complex periodic and quasiperiodic patterns found in systems such as twofrequency forced Faraday waves, we study the interaction of two spatially periodic modes that are nearly resonant. Within the framework of two coupled onedimensional Ginzburg–Landau equations we investigate analytically the stability of the periodic solutions to general perturbations, including perturbations that do not respect the periodicity of the pattern, and which may lead to quasiperiodic solutions. We study the impact of the deviation from exact resonance on the destabilizing modes and on the final states. In regimes in which the mode interaction leads to the existence of traveling waves our numerical simulations reveal localized waves in which the wavenumbers are resonant and which drift through a steady background pattern that has an offresonant wavenumber ratio.