arXiv subjects

Mary Silber

Publications and source records attributed to Mary Silber.

31 records · Page 2Linked to original sources

arXiv2005

A Weakly Nonlinear Analysis of Impulsively-Forced Faraday Waves

Parametrically-excited surface waves, forced by a periodic sequence of delta-function impulses, are considered within the framework of the Zhang-Viñals model (J. Fluid Mech. 1997). The exact impulsive-forcing results, in the linear and weakly nonlinear regimes, are compared with numerical results for sinusoidal and multifrequency forcing. We find surprisingly good agreement between impulsive forcing results and those obtained using a two-term truncated Fourier series representation of the impulsive forcing function. As noted previously by Bechhoefer and Johnson (Am. J. Phys. 1996), in the case of two equally-spaced impulses per period there are only subharmonic modes of instability. The familiar situation of alternating subharmonic and harmonic resonance tongues emerges for unequally-spaced impulses. We extend the linear analysis for two impulses per period to the weakly nonlinear regime for one-dimensional waves. Specifically, we derive an analytic expression for the cubic Landau coefficient in the bifurcation equation as a function of the dimensionless fluid parameters and spacing between the two impulses. As the capillary parameter is varied, one finds a parameter region of wave amplitude suppression, which is due to a familiar 1:2 spatio-temporal resonance between the subharmonic mode of instability and a damped harmonic mode. This resonance occurs for impulsive forcing even when harmonic resonance tongues are absent from the neutral stability curve. The strength of this resonance feature can be tuned by varying the spacing between the impulses. This finding is interpreted in terms of a recent symmetry-based analysis of multifrequency forced Faraday waves by Porter, Topaz and Silber (Phys. Rev. Lett. 2004, Phys. Rev. E 2004).

nlin.PS↗

arXiv2004

Multi-frequency control of Faraday wave patterns

We show how pattern formation in Faraday waves may be manipulated by varying the harmonic content of the periodic forcing function. Our approach relies on the crucial influence of resonant triad interactions coupling pairs of critical standing wave modes with damped, spatio-temporally resonant modes. Under the assumption of weak damping and forcing, we perform a symmetry-based analysis that reveals the damped modes most relevant for pattern selection, and how the strength of the corresponding triad interactions depends on the forcing frequencies, amplitudes, and phases. In many cases, the further assumption of Hamiltonian structure in the inviscid limit determines whether the given triad interaction has an enhancing or suppressing effect on related patterns. Surprisingly, even for forcing functions with arbitrarily many frequency components, there are at most five frequencies that affect each of the important triad interactions at leading order. The relative phases of those forcing components play a key role, sometimes making the difference between an enhancing and suppressing effect. In numerical examples, we examine the validity of our results for larger values of the damping and forcing. Finally, we apply our findings to one-dimensional periodic patterns obtained with impulsive forcing and to two-dimensional superlattice patterns and quasipatterns obtained with multi-frequency forcing.

nlin.PS↗

arXiv2003

Faraday Wave Pattern Selection Via Multi-Frequency Forcing

We use symmetry considerations to investigate how damped modes affect pattern selection in multi-frequency forced Faraday waves. We classify and tabulate the most important damped modes and determine how the corresponding resonant triad interactions depend on the forcing parameters. The relative phase of the forcing terms may be used to enhance or suppress the nonlinear interactions. We compare our predictions with numerical results and discuss their implications for recent experiments. Our results suggest how to design multi-frequency forcing functions that favor chosen patterns in the lab.

nlin.PS↗

arXiv2003

Resonant triad dynamics in weakly damped Faraday waves with two-frequency forcing

Many of the interesting patterns seen in recent multi-frequency Faraday experiments can be understood on the basis of three-wave interactions (resonant triads). In this paper we consider two-frequency forcing and focus on a resonant triad that occurs near the bicritical point where two pattern-forming 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 coefficients derived numerically from a quasipotential formulation of the Faraday problem due to Zhang and Vinals.

nlin.PS↗

arXiv2002

Broken symmetries and pattern formation in two-frequency forced Faraday waves

We exploit the presence of approximate (broken) symmetries to obtain general scaling laws governing the process of pattern formation in weakly damped Faraday waves. Specifically, we consider a two-frequency forcing function and trace the effects of time translation, time reversal and Hamiltonian structure for three illustrative examples: hexagons, two-mode superlattices, and two-mode rhomboids. By means of explicit parameter symmetries, we show how the size of various three-wave resonant interactions depends on the frequency ratio m:n and on the relative temporal phase of the two driving terms. These symmetry-based predictions are verified for numerically calculated coefficients, and help explain the results of recent experiments.

nlin.PS↗

arXiv2001

Resonances and superlattice pattern stabilization in two-frequency forced Faraday waves

We investigate the role weakly damped modes play in the selection of Faraday wave patterns forced with rationally-related frequency components m*omega and n*omega. 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-m|*omega and the "sum frequency" (n+m)*omega. We then perform a weakly nonlinear analysis using equations of Zhang and Vinals (1997, J. Fluid Mech. 336) which apply to small-amplitude waves on weakly inviscid, semi-infinite fluid layers. For weak damping and forcing and one-dimensional 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 near onset. For stronger damping and forcing we numerically compute these same parameters, as well as the cubic cross-coupling coefficient for competing waves travelling at an angle theta 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 SL-I type observed by Kudrolli, Pier and Gollub (1998, Physica D 123).

nlin.PS↗

arXiv2000

Two-frequency forced Faraday waves: Weakly damped modes and pattern selection

Recent experiments (Kudrolli, Pier and Gollub, 1998) 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 $D_6\dot{+}T^2$-equivariant bifurcation problem, and may be stable if the nonlinear coefficients of the bifurcation problem satisfy certain inequalities (Silber and Proctor, 1998). 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 Vinals (1997, J. Fluid Mech. 336) 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 codimension-two 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.

nlin.PS↗

arXiv1998

Parametrically Excited Surface Waves: Two-Frequency Forcing, Normal Form Symmetries, and Pattern Selection

Motivated by experimental observations of exotic standing wave patterns in the two-frequency Faraday experiment, we investigate the role of normal form symmetries in the pattern selection problem. With forcing frequency components in ratio m/n, where m and n are co-prime integers, there is the possibility that both harmonic and subharmonic waves may lose stability simultaneously, each with a different wavenumber. 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 results within the example of one-dimensional surface wave solutions of the Zhang-Vinals model of the two-frequency Faraday problem. In one-dimension, a 1:2 spatial resonance takes the place of a resonant triad in our investigation. We find that when the bifurcating modes are in this spatial resonance, it dramatically effects the bifurcation to subharmonic waves in the case of forcing frequencies are in ratio 1/2; this is consistent with the results of Zhang and Vinals. In sharp contrast, we find that when the forcing frequencies are in ratio 2/3, the bifurcation to (sub)harmonic waves is insensitive to the presence of another spatially-resonant bifurcating mode.

patt-sol↗

arXiv1998

Nonlinear Competition Between Small and Large Hexagonal Patterns

Recent experiments by Kudrolli, Pier and Gollub on surface waves, parametrically excited by two-frequency forcing, show a transition from a small hexagonal standing wave pattern to a triangular ``superlattice'' pattern. We show that generically the hexagons and the superlattice wave patterns bifurcate simultaneously from the flat surface state as the forcing amplitude is increased, and that the experimentally-observed transition can be described by considering a low-dimensional bifurcation problem. A number of predictions come out of this general analysis.

patt-sol↗

arXiv1998

Simple and Superlattice Turing Patterns in Reaction-Diffusion Systems: Bifurcation, Bistability, and Parameter Collapse

This paper investigates the competition between both simple (e.g. stripes, hexagons) and ``superlattice'' (super squares, super hexagons) Turing patterns in two-component reaction-diffusion systems. ``Superlattice'' patterns are formed from eight or twelve Fourier modes, and feature structure at two different length scales. Using perturbation theory, we derive simple analytical expressions for the bifurcation equation coefficients on both rhombic and hexagonal lattices. These expressions show that, no matter how complicated the reaction kinectics, the nonlinear reaction terms reduce to just four effective terms within the bifurcation equation coefficients. Moreover, at the hexagonal degeneracy -- when the quadratic term in the hexagonal bifurcation equation disappears -- the number of effective system parameters drops to two, allowing a complete characterization of the possible bifurcation results at this degeneracy. The general results are then applied to specific model equations, to investigate the stability of different patterns within those models.

patt-sol↗

arXiv1997

New stability results for long-wavelength convection patterns

We consider the transition from a spatially uniform state to a steady, spatially-periodic pattern in a partial differential equation describing long-wavelength convection. This both extends existing work on the study of rolls, squares and hexagons and demonstrates how recent generic results for the stability of spatially-periodic patterns may be applied in practice. We find that squares, even if stable to roll perturbations, are often unstable when a wider class of perturbations is considered. We also find scenarios where transitions from hexagons to rectangles can occur. In some cases we find that, near onset, more exotic spatially-periodic planforms are preferred over the usual rolls, squares and hexagons.

patt-sol↗

arXiv1997

Instabilities of periodic orbits with spatio-temporal symmetries

Motivated by recent analytical and numerical work on two- and three-dimensional convection with imposed spatial periodicity, we analyse three examples of bifurcations from a continuous group orbit of spatio-temporally symmetric periodic solutions of partial differential equations. Our approach is based on centre manifold reduction for maps, and is in the spirit of earlier work by Iooss (1986) on bifurcations of group orbits of spatially symmetric equilibria. Two examples, two-dimensional pulsating waves (PW) and three-dimensional alternating pulsating waves (APW), have discrete spatio-temporal symmetries characterized by the cyclic groups Z_n, n=2 (PW) and n=4 (APW). These symmetries force the Poincare return map M to be the nth iterate of a map G: M=G^n. The group orbits of PW and APW are generated by translations in the horizontal directions and correspond to a circle and a two-torus, respectively. An instability of pulsating waves can lead to solutions that drift along the group orbit, while bifurcations with Floquet multiplier +1 of alternating pulsating waves do not lead to drifting solutions. The third example we consider, alternating rolls, has the spatio-temporal symmetry of alternating pulsating waves as well as being invariant under reflections in two vertical planes. When the bifurcation breaks these reflections, the map G has a "two-symmetry", as analysed by Lamb (1996). This leads to a doubling of the marginal Floquet multiplier and the possibility of bifurcation to two distinct types of drifting solutions.

patt-sol↗

arXiv1993

Temporal Forcing of Small-Amplitude Waves in Anisotropic Systems

We investigate the effect of resonant temporal forcing on an anisotropic system that exhibits a Hopf bifurcation to obliquely traveling waves in the absence of this forcing. We find that the forcing can excite various phase-locked standing-wave structures: rolls, rectangles and cross rolls. At onset, at most one of the two - rolls or rectangles - is stable. The cross rolls can arise in a secondary bifurcation and can be stable. Experimentally, they would appear as a periodic switching between a structure in which the `zig'-component dominates and one with dominating `zag'-structure. Since there are two symmetry-related states of this kind one may expect disordered structures to arise due to the break-up of the pattern into domains. The results are consistent with recent experiments on electro-convection in nematic liquid crystals by de la Torre and Rehberg. We also apply the general analysis to a model of the behavior near a Lifshitz point, where the angle of obliqueness vanishes. This analysis indicates that phase-locked standing rectangles are always unstable in this parameter regime.

patt-sol↗