Chad M. Topaz and Mary Silber- Resonances and superlattice pattern stabilization in two-frequency forced Faraday waves

8/3/2019 Chad M. Topaz and Mary Silber- Resonances and superlattice pattern stabilization in two-frequency forced Faraday

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Resonances and superlattice pattern

stabilization in two-frequency

forced Faraday waves

Chad M. Topaz , Mary Silber

Department of Engineering Sciences and Applied Mathematics

Northwestern University, Evanston, IL, 60208, USA

Abstract

We investigate the role weakly damped modes play in the selection of Faraday wavepatterns forced with rationally-related frequency components m and n. We usesymmetry considerations to argue for the special importance of the weakly dampedmodes oscillating with twice the frequency of the critical mode, and those oscillat-ing primarily with the difference frequency |n m| and the sum frequency(n + m). We then perform a weakly nonlinear analysis using equations of Zhangand Vinals [1] which apply to small-amplitude waves on weakly inviscid, deep fluid

layers. For weak damping and forcing and one-dimensional waves, we perform aperturbation expansion through fourth order which yields analytical expressions foronset parameters and the cubic bifurcation coefficient that determines wave am-plitude as a function of forcing. For stronger damping and forcing we numericallycompute these same parameters, as well as the cubic cross-coupling coefficient forcompeting standing waves oriented at an angle relative to each other. The reso-nance effects predicted by symmetry are borne out in the perturbation results forone spatial dimension, and are supported by the numerical results in two dimen-sions. The difference frequency resonance plays a key role in stabilizing superlatticepatterns of the SL-I type observed by Kudrolli, Pier and Gollub [2].

Key words: Faraday waves, pattern selection, superlattice pattern, resonant triadsPACS: 05.45.-a, 47.54.+r

Corresponding author. Email to: chad [email protected].

Preprint submitted to Physica D 19 April 2002


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1 Introduction

The Faraday wave system provides the canonical example of how spatiotem-poral patterns form through a parametric instability. In this system, a fluid

subjected to a time-periodic vertical acceleration of sufficient strength under-goes an instability to standing waves on the free surface. In his original exper-iment [3] Faraday observed that the standing waves had half the frequency ofthe forcing; this is the typical subharmonic response. Other experimentalistssubsequently observed familiar patterns such as stripes, squares, and hexagons(see [4] for a review).

Recent experiments have utilized a two-frequency forcing function, which wemay write in the following forms:

g(t) = gz[cos()cos(mt) + sin()(nt + )] (1)= gm cos(mt) + gn(nt + )

= Gm eimt +Gn e

int +c.c.

Here m and n are co-prime integers, so that the forcing function is periodicwith period T = 2/. An interesting feature of this forcing function is thatthe primary instability leading to Faraday waves may be either harmonic orsubharmonic (with respect to T) depending on the value of and the paritiesof m and n. For instance, if the cos(mt) component is dominant and if mis even (odd) then the bifurcation will be to harmonic (subharmonic) waves.

This was demonstrated by the (numerical) linear stability analysis of Bessonet al. [5]. For m and n not both odd, there is a codimension-two point in thegz - parameter space (or alternatively, in gm - gn space) at which harmonicand subharmonic instabilities occur simultaneously at different spatial wavenumbers. The corresponding value = bc is called the bicritical point. Ex-periments performed near the bicritical point have produced exotic patterns,including triangles [6], quasipatterns [2,7] and superlattice patterns [2,810].

The term superlattice pattern refers to a periodic pattern that has spatialstructure on more than one length scale typically, a small scale structureand a large scale spatial periodicity. A wealth of recent experimental work

has produced superlattice patterns. In addition to the Faraday experimentswith two-frequency forcing mentioned above, superlattice patterns have beenobserved in nonlinear optical systems [11], in vertically vibrated Rayleigh-Benard convection [12,13], in Faraday waves with single frequency forcing atvery low frequencies [14], in granular layers forced with two frequencies [15],and in ferrofluids driven by an ac magnetic field [16]. The superlattice patternscome in a variety of types, and may be comprised of different numbers ofcritical modes having different spatial arrangement and temporal dependence.

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We mention two types of superlattice patterns here. One type is called super-lattice two or SL-II in [2]. Patterns of this type have been shown to arise in aspatial period-multiplying bifurcation from hexagons; see [17,18] for a bifurca-tion analysis of these patterns. In contrast, superlattice one or SL-I patternsexist as primary branches bifurcating directly from the trivial solution [19]. For

SL-I Faraday wave patterns, one of the length scales is set (approximately) bythe dominant frequency component in (1). Curiously, the second length scalein the pattern is not set directly by the other forcing frequency. For instance,a numerical linear analysis using the parameters corresponding to the experi-mental SL-I pattern from [2] indicates that the two critical wave numbers arein a ratio of 1.22 at the bicritical point, but the ratio of the two length scalesin the pattern is actually observed to be

7 2.65. The identification of a

mechanism for the selection of the second length scale is our motivation forthis paper.

Many studies of Faraday wave pattern formation [1,610,14,2024] focus on

the effect that resonant triad interactions have on the nonlinear pattern se-lection problem. The triads of spatially resonant modes satisfy k1 + k2 = k3,where |k1| = |k2| is the wave number of one of the instabilities at the bicriticalpoint, and |k3| is the wave number of the other. In [25] temporal symmetryarguments were used to show that this triad interaction affects pattern selec-tion on the subharmonic side of the bicritical point, but not on the harmonicside. In [26] it was shown that weakly damped harmonic modes not associ-ated with the bicritical point can affect harmonic and subharmonic patternselection. Explicit numerical calculations were performed to demonstrate thatthe presence of a weakly damped harmonic mode influences pattern selection

by affecting the cubic cross-coupling coefficient in the bifurcation equationsdescribing the dynamics of competing waves.

In this paper, too, our goal is to investigate the role weakly damped modesplay in pattern selection. We use symmetry considerations to explain the spe-cial importance of particular weakly damped harmonic modes in terms oftheir contributions to cubic cross-coupling coefficients in the relevant bifurca-tion equations. Specifically, we find that the most important weakly dampedmodes are i) the mode oscillating with dominant frequency m, which is thetemporal harmonic of the Faraday-unstable mode, ii) the difference frequencymode oscillating with dominant frequency

|m

n

|, and iii) the sum fre-

quency mode oscillating with dominant frequency (m + n). (Here, we haveassumed without loss of generality that the cos(mt) component in (1) is thedominant one. Additionally, we exclude the forcing frequency ratios m/n withm + n 5 which yield stronger resonances). Our symmetry arguments areborne out in a weakly nonlinear analysis of equations derived by Zhang andVinals [1] which apply to weakly inviscid, deep fluid layers. For weak dampingand forcing and one-dimensional surface waves, we perform a perturbationexpansion through fourth order which yields analytical expressions for onset

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parameters and the cubic self-interaction coefficient that determines wave am-plitude as a function of forcing amplitude near onset. For stronger dampingand forcing we compute these same parameters numerically as well as the cu-bic cross-coupling coefficient B() for competing waves oriented at an angle relative to each other. From the analytical expressions for the one-dimensional

case, we are able to quantify the effect of the key resonances and see how theirexistence depends on the forcing frequency ratio m/n. For the two dimen-sional case, our numerical results show that the resonance effects follow thesame scaling laws as in the one-dimensional case. A simple argument, validfor weak damping and forcing which relies only on the inviscid dispersion re-lation, allows us to predict the spatial angles at which the resonances occur,and to see how their existence depends on m, n and a dimensionless fluidgravity-capillarity parameter. A bifurcation analysis reveals that the differ-ence frequency resonance can help stabilize an SL-I pattern whose large-scaleperiodicity depends on the wavelength associated with the difference frequencymode. We note that the difference frequency mode has been observed to play

an important role in experiments where other complex Faraday wave patternsare formed [810].

This paper is organized as follows. In section 2.1 we review basic ideas aboutresonant triad interactions and their potential for affecting SL-I pattern se-lection. In section 2.2 we use symmetry arguments to identify which weaklydamped modes we expect to be most important in terms of their contributionsto the cross-coupling coefficient B(). Section 3 contains the weakly nonlin-ear analysis of the Zhang-Vinals equations for weak damping and forcing andone-dimensional waves, which leads to approximate formulas for the criticalforcing and wave number, and the cubic self-interaction coefficient. The per-turbation results for onset parameters are discussed in 4.1. The expressionfor the self-interaction coefficient and numerical results for the cross-couplingcoefficient are examined in sections 4.2.1 and 4.2.2 respectively, with specialattention given to the role played by resonant triads and the implications forSL-I pattern selection. We summarize our main results in section 5.

2 Background

2.1 Resonant triads, standing wave equations and pattern stability

For Faraday waves on a domain of infinite horizontal extent there is no pre-ferred direction, so each wave number is associated with a circle of wave vectorsin Fourier space. One of the simplest mechanisms through which waves on twodifferent Fourier circles may interact is a resonant triad interaction. Such res-onant triads consist of three wave vectors that determine an associated angle

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Fig. 1. Spatially resonant triads of three wave vectors and their associated resonant

angle r. (a) r < 2/3. (b) r > 2/3.

r [0, ).

Two examples of spatially resonant triads are shown in figure 1. In both cases,the spatial resonance condition is

k1 + k2 = k3. (2)

Here, k1 and k2 are wave vectors associated with neutrally stable modes of theFaraday instability, so that

|k1

|=

|k2

|= kc. The wave vector

|k3

|is associated

with a damped harmonic mode. The resonant angle r satisfies

cos

r2

=

|k3|2|k1| . (3)

For the first case where 0 r < 2/3, we have that |k3| > |k1|. It is alsopossible to have a resonant triad where |k3| < |k1|, as shown in the second casewhere 2/3 < r < . We exclude the hexagonal case for which |k3| = |k1|and r = 2/3.

We follow [25,26] and consider equations describing the slowly-varying ampli-tudes of the modes with wave vectors k1 . . . k3, which we assume satisfy thespatial resonance condition (2):

Z1 = 1Z1 + 1Z2Z3 + (A|Z1|2 + b|Z2|2 + C|Z3|2)Z1 (4)Z2 = 1Z2 + 1Z1Z3 + (A|Z2|2 + b|Z1|2 + C|Z3|2)Z2Z3 = 2Z3 + 2Z1Z2 + (D|Z1|2 + D|Z2|2 + E|Z3|2)Z3.

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Here Z1, Z2, and Z3 are the slowly varying amplitudes of the Fourier modeswith wave vectors k1, k2, and k3 respectively, and the dot represents differ-entiation with respect to the slow time scale. All coefficients are real-valued.If the Z3 mode is in fact damped (i.e. 2 < 0) and the Z1 and Z2 modes areneutrally stable (i.e. 1 = 0) then a further center manifold reduction may be

performed to the critical Z1 and Z2 modes. Then Z3 satisfies

Z3 = 22

Z1Z2 + . . . , (5)

and the (unfolded) bifurcation problem, to cubic order, is

dZ1dT

= 1Z1 + A|Z1|2Z1 + B(r)|Z2|2Z1 (6)dZ2dT = 1Z2 + A|Z2|

2

Z2 + B(r)|Z1|2

Z2,

where

B(r) = b 122

. (7)

The dependence on the resonant angle r indicates that the cross-couplingcoefficient B() for competing waves oriented at an angle relative to eachother is evaluated at the angle of spatial resonance.

For appropriately chosen angles r, the bifurcation problem (6) describes dy-namics on an invariant subspace of the twelve-dimensional problem which de-scribes the competition of simple rolls, simple hexagons, certain rhombic pat-terns, and SL-I patterns. We are ultimately concerned with the cross-couplingcoefficient B(), which plays a key role in determining the relative stabilityof these patterns. In particular, the stability properties of SL-I patterns withcharacteristic angles h are enhanced when B(h) is small in magnitude; see[26] for a detailed discussion.

In [27], symmetry arguments are used to derive a scaling law for the quadratic

coefficients 1 and 2 in (4) when the resonant triad applies to the bicriticalpoint, and for the case of weak damping and forcing. For instance, it is shownthat for m odd and n even, and for the case that the cos(mt) forcing frequency

component dominates, 1 and 2 are each proportional to gn22

m gm12

n . Thus form + n 5 , the quadratic terms in (4) are quite small for weak forcingand damping. Their contribution to the pattern selection problem can onlybe made significant by getting sufficiently close to the bicritical point where2 0.

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Here we will focus on resonant triads other than those associated with thebicritical point. In particular, we will identify resonant triads for which thequadratic terms scale (at most) linearly with g, and for which the scaling isindependent of m and n. Thus away from the bicritical point and for weaklyforced waves, we expect these triads to play a more important role in pattern

formation if m + n > 5.

2.2 Determination of important resonances for weak damping and forcing

Our goal in this section is to examine resonant triads from a symmetry per-spective with a special emphasis on temporal symmetries. Without loss ofgenerality, we assume (unless otherwise specified) that the cos(mt) forcing isof greater significance than the cos(nt) forcing. For weak damping, the crit-ical Faraday waves oscillate with a dominant frequency component of m/2.

We also consider weakly damped waves of frequency > 0 ( = m/2), tobe determined such that they lead to the largest possible contribution to thecross-coupling coefficient B() when slaved away; cf. (5) - (7) in section 2.1.

We follow [27] and focus on travelling waves, on which the action of time-translation is transparent. The travelling wave bifurcation equations are thenreduced to those describing the standing wave problem. Specifically, we expandthe fluid surface height h(x, t), x R2, in terms of the following six travellingwaves:

z1 ei(k1x+

1

2mt)

+w1 ei(k1x

1

2mt)

+z2 ei(k2x+

1

2mt)

(8)+ w2 e

i(k2x1

2mt) +z3 e

i(k3x+t) +w3 ei(k3xt) + c.c.

Here, zj and wj , j = 1, 2, 3, are the slowly varying amplitudes of the travellingwaves. The wave vectors k1 . . . k3 are assumed to satisfy the spatial resonancecondition (2). The frequency and the wave number |k3| are related by adispersion relation. In writing (8) we have assumed the problem is posed onan unbounded horizontal domain and then restricted our attention to solutionsthat are periodic on a rhombic lattice. Spatial translation symmetry acts on(zj, wj), j = 1, 2, 3, as

T(1, 2) : (zj, wj) (zj, wj) eij , 3 1 + 2 (9)

where (1, 2) T2. A rotation by , denoted by R, acts as

R : (zj, wj) (wj, zj), j = 1, 2, 3. (10)

and a reflection in the plane containing k3, denoted by , acts as

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: (z1, w1) (z2, w2) (11)(z3, w3) (z3, w3).

Furthermore, there is a time-translation symmetry which acts on the forcing

parameters in (1) and on the complex travelling wave amplitudes in (8):

Tt : (z1, z2, z3) (z1 ei 12mt, z2 ei 12mt, z3 eit) (12)(w1, w2, w3) (w1 ei 12mt, w2 ei 12mt, w3 eit)

(Gm, Gn) (Gm eimt, Gn eint).

We now determine which quadratic terms will be allowed in the travelling waveamplitude equations, anticipating that these terms will lead to contributionsto B(). For example, from the spatial translation symmetry (9), the only

quadratic terms that are allowed in the z1 equation are z2z3, z2w3, w2z3 andw2w3.

We now consider the restrictions placed by the time translation symmetry (12)in order to determine which are allowed. We expect the largest contributionsto B() in (6) to occur when the coefficients of quadratic terms are independentof the forcing amplitudes Gm and Gn at leading order, at least for small forcing.In this case, there is only one quadratic term that is permitted, namely

i. w2w3 with = m.

The next largest contributions to B() occur when the coefficients of thequadratic terms in (4) are proportional to one power of Gm or Gn at leadingorder. In this case, the permitted equivariant terms in the z1 equation are

ii. Gmz2z3 with = 2miiia. Gnz2z3 with = (m n) if m > niiib. Gnz2w3 with = (n m) if n > m

iv. Gnz2z3 with = (m + n)v. Gnw2z3 and Gnw2w3 with = n.

We may immediately dispense with several of these cases. The resonance in

case ii. is not relevant for our investigation of Faraday waves because theweakly damped mode oscillating with frequency 2m is at sufficiently highwave number that the spatial resonance condition (2) cannot be satisfied forthe inviscid dispersion relation. The resonance in case v. does not result ina contribution to B() at linear order in Gm, Gn. This may be understoodby considering the effects of an approximate time reversal symmetry and anapproximate Hamiltonian structure [28] and has been verified by an explicitperturbation calculation similar to those performed in section 3.

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We refer to iiia. and iiib. as cases of difference frequency resonance. We nowexamine the amplitude equations for case iiia. (case iiib. is analogous) whichare determined by the symmetries (9) - (12). The cubic truncation takes theform

z1 = 1z1 + Gmw1 + Gn1z2z3 + r0w1z2w2 (13)

+ (r1|z1|2 + r2|z2|2 + r3|z3|2 + r4|w1|2 + r5|w2|2 + r6|w3|2)z1z3 = 2z3 + Gn2z1z2 (14)

+ (r7|z1|2 + r7|z2|2 + r8|z3|2 + r9|w1|2 + r9|w2|2 + r10|w3|2)z3.

Related equations for z2, w1, w2, and w3 can be obtained from the discretespatial symmetries (10) and (11). The Gmw1 term in (13) is the usual para-metric forcing term. We have dropped linear and quadratic terms that scalehigher than linearly in Gm and Gn. We have also dropped any cubic termswhose coefficients depend on these parameters.

Since the resonant z3 mode is damped (2 < 0), we may slave it away so that(13) becomes

z1 = 1z1 + Gmw1 + r0w1z2w2 (15)

+

r1|z1|2 +

r2 |Gn|

2122

|z2|2 + r4|w1|2 + r5|w2|2

z1.

and equations related by the discrete spatial symmetries (10) and (11).

For sufficiently large forcing |Gm|, the trivial solution of (15) loses stability.A center manifold reduction to standing waves equations of the form (6) maybe formed at the critical forcing strength. The cross-coupling coefficient B()in (6) then includes a contribution proportional to |Gn|2/(Re 2) that resultsfrom slaving the difference frequency mode.

Similar arguments can be made for case iv., in which the resonant mode oscil-lates at the so-called sum frequency (m + n). For this case, too, the slavedmode results in a contribution to B() that is proportional to |Gn|2/(Re ),where Re represents the damping of the slaved resonant mode.

Case i. corresponds to the well-known 1:2 temporal resonance, which is presentfor single frequency forcing [1]. An analysis similar to that performed abovereveals that slaving of the damped mode oscillating with dominant frequencym results in a contribution to B() which is independent ofGm and Gn, andis inversely proportional to Re(). Thus we expect that this contribution willbe larger than that due to the sum or difference frequency resonances.

For weak damping and forcing, then, we expect the standing wave modes

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with frequency m, |m n|, and (m + n) to be the most important weaklydamped modes in terms of their contributions to B() when m + n > 5.Due to the constraints imposed by temporal symmetries, any other resonantmodes will necessarily have higher powers of g in front of the quadratic termsin their travelling wave equations, and thus will result in smaller contribu-

tions to B() for weak forcing. The exception is the wave oscillating withdominant frequency 1

2n. Because this wave is forced directly by the n forc-

ing component, its damping becomes arbitrarily small as the bicritical pointis approached, and its slaving can result in a large contribution to B() asdemonstrated in [26]. Since our analysis assumes that the resonant modeshave finite damping (i.e. we are bounded away from the bicritical point), thiscase is excluded here.

3 Perturbation analysis for one dimensional waves

Using the ideas discussed in the previous section, we now perform a perturba-tion analysis on the Zhang-Vinals Faraday wave equations (which we introducein section 3.1) to obtain quantitative results for Faraday waves in one spatialdimension for m + n > 5.

Since there is no spatial angle to tune in one dimension, the only possibleresonant triad interaction is the 1:2 interaction which occurs when a standingwave with critical wave number k and its spatial harmonic with wave number2k fulfill one of the temporal resonance conditions from section 2.2. This situa-

tion may be achieved by varying fluid parameters in the dispersion relation. Inparticular, in our calculations we vary a dimensionless capillarity parameter.Near those special values of the capillarity parameter where a temporal reso-nance occurs, we expect additional contributions to the cubic self-interactioncoefficient A in the standing wave bifurcation equation

dZ1dT

= 1Z1 + A|Z1|2Z1, (16)

which is simply (6) restricted to one spatial dimension.

3.1 The Zhang-Vinals Hydrodynamic Equations

Zhang and Vinals derive from the Navier Stokes equations reduced equationsfor Faraday waves in [1]. This derivation is accomplished by focusing on fluidsof low viscosity and making a quasipotential approximation. The resultingequations apply to weakly damped, small amplitude surface waves on a deep

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layer of fluid. The system consists of two evolution equations for the surfaceheight h(x, ) and surface velocity potential (x, ), where x R2 is thehorizontal coordinate. We assume periodic boundary conditions.

The Zhang-Vinals equations are

( 2)h D = F(h, ) (17)( 2)

02 G()

h = G(h, ), (18)

where the nonlinear terms are

F(h, ) = (h) + 122(h2 D ) D(h D ) (19)

+

D

h

D(h

D ) + 1

2h22

G(h, ) = 12

(D)2 12

()2 ( D)h2 + D(h D) (20) 1

20

(h)(h)2

.

(For brevity, we drop the h and dependence ofFand G from now on.) Theoperator D is a nonlocal operator that multiplies each Fourier component ofa field by its wave number, e.g. D eikx = |k|eikx. Here time has been scaledby so that the dimensionless two-frequency acceleration is

G() = G0 [fm cos(m) + fn cos(n + )] (21)= G0 f[cos()cos(m) + sin()cos(n + )] .

The damping number (), capillarity number (0), gravity number (G0), anddimensionless accelerations (fm and fn) are related to the forcing function (21)and the fluid parameters by

2k2

, 0

k32

, G0 g0k

2, fm gm

k2

, fn gnk

2. (22)

Here is the kinematic viscosity, is the surface tension, is the fluid den-sity, and the wave number k is chosen to satisfy the gravity-capillary wavedispersion relation

g0k + k3

=m

2

2. (23)

Note that (22) and (23) imply a relationship between the gravity number and

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the capillarity number, namely

G0 + 0 =m2

4. (24)

Following [25], we express the governing equations in the following alternativeform. We apply ( 2) to (17) to obtain

( 2)2h ( 2) D = ( 2)F. (25)In (25), we substitute for ( 2) by using (18). The resulting equationand equation (17) (which we rearrange) constitute the system of equationsthat we use in our perturbation analysis:

( 2)2 D 02 G()h = ( 2)F+ D G (26)D = ( 2)h F. (27)3.2 Outline of the calculation

We calculate from the Zhang-Vinals equations (26) - (27) systems of ODEsfor travelling waves in one spatial dimension that are valid for the differentcases of spatiotemporal resonance described in section 2.2. Our perturbationcalculations are performed for small amplitude waves in the limit of weakdamping and forcing (, fm, fn 1).

For our calculations, we focus on counter-propagating travelling waves havingcritical wave number k (to be determined) which are assumed to be subhar-monic to the dominant forcing component cos(m) and thus, to leading order,have frequency m/2. We refer to these waves as the basic waves. Further-more, we insist that fn not exceed the critical value at which standing wavesof dominant frequency n/2 bifurcate. Thus, we do not include these wavesin our nonlinear calculation, and our nonlinear analysis is restricted to theparameter region

0 < fm 1, 0 < fn < fcrit.n 1 (28)

which is bounded away from the bicritical point.

To facilitate our analysis, we perform four separate calculations, each of whichpertains to a different possible case of spatiotemporal resonance. The first caseis a nonresonant case, for which we retain only the basic waves in the leading

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order solution of the perturbation problem. Then we consider cases in whichthe basic waves are (nearly) temporally resonant with their spatial harmon-ics having wave number 2k. Based on the arguments in section 2, the threeresonant frequencies we consider are m (1:2 temporal resonance), |m n|(difference frequency resonance) and m + n (sum frequency resonance). Each

resonance may be achieved by choosing a particular value of the capillarityparameter 0. For each of these resonant cases, we retain both the basic wavesand the resonant waves at leading order. For all cases, resonant terms at sub-sequent orders in the perturbation calculation lead to solvability conditions,from which we obtain travelling wave amplitude equations, and by a furthercenter manifold reduction, standing wave amplitude equations.

3.3 No resonance

We use the following scaling:

= 1, fn = f1n, fm = f

1m +

3f3m + . . . (29)

k = k0 + 2k2 + . . . , + T1 + 2T2 + 3T3 + . . .

h = h1 + 2h2 +

3h3 + . . . , = 1 + 22 +

33 + . . .

where 0 < 1. The fields h and are functions of the spatial variable x,the fast time , and the slow times Tj .

The expressions for fm and k indicate expansions of the critical wave number

and forcing value. We find that terms proportional to 2

in fm and in k arenot necessary. The wave number, forcing, time derivative, and the two fieldsare expanded through O(3) because we carry out the perturbation calculationto O(4). This higher order calculation is needed since A, the cubic coefficientin the standing wave equation (16), turns out to be an O() quantity. (This isrelated to a weakly broken time reversal symmetry as discussed in [27].)

At O(), (26) isL0h1 = 0 (30)

where

L0 2 + D(G0 02x). (31)Equation (30) has an infinite-dimensional solution space consisting ofall planewaves eik0x+i(k0) that satisfy the dispersion relation

2(k0) = G0k0 + 0k30. (32)

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Thus, h1 should consist of a superposition of these plane waves, i.e. h1 =k0 z(k0) e

ik0x+i(k0), where the wave number k0 may be any wave numberthat fits into our periodic domain. However, at O(2), all of the amplitudesz(k0) are damped on the slow time scales, except for the case (k0) = m/2,k0 = 1. Using this a posteriori justification, we choose h1 to include only those

solutions which may grow on the slow time scales. Therefore, h1 consists ofone set of counter-propagating waves:

h1 = z1 eikx+im

2 +w1 e

ikxim2+ c.c. (33)

where z1 and w1 are functions ofT1, T2, and T3.

At O(2), O(3) and O(4) we apply solvability conditions which yield therespective equations

z1T1 = 1z1 + i1w1 (34)z1T2 = i2z1 + ic1|z1|

2z1 + ic2|w1|2z1 (35)z1T3 = 3z1 + i3w1 + c3|z1|

2z1 + c4|w1|2z1 (36)+ ic5|w1|2z1 + ic6|w1|2w1 + ic7z21w1

and similar equations for w1 which are related by the spatial reflection sym-metry

x x : (z1, w1) (w1, z1). (37)

The coefficients in (34) - (36) are given in the appendix.

We reconstitute the time derivative and amplitudes in the travelling waveequations by multiplying (34), (35), and (36) by 2, 3, and 4 respectively,adding the results, and letting z1 z1, w1 w1, and T1 + 2T2 + 3T3 T. We obtain

dz1dT = (+ i)z1 + iw1 + (d1 + ic1)|z1|2z1 + (d2 + ic2)|w1|2z1 (38)+ id3|z1|2w1 + id4|w1|2w1 + id5z21w1

and a similar equation for dw1dT

related by (37). The coefficients in the recon-stituted equations are

= 1 + 33, =

22, = 1 + 33, d1 = c3 (39)

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d2 = c4, d3 = c5, d4 = c6, d5 = c7.

(Note that 1 = , etc. so that the s drop out of the final equation.)

Now we solve for k2, the correction to the critical wave number, and f1m and

f3m

in (29), the forcing amplitudes associated with onset. The condition forneutral stability of the flat state follows from (38) and is

2 = 2 + 2. (40)

At leading order, O(2), we solve (40) for f1m, to find that

f1m = 2m1. (41)

At O(4), we find that

f3m =k22(m

2 + 80)2

16m1+

k2(f1n)

2(m2 + 80)

4m1(n2 m2) (42)

1(f1n)

2(7n2m2 + n4 4m4)2mn2(n2 m2)2 +

1k2(7m2 80)

4m

+(f1n)

4

4m1(n2 m2)2 9314m

which is minimized at

k2 = 22

1(7m2

80)(m2 + 80)2 2(f1

n)2

(m2 + 80)(n2 m2) (43)

and so

f3m = 31(29m

4 + 16m20 + 32020)

2m(m2 + 80)2(44)

+21(f

1n)

2m(m4 + 8m20 16n20 2n4)n2(m2 + 80)(n2 m2)2 .

To reconstitute the expressions for critical forcing and wave number, we recall(29) to obtain

fcm = 2m3(29m4 + 16m20 + 320

20)

2m(m2 + 80)2(45)

+2(fn)

2m(m4 + 8m20 16n20 2n4)n2(m2 + 80)(n2 m2)2

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k = 1 22(7m2 80)

(m2 + 80)2 2(fn)

2

(m2 + 80)(n2 m2) . (46)

The superscript c indicates that fm has been set to its critical value.

Due to the restrictions (28) which we placed on the forcing amplitudes, thecalculation performed above gives us information only about one side of thelinear stability boundary. In order to obtain expressions for the entire linearstability boundary in fm - fn space, we perform a similar linear calculation forthe case that the dominant forcing component is cos(n). The critical forcingand wave number in this case are

fcn = 2nkn (47)

(fm)2nk2n(3n4 + 8n2G0kn + 4m2n2 16knm2G0 + 2m4)

2m2(G0 + 30k2n)(n2 m2)2

3

(n2

+ 4G0kn)(53n4

176n2

G0kn + 320G20k

2n)

320n(12G00k2n 4G20 90knn2)k = kn +

(fm)2k2n

2(G0 + 3k2n0)(n2 m2) +

2k3n(G0kn + 3k3n0 2n2)

2(6G00k2n + G20 + 9

20k

4n)

. (48)

Here kn satisfies

2(kn) =

n

2

2, (49)

where (k) represents the natural frequency given by the dispersion relation

2(k) = G0k + 0k3. (50)

These linear results are discussed in section 4.1.

We now return to the case that the cos(m) forcing dominates and continueour calculation in order to determine the cubic coefficient in the standing waveequation (16). We reduce (38) to the a bifurcation equation for the amplitudeZ1 of the critical mode to obtain the standing wave amplitude equation:

dZ1dT

= Z1 + Anonres|Z1|2Z1. (51)

The cubic coefficient Anonres is calculated through its leading order, namelyO(). We find

Anonres = 2

c3 + c4 c5 c6 + c7 2(c1 + c2)

1

. (52)

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We substitute for 2 and c1 . . . c7 in (52) and reconstitute to obtain

Anonres =3(5m2 + 20)

2m2+

181m2

10(m2 + 80) 28m

2

m2 + 120(53)

+ 37m2

5(m2 120) 16m

4

(m2 120)2 .

Note that Anonres diverges as 0 m2/12. This divergence reflects the factthat the second spatial harmonic of the critical mode is resonantly excitedwhen 0 = m

2/12. We perform the necessary calculation for this case next.

3.4 1:2 spatiotemporal resonance

Now we perform a calculation to handle the case of resonance involving thetemporal harmonic; this resonance occurs when the spatial harmonic of thebasic waves oscillates with frequency m. The condition is

2(2k0) = m2 (54)

where is given by the dispersion relation (50). Solving (54) for 0, we seethat the 1:2 spatiotemporal resonance occurs for 0 = 1:2 = m

2/12, which isthe value of 0 at which the nonresonant calculation in section 3.3 diverges.

The analysis here is similar to that of section 3.3, except that we now includethe resonant mode in our calculation. Thus,

h1 = z1 eikx+im

2 +w1 e

ikxim2+z3 e

2ikx+im +w3 e2ikxim + c.c. (55)

Additionally, since we are interested only in the parameter region near theresonance, we expand around the resonant value of the capillarity number:

0 = 1:2 +

1:2 (56)

A solvability condition at O(2) yields

z1T1 = 1z1 + i1w1 + ie1z1z3 (57)z3T1 = (4 + i4)z3 + ie2z

21

17


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and equations related by a spatial reflection symmetry similar to (37). Thecoefficients are given in the appendix.

The leading order term in the standing wave cubic coefficient A1:2 dependsonly on terms in (57) and thus may be determined without carrying the per-

turbation calculation any further. A reduction of (57) to the standing waveequation (16) reveals that the leading term in A1:2 is an O(1) quantity givenby

A1:2 = 2e1e24(24 +

24)

. (58)

We substitute for the coefficients to obtain

A1:2 = m4

9(0 1:2)2 + 162m2 , (59)

which is valid for 0 sufficiently close to 1:2.

3.5 Difference frequency resonance

Now we perform a calculation to handle the case of resonance involving thedifference frequency mode, which occurs when the spatial harmonic of thebasic waves oscillates with frequency |m n|. This condition may be writtenas

2(2k0) = (m n)2 (60)

where is given by the dispersion relation (50). Solving (60) for 0, we seethat the difference frequency resonance occurs for

0 = diff =1

6n2 1

3nm +

1

12m2. (61)

The calculation is similar to that of the previous section. We let

0 = diff + diff. (62)Now, h1 is given by

h1 = z1 eikx+im

2 +w1 e

ikxim2 (63)

+ z3 e2ikx+i(mn) +w3 e

2ikxi(mn) + c.c.

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The solvability conditions at O(2) and O(3) yield

z1T1 = 1 + z1 + i1w1 (64)

z3T1 = (4 + i4)z3z1T2 = i2z1 + ir1z1z3 + ic1|z1|

2z1 + ic2|w1|2z1+ ic8|z3|2z1 + ic9|w3|2z1

z3T2 = i5z3 + ir2z

21 + ic10|z1|2z3 + ic11|w1|2z3

+ ic12|z3|2z3 + ic13|w3|2z3.

and equations for w1 and w3 related by a spatial reflection symmetry. The

coefficients are given in the appendix. The values for 2, c1, and c2 are givenby (A.2), (A.3), and (A.4) evaluated at 0 = diff.

It is not necessary to carry the perturbation calculation further to determinethe standing wave cubic coefficient Adiff at leading order. The coefficient Adiffhas two types of contributions. One type is unrelated to the slaved differencefrequency mode and is equal to Anonres evaluated at 0 = diff. The othertype is due to the quadratic terms in (64) and results from the slaving of thedamped difference frequency mode. We find that

Adiff = Anonres(0 = diff) + Adiff (65)

where

Adiff = 24r1r224 + 24 . (66)

We substitute for the coefficients to obtain

Adiff =

m(fn)2(m2 4mn + 2n2)2

n2 [162(n

m)2 + 9(0

diff)2] (n

m)(n

2m)2

. (67)

3.6 Sum frequency resonance

The condition for the sum frequency resonance is

2(2k0) = (m + n)2 (68)

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where is given by the dispersion relation (50). Solving (68) for 0, we seethat the sum frequency resonance occurs for

0 = sum =1

6n2 +

1

3nm +

1

12m2. (69)

The calculation is almost identical to that of the previous section, and theresult may be obtained by letting n n (and thus diff sum) in (67) tofind Asum. We find

Asum = Anonres(0 = sum) + Asum. (70)where

Asum =

m(fn)2(m2 + 4mn + 2n2)2n2 [162(n + m)2 + 9(

0 sum

)2] (n + m)(n + 2m)2. (71)

4 Results

4.1 Linear results

We now discuss results that apply to the linear stability of the trivial solutionof the Faraday wave problem, i.e. the flat interface state. Figures 2 and 3

contain sample results for the case m = 4, n = 9, 0 = 2, and various valuesof the damping parameter . The data are computed both numerically andfrom the analytical expressions (45) - (48). Figure 2 shows the linear stabilityboundary in fm - fn space. Figure 3 shows the critical wave number as afunction of the quantity . Note that increasing corresponds to marchingcounterclockwise around the linear stability boundary of figure 2.

The expressions for critical acceleration and wave number were derived in sec-tion 3.3 by performing a perturbation expansion on the Zhang-Vinals equa-tions (26) - (27) for small amplitude waves and weak damping and forcing.For arbitrary damping and forcing, the linearization of (26) - (27) is a damped

Mathieu equation for each Fourier mode pk()eikx:

pk + 2k2pk +

2k4 + 2(k)

pk (72)

= k {fm cos(m) + fn cos(n + )}pk.

where the natural frequency (k) satisfies the dispersion relation (50). Here wefocus on (45) and (46) which apply when the bifurcation is due to the cos(m)

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0 0.4 0.8 1.2 1.60

2

4

6

8

fm

fn

= 0.01

= 0.05

= 0.1

= 0.2

Fig. 2. Linear stability boundary in fm - fn space, the parameter space of the twoacceleration amplitudes in (21). For a given value of the damping parameter , theflat interface state is unstable above and to the right of the corresponding curve.

Dotted lines are numerical data; solid lines correspond to the analytical expressions(45) and (47). The two are distinguishable on this graph only for = 0.2. The otherparameters are m = 4, n = 9, = 0 in (21), and 0 = 2 in (26) - (27).

forcing. This bifurcation corresponds to crossing through the right side of thelinear stability region. Similar statements hold for crossing through the top ofthe linear stability region, when the bifurcation is due to the cos(n) forcing,in which case (47) - (48) are the relevant quantities.

At leading order, the critical forcing (45) is proportional to the damping .There are two correction terms. One correction term is proportional to 3 and

is independent offn. This term always has an overall negative sign and hencelowers fcm. The other correction term is proportional to (fn)2 and is due to

the second forcing component. The overall sign of this term is determined bythe quantity

s m4 + 8m20 16n20 2n4. (73)

If s < 0, the (fn)2 term has an overall negative sign, and thus the second

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0 10 20 30 40 50 60 70 80 900.94

0.96

0.98

1

1.02

kc

= 0.01

= 0.1

= 0.2

= 0.05

Fig. 3. Critical wave number kc as a function of , shown here for < bc. Thecritical wave number decreases as the bicritical point is approached. Dotted linesare numerical data; solid lines correspond to the analytical expression (46). Theparameters used are the same as those in Figure 2.

forcing component cos(n) is destabilizing; that is to say, it pushes the bifur-cation to occur at a smaller value of fm. However, ifs > 0, the second forcingcomponent actually stabilizes the flat fluid surface beyond those values of fmwhere it would have otherwise gone unstable.

By analyzing the expression for s, remembering that 0 is restricted to theinterval 0 < 0 < m

2/4, we see that there are three possible cases:

(1) If m/n < 4

2, the second frequency component is destabilizing for allvalues of

0.

(2) If 42 < m/n < 23 + 1310, the second frequency component is stabiliz-ing for 0 < c =

m42n4

16n28m2.

(3) Ifm/n >

23 +

13

10, the second frequency component is stabilizing for

all values of 0.

In short, the secondary forcing component is stabilizing if it is at sufficientlylow frequency compared to the dominant forcing component. (However, for

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weak damping and forcing, the effect of the f2n term is quite small.)

The bicritical point bc may be determined from (45) and (47), the expressionsfor critical forcing. To leading order, it is given by the simple expression

bc = arctannknm (74)where kn is determined by the dispersion relation (49). Note that to leadingorder, bc depends on m, n, and the capillarity number 0, and is independentof damping and forcing. Using the bounds on kn that are set by the dispersionrelation, we see that for a given ratio m/n, bc takes on extreme values of

1bc = arctan

n

m

3at 0 = 0 (gravity waves) (75)

2bc = arctan n

m

5/3at 0 = m

2/4 (capillary waves).

For m < n, 1bc is a maximum and 2bc is a minimum; the reverse is true for

m > n. As 0 is changed, bc varies smoothly and monotonically between thetwo extrema. Examples are shown in figure 4 for m = 4 and various values ofn.

The critical wave number, to leading order, is one. This is simply the dimen-sionless wave number determined by the dispersion relation 2(k) = (m/2)2,where (k) is given by (50). There are two correction terms. One is propor-

tional to 2 and has an overall negative sign. The other is proportional to f2n.The overall sign of this term is given by the sign of m n. Therefore, thepresence of the second forcing component shifts the wave number in such away as to repel it from the other instability associated with the bicriticalpoint. This effect was observed in the experiments in [10] and can also be seenin figure 3, which shows the critical wave number for < bc.

Finally, we discuss the fast-time dependence of the Faraday-unstable mode,which we write as p(). As demonstrated in [5], p() will be harmonic orsubharmonic to the period of the forcing function (21), namely 2. Previouswork has depended on a numerically determined (truncated) Fourier series atsome point in the linear or nonlinear analysis. For instance, in [5,25,26,29] thetime dependence of the critical mode is written as

p() =N

j=N

aj eij (76)

for the harmonic case. Then, the coefficients aj are determined numerically.

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0 1 2 3 420

30

40

50

60

70

80

90

0

bc

n = 3

n = 5

n = 7

n = 9

Fig. 4. Bicritical point bc (in degrees) versus capillarity parameter 0 for m = 4.Lines correspond to the expression in (74). Symbols correspond to a numericalcomputation with = 0.1 (o) and = 0.4 (x).

This method assumes no a priori information about the relative importanceof the frequency components kept in the expansion.

Our analysis determines the relative importance of the frequency componentsin p() for arbitrary m and n in (21). For our perturbation expansion insection 3 we assumed that at leading order, the Faraday waves have frequency12

m. At second order in the expansion we captured the frequency components

|n 1

2m|, n +1

2m, and3

2m. At third order, we captured the components|n 32

m|, |2n 12

m|, n + 32

m, 2n + 12

m, and 52

m.

These results are consistent with what we find numerically. Figure 5 shows thenine temporal Fourier coefficients |aj| that are largest, arranged in decreasingorder of their magnitude. Note that even near the bicritical point, where thisdata was obtained, the 12m frequency component is order 1/ larger than anyof the other components.

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2 7 11 6 3 16 15 20 1010

5

104

103

102

101

100

j

|aj|

Fig. 5. Magnitudes of the nine most significant fast-time frequency components ina neutrally stable Faraday mode near the bicritical point. The vertical axis (notelog scale) shows the magnitude |aj | of the frequency component eij, normalizedso that the largest component has magnitude one. The horizontal axis shows theFourier index j. The components have been arranged in decreasing order of theirmagnitude. Squares correspond to data from a numerical computation. Circles followfrom the perturbation analysis in section 3. The parameters used are m = 4, n = 9,fn = 3.61, and = 0 in (21), and = 0.1 and 0 = 2 in (26) - (27). These data arefor the critical mode, with wave number k; the spatially resonant mode with wavenumber 2k will be dominated by a different frequency component determined bythe dispersion relation (50) (e.g. |m n| if 0 = diff).

4.2 Nonlinear results

4.2.1 One spatial dimension

In this section we discuss the nonlinear results of section 3 for the cubic coeffi-cient A in (16). We have checked our perturbation results with numerical com-putations. Figure 6 shows a sample result ofA versus the capillarity parameter0 for m/n = 4/9. The solid line corresponds to an expression which matchesAnonres in (53) to Adiff in (65) and thus is valid for all values of 0 away fromthe 1:2 resonance. This expression diverges at 0 = 1:2 = m

2/12 as discussed

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0 1 2 3 4

20

15

10

5

0

0

A

Fig. 6. Cubic coefficient A in (16) as a function of the capillarity parameter 0.The dots correspond to a numerical computation. The dotted line corresponds tothe expression for A1:2 in (59). The solid line corresponds to an expression whichasymptotically matches Anonres and Adiff (details not given). The large dip at

0 1:2 = 4/3 is due to the 1:2 resonance discussed in section 2.2. The smallbump around 0 diff = 17/6, at which the one-dimensional waves have theirlargest amplitude, is due to the difference frequency resonance, also discussed insection 2.2. The parameters used are m = 4, n = 9, = 75 and = 0 in (21), and = 0.05 in (26) - (27).

in section 3. The dotted line corresponds to the expression A1:2 in (59). Addi-tionally, we have calculated the relative error in the perturbation results for Aas a function of the damping . For instance, for m/n = 4/9 and = 75 < bc,as is varied from 0.05 to 0.25, the relative error in A(0 = diff) increasesfrom 0.001 to 0.25, and the relative error in A(0 = 1:2) increases from 0.05to 0.43.

Anonres, the value of the cubic coefficient away from the 1:2, difference, andsum frequency resonances, was computed in section 3.3 and is given by (53).Anonres is proportional to the damping parameter . Furthermore, Anonres isalways negative indicating that in the nonresonant regime, the bifurcationfrom the flat state is always supercritical.

A1:2, the value of the cubic coefficient near the 1:2 temporal resonance, is

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given by (59). This quantity was derived for |0 1:2| O(). Thus, in theregion of validity, A1:2 O(1). This is significantly larger in magnitudethan Anonres, which is O(). Furthermore, A1:2 is negative, again indicating asupercritical bifurcation. This large negative contribution is manifest as thelarge dip around 0 1:2 = 4/3 in figure 6. A1:2 has a global minimumat (0, A1:2) = (1:2,

m2

16), so that exactly at the 1:2 resonance, the valueof the coefficient A is inversely proportional to the damping, as predicted bythe symmetry arguments of section 2.2. Thus, near the 1:2 resonance, one-dimensional waves will decrease significantly in amplitude.

Adiff, the value of the cubic coefficient near the difference frequency resonance,is given by (65). The condition for difference frequency resonance is 0 = diff,where diff is given by (61). Since 0 is restricted to the range [0, m

2/4], thiscondition can only be met for certain m/n ratios. Specifically, diff [0, m2/4]only for

m/n M1 M2, M1 = [2 1, 2 2], M2 = [2 + 2, ). (77)

Thus, while the 1:2 resonance was relevant for all possible forcing frequencyratios m/n, this is not the case for the difference frequency resonance. Thedifference frequency resonance results in a contribution to A, namely Adiffgiven by (67), and thus Adiff has a local extremum at 0 = diff. The signof Adiff is given by the sign of n m. If the secondary forcing component isat a higher frequency than the primary, i.e. if m/n M1, then the differencefrequency resonance results in a positive contribution to A. The extremum isa local maximum, and the amplitude of the supercritical waves increases as

the resonance is approached. This is demonstrated by the small bump around0 diff = 17/6 in figure 6. Ifm/n M2, then the contribution is negative.The extremum is a local minimum, and the amplitude of the waves decreases.In either case, the extra contribution to A(0 = diff) that is due to thedifference frequency resonance is proportional to (fn)

2/ as predicted by thesymmetry arguments of section 2.2, and thus is a significantly smaller effectthan the 1:2 resonance.

For the case that m/n M1, when A has a local maximum, it is possiblefor this maximum to actually cross the A = 0 axis and become positive, thuscausing the bifurcation to become subcritical. An example of a subcritical

bifurcation may be obtained with the parameters m/n = 49/100, 0 = diff =2801/12 233.4, 1 = 0.01 and fn = 3.9 < 3.94 fcn, in which case A =0.57 > 0.

Now we turn to the results for the sum frequency resonance. Asum is given by(70). The condition for sum frequency resonance is 0 = sum, where sum isgiven by (69). Similar to the difference frequency resonance case, this conditionwill only be met for certain m/n ratios. Specifically, the sum frequency mode

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Fig. 7. Regions of forcing frequency ratio m/n in which the 1:2, difference, andsum frequency resonances are possible for one-dimensional waves. The plus (+) andminus () signs indicate whether the resonance results in a positive or negativecontribution to the cubic coefficient A in (16). In the case of () the bifurcationto one-dimensional waves is necessarily supercritical. Note that only points corre-sponding to rational numbers on the number line are meaningful.

resonance is possible only for

m/n

2 + 1. (78)

Thus, the sum frequency resonance can only be realized when the second forc-ing component is at sufficiently low frequency. The sum frequency resonanceresults in a contribution to A, namely Asum, which is given by (71). Asum hasa local extremum at 0 = sum. Like the difference frequency case, the con-tribution to A due to the sum frequency resonance is proportional to (fn)

2/.Unlike the difference frequency case, the Asum contribution always has a neg-ative sign. However, this contribution is generally not significant because thealgebraic prefactor in Asum is small for typical values of m/n for which thesum frequency resonance is possible.

A partial summary of the results for 1-d resonances may be found in figure 7.This number line shows the regions of forcing frequency ratio m/n in whicheach type of resonance is possible. The plus (+) and minus () signs indicatewhether the resonance results in a positive or negative contribution to thecoefficient A, and hence whether it makes the supercritical waves larger (+)or smaller () in amplitude.

4.2.2 Two spatial dimensions

We now present nonlinear results for Faraday waves in two spatial dimensions.

We have computed the cross-coupling coefficient B() in (6) using the methodin [26]. We interpret features of B() in light of the resonances discussed insection 2.2. Many of these features may be understood by means of a simpleargument which is valid for weak damping and forcing. We simply solve thespatial resonance condition (2) for 1:2, sum or diff, which are the angles atwhich the 1:2, sum frequency, and difference frequency resonances occur. Todo this, we must set |k3| = k() where k() is the inverse of the dispersionrelation (50) and = m, m + n or |m n| depending on the resonance under

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0 10 20 30 40 50 60 70 80 9012

10

8

6

4

2

0

B()

Fig. 8. Cross-coupling coefficient B() in (6). The solid line corresponds tom/n = 8/9 in (21); the dotted and dashed lines correspond to m/n = 8/11 and8/21 respectively. For each curve, the parameter is chosen to obtain a harmonicinstability near the bicritical point. The other parameters are = 0 in (21), and0 = 14 and = 0.1 in (26) - (27). The large dip at = 1:2

70 is due to the 1:2

temporal resonance discussed in section 2.2 and is independent of the second forcingcomponent. The small spike is due to the difference frequency resonance discussedin section 2.2. We have removed from this plot the region near = 60 where B()necessarily diverges; a calculation for the hexagonal lattice is required here.

consideration. A number of results immediately follow:

The 1:2 resonance is possible only for 0 m2/12 = 1:2. The difference frequency resonance is possible only for

m

1

22m2 + 240

n

m +

1

22m2 + 240. (79)

The sum frequency resonance is possible only for n m + 12

2m2 + 240.

From these statements, we also see that

The ranges of 1:2, sum and diff are restricted. There are some forcing frequency ratios m/n for which the sum and differ-

ence frequency resonances are not possible for any value 0.

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An example is given in figure 8, which shows the cross-coupling coefficient B()computed for forcing frequency ratios m/n = 8/9, 8/11 and 8/21 for fixed fluidparameters; is chosen in each case to obtain a harmonic instability near thebicritical point. The large dip at = 1:2 70 is a consequence of the 1:2resonance. At this angle, there is a resonant triad comprised of two Faraday-

unstable modes with dominant frequency m/2 and the weakly damped modeoscillating primarily with the harmonic frequency m. As expected from theanalysis of section 2.2, near this angle, the weakly damped mode contributesto B(), which here is manifest as the large dip. This phenomenon is similarto the 1:2 resonance in one spatial dimension, which resulted in a large dip inthe cubic self-interaction coefficient A in (16). It follows from the dispersionrelation that 1:2 will depend on m and 0 but will be largely independent ofthe parameters n, fn, and . The independence with respect to n is evidentin figure 8, in which the dip occurs at the same angle for m/n = 8/9, 8/11,and 8/21.

For all numerical calculations that we performed, the 1:2 resonance resultedin a large negative contribution to the cross-coupling coefficient B(). As dis-cussed in section 2.1, this type of contribution is destabilizing for superlatticepatterns with characteristic angles h near 1:2. Our numerical results (notshown) indicate that the magnitude of the dip caused by the 1:2 resonancefollows the scaling law that we deduced from symmetry considerations in sec-tion 2.2, and that we derived for one-dimensional waves: namely, that thecontribution from the weakly damped mode scales like 1/.

The sum frequency resonance angle sum may also be predicted by the weakdamping argument. However, unlike the 1:2 resonance described above and thedifference frequency resonance described below, the sum frequency resonancefor two dimensional waves is quite difficult to detect numerically for typicalvalues ofm/n and for small . This is consistent with the result for one spatialdimension, and consistent with the fact that the mode oscillating at the sumfrequency has a larger wave number and thus is more strongly damped.

Finally, we turn to results for the difference frequency resonance. The effectof the difference frequency resonance may be seen in figure 8, and is manifestas a spike in the plot of B(). Let us first concentrate on the solid curve in

figure 8, which corresponds to a forcing frequency ratio of m/n = 8/9. Forthis case, at = diff 17 there is a resonant triad composed of two modeswith dominant frequency m/2 and the weakly damped mode oscillating withdominant frequency |nm|. As expected from the analysis of section 2.2, nearthis angle, the weakly damped mode contributes to B(), which causes thespike. This phenomenon is similar to the difference frequency resonance in onespatial dimension, which resulted in a contribution to the cubic self-interactioncoefficient A.

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0 2 4 6 8 10 12 14 160

10

20

30

40

50

0

diff

n = 9

n = 11

Fig. 9. Angle of spatial resonance diff versus capillarity number 0. Lines cor-respond to a prediction of diff based on the dispersion relation (50) and on thetrigonometric relation (3). Symbols correspond to a numerical calculation of diff:

= 0.2 (x), = 0.8 (

). The other parameters are m = 8, = 50

and = 0in (21).

As with the case of 1:2 resonance, the simple argument we have used to predictthe resonance angle diff relies only on the dispersion relation. We thus expectthat diff will depend on m, n, and 0 but will be largely independent of theparameters fn and . The dependence on the second forcing frequency n isevident in figure 8, in which shifting from n = 9 to n = 11 causes the spike toshift from diff 17 to diff 47.

Figure 9 shows the angle of spatial resonance diff versus the capillarity pa-rameter 0 for the forcing frequency ratios m/n = 8/9 and 8/11 and for

various values of . The solid lines represent the prediction of diff based onthe dispersion relation, while the points represent data from a full numericalcomputation of B().

Another result that follows from the dispersion relation is that if the secondforcing frequency n is sufficiently different from m, the difference frequencyresonance will not be possible for any value of 0. This phenomenon is demon-strated in figure 8. The forcing frequency ratio m/n = 8/21 violates the con-

31


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dition (79) for all allowed 0, and the corresponding B() curve (dashed line)displays only the 1:2 resonance effect.

Now we discuss the magnitude and direction of the difference frequency mode

resonance effect. In contrast to the 1:2 resonance, we find that the differencefrequency resonance may result in a spike or a dip. Limited numerical resultsfor the sign of the resonance effect agree with the result for one spatial dimen-sion discussed in section 4.2.1. In particular, we have performed computationsat = 0.1 for the forcing frequency ratios m/n = 8/7, 8/9, 8/11, 10/7, 10/9and 10/11, each for values of 0 ranging between 0 and max = m

2/4. In allcases, we observe that ifn < m then the difference frequency resonance resultsin a dip at = diff; if n > m, it results in a spike.

As in the one-dimensional case, the magnitude of the difference frequency res-

onance effect follows the scaling law that we deduced from symmetry consid-erations in section 2.2, namely that the contribution from the weakly dampedmode scales like (fn)

2/. This scaling may be seen in figure 10. We compute themagnitude of the effect by finding B(diff) Bfn=0(diff), where Bfn=0(diff)is the value of the cross-coupling coefficient at the resonant angle computedwithout the second forcing component. We plot the size of the spike versus(fn)

2/.

As discussed in section 2.1, a spike occurring at spatial angle = diff will helpstabilize SL-I patterns with characteristic angles h near diff. To demonstrate

this effect, we consider an example for m/n = 8/11 forcing, with = 0.2, and0 = 13 and focus on the case of a harmonic instability. These dimensionlessparameters can be realized, for instance, by a fluid with surface tension =4.2 dyn/cm, density = 1.0 g/cm3, and kinematic viscosity = 0.01 cm2/sbeing forced with base frequency /(2) = 16.2 Hz. (The fluid properties hereare similar to those of water, but with lower surface tension. This situationmight be achieved by the use of a surfactant).

When = 60.5, there is a spike in B() at diff = 46.9, which is close to the

value of 47.0 that is predicted by the dispersion relation. We have performed

a limited bifurcation analysis similar to that in [26]. The stability of super-hexagon and supertriangle SL-I patterns is investigated within the setting ofa 12-dimensional bifurcation problem that also admits stripes, hexagons, andthree distinct rhombic patterns. An SL-I pattern with lattice angle h 47is stable for a small range of f above onset. A higher order calculation is nec-essary to determine whether it is the superhexagon or supertriangle variety ofSL-I pattern that is stabilized (these two different types of SL-I patterns havedifferent phases associated with the complex amplitudes).

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0 10 20 30 400

0.5

1

(fn

)2/

B(

diff

)

Bf

n=0

(

diff

)

n = 9

n = 11

Fig. 10. B(diff) Bfn=0(diff), the magnitude of the difference frequency spike,versus (fn)

2/. The damping parameter is varied between 0.01 and 0.1, and thestrength of the second forcing frequency fn is varied between 0 (which corresponds tosingle frequency forcing) and fcn. Best-fit lines are also shown. The other parameters

are m = 8 and = 0 in (21), and 0 = 14 in (26) - (27).

5 Conclusions

In this paper we have examined the role that weakly damped modes play in thepattern selection process for Faraday waves forced with frequency componentsm and n. Our symmetry arguments predict that the modes oscillating pri-marily with the frequency m, the difference frequency |nm|, and the sumfrequency (n + m) will be the most important in terms of their contributionto the cubic coefficients A and B() in the standing wave equations (6). The

symmetry considerations also provided scaling laws for the magnitude of theseresonance effects.

Starting with the Zhang-Vinals Faraday wave equations, we performed aweakly nonlinear analysis for weak damping and forcing and for m + n > 5in order to calculate expressions for the self-interaction coefficient A. We ob-tained expressions for the critical forcing acceleration and wave number, andanalyzed them to elucidate the role played by the secondary forcing compo-

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Fig. 11. Superhexagon (left) and supertriangle (right) patterns with characteristicangle h 47. For m/n = 8/11 forcing with = 0.2 and 0 = 13, both patterns areunstable for = 0

. For = 60.5

< bc, one of these patterns is stabilized by thedifference frequency resonance effect; a higher order calculation is needed to deter-mine which one. The patterns shown were created by an appropriate superpositionof the twelve critical Fourier modes [19].

nent in the linear instability. We also were able to identify the most importantfrequency components in the unstable eigenmode in terms of the integers mand n. We then analyzed the expression for the cubic coefficient A, and de-termined the sign and scaling of contributions due to the various resonanceeffects.

We then used the Zhang-Vinals equations to numerically calculate the cross-coupling coefficient B() according to the method in [26]. The predictions ofthe symmetry arguments were manifest. The results for B() are of particu-lar interest since this coefficient is crucial in determining the stability of SL-Ipatterns like those observed in [2]. We made use of an argument, valid forweak damping, which relies only on the dispersion relation to successfully pre-dict the resonant angle. While our symmetry arguments predict the scalingof the resonance effects and the dispersion relation predicts the angle, neitherargument predicts the sign of the contribution to B(). Our numerical calcu-lation revealed that the 1:2 resonance results in a dip, and thus is destabilizingfor SL-I patterns. However, the difference frequency resonance in some cases

results in a spike, which can help stabilize SL-I patterns with characteristicangles near the resonant angle diff. This was demonstrated by means of asimple bifurcation example.

We may now speculate on the role of the bicritical point in stabilizing SL-Ipatterns. It is been observed that SL-I patterns occur in experiments onlyfor parameters near the bicritical point. It is tempting to believe, then, thatthe weakly damped mode associated with the secondary forcing component is

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0 10 20 30 40 50 60 70 80 9016

12

8

4

0

B()

Fig. 12. Cross-coupling coefficient B() in (6) computed for the case of afour-frequency forcing function analogous to (21), with frequency components(m,n,p,q) = (8, 9, 11, 13). For this calculation, the m = 8 forcing component dom-inates, but we are near a quad-critical point so that the other three frequency

components are strong. The three spikes at = 13

, 47

, 73

are due to reso-nances with the weakly damped modes oscillating with the difference frequenciesn m, p m and q m. The other parameters used are 0 = 12.4 and = 0.25in (26) - (27). The stability of the SL-I patterns with h 13 is enhanced by theadditional spikes at 60 h; see [26] for a detailed discussion of how these twoadditional symmetry-related angles are relevant.

somehow responsible for the pattern. Here we have shown that this interpre-tation is not necessarily the correct one. Proximity to the bicritical point (i.e.making fn as large as possible before switching over to the other instability)

maximizes the strength of the difference frequency mode. As we have seen,this mode can help stabilize the SL-I pattern.

It will be interesting to consider Faraday waves forced with more than two fre-quency components. In this case, more difference frequency mode resonanceswill be possible. As demonstrated in figure 12, if the parameters are chosenproperly, multiple spikes in the cross-coupling coefficient B() might conspireto further enhance the stability of a particular superlattice pattern.

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A Appendix A

We now give the expressions for the coefficients in the travelling wave equations(34) - (36), (57) and (64) which we computed in section 3.

1 = f1m

2m(A.1)

2 =k2(80 + m

2)

4m+

3(f1m)2

8m3+

(f1n)2

2m(n2 m2) (A.2)

c1 =2m4 15m20 + 3620

2m(m2 120) (A.3)

c2 = 2m4 + 15m20 + 36

20

m(m2 + 120) (A.4)

3 = 21k2 (A.5)

3 = 9(f1m)

3

32m5+

f1m(f1n)

2(m4 m2n2 n4)2n2m3(n2 m2)2 (A.6)

f1mk2(80 + 3m

2)

4m3 f

3m

2m

c3 = 1(7m4 48m20 + 14420)(m2 120)2 (A.7)

c4 =61(m

2 + 40)

m2

+ 120(A.8)

c5 =3f1m(4m

8 47m60 + 516m420 + 2160m230 + 864040)4m3(m2 + 120)(m2 120)2 (A.9)

c6 =3f1m(4m

6 63m40 240m220 72030)8m3(m2 + 120)(m2 120) (A.10)

c7 = f1m(4m

6 39m40 + 144m220 + 43230)8m3(m2 + 120)(m2 120) (A.11)

e1 =m

2(A.12)

4 = 41 (A.13)

4 = 31:2m

(A.14)

e2 =m

4(A.15)

4 = 3diffn m (A.16)

r1 =ei f1n(2n

2 4nm + m2)2n(m n)(2m n) (A.17)

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c8 =48n6 72n3m3 + 204m2n4 176n5m

4nm(m n)(m2 10nm + 6n2) (A.18)

m6 + 8nm5 8m4n2

4nm(m n)(m2 10nm + 6n2)

c9 = 48n6

2200n3m3 + 1324m2n4

400n5m

4m(6n2 14nm + 5m2)(2m2 3nm + n2) (A.19)

143m6 824nm5 + 1912m4n2

4m(6n2 14nm + 5m2)(2m2 3nm + n2)

5 =92diff

2(m n)3 (f1m)

2

(3m 2n)(m 2n)(m n) (A.20)

(f1n)

2

(2m n)(2m 3n)(m n) +k2(7m

2 22nm + 11n2)6(m n)

r2 =ei mf1n(m

2 4nm + 2n2)4n(m

n)2(2m

n)

(A.21)

c10 = c8m

m n (A.22)

c11 =c9m

m n (A.23)

c12 =2(2n4 8n3m + 9m2n2 2m3n + m4)

(n m)(3n2 6nm + m2) (A.24)

c13 =4(119m2n2 62m3n + 11m4 + 22n4 88n3m)

(n m)(5n2 10nm + 3m2) (A.25)

Acknowledgments

We have benefitted from numerous detailed discussions with Jeff Porter. Wealso thank Hermann Riecke and Paul Umbanhowar for helpful conversation.The research of MS is supported by NSF grant DMS-9972059 and by NASAgrant NAG3-2364.

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Documents

Chad M. Topaz and Mary Silber- Resonances and superlattice pattern stabilization in two-frequency forced Faraday waves