Theoretical Model for Faraday Waves with Multiple-Frequency Forcing

4 Pages·0.15 MB·English

by  Ron Lifshitz

Theoretical Model for Faraday Waves with Multiple-Frequency Forcing
Ron Lifshitz and Dean M. Petrich
Condensed Matter Physics 114-36, California Institute of Technology, Pasadena, CA 91125, USA
(Recieved 4 April1997)
A simple generalization of the Swift-Hohenberg equation is proposed as a model for the pattern-
forming dynamics of a two-dimensional field with two unstable length scales. The equation is used
8 tostudythedynamicsofsurfacewavesinafluiddrivenbyalinearcombination oftwofrequencies.
9 Themodelexhibitssteady-statesolutionswithtwo-,four-,six-,andtwelve-foldsymmetricpatterns,
9 similar tothe periodic and quasiperiodic patterns observed in recent experiments.
1
PACS numbers: 47.54.+r, 47.35.+i, 47.20.Ky, 61.44.Br
n
a
J
Parametric excitations of surface waves have been ex- thetwoforcingfrequencies,andthereforethewave-
9
2 tensivelystudiedsincetheirfirstdiscoverybyFaraday[1] lengths involved in the selected pattern lie in nar-
over a century and a half ago. In the basic experimental row bands about two critical wavelengths.

] setup an open container of fluid is subjected to vertical
t 2. The driving used in the experiments is such that
f sinusoidal oscillations, which periodically modulate the
o the up-down symmetry, taking u to u, is bro-
effective gravity. When the driving amplitude a exceeds
s kenallowinginteractionsamongtriplets−ofstanding
. a critical threshold ac a standing-wave instability occurs
t plane-waves to exist. These triad interactions are
a with temporal frequency ω one half that of the driving
m frequency. The characteristic spatial wavelength of the the only stabilizing mechanism for non-trivial pat-
terns in our rotationally-invariantmodel equation.
- standing-wavepattern is selected through the dispersion
d relationω(k)ofthefluid. Onetypicallyobservespatterns
Wecapturetheessentialdynamicswithasinglefieldand
n
of stripes or squares in such experiments. It is only in
o withouta priori specifying anyangle-dependentinterac-
recentyearsthatavarietyofadditionalpatterns—some
c tionsamongcriticalmodes. Thisallowsforameaningful
[ with quasiperiodic rather than periodic long range order comparisonofthestabilityofdifferentN-foldsymmetric
— have been observed [2–6]. We shall focus here on a states. Wefindpatternsof2-,4-,6-,and12-foldsymme-
2
particularsetofexperiments,performedbyEdwardsand
v try that are globally stable, but none with 8- or 10-fold
Fauve [3], in which a fluid was driven by a linear combi-
0 symmetry, which is in agreement with the experimental
6 nationoftwo frequencies,formingperiodicpatternswith observations of Edwards and Fauve [3].
0 2-, 4-, and 6-fold symmetry, and quasiperiodic patterns Thesupercriticalinstabilityofahomogeneousstateto
4 with 12-fold symmetry.
a striped state is often modeled by the Swift-Hohenberg
0 Previoustheoreticalwork[6–11]hasfocusedmainlyon
equation [12]
7 adescriptionthroughamplitudeequationswithanangle-
t/9 Sduepchenadnenintteirnatcetriaocnt,iownhiβch(θiisj)eitbheetrwpeeonstuplaaitresdoofrdmeoridveesd. ∂tu=εu−(∇2+1)2u−u3, (1)
a
m fromtheunderlyingmicroscopicdynamics,canbechosen which is variational,
to stabilize N-fold symmetric patterns for arbitrary N.
d- Mu¨ller [10]has alsouseda setoftwocoupledpartialdif- ∂tu=−δF/δu, (2)
n ferential equations, where the pattern of a primary field
drivingthefieldu(x,y,t)towardsaminimumoftheLya-
o is stabilized by coupling to a secondary field which pro-
c vides an effective space-dependent forcing. Newell and punov functional (effective “free energy”) —
:
v Pomeau[11]havecoupledmultiplefieldsinasimilarway. 1 1 1
2 2 2 4
i In both cases the coupling between the different fields is = dxdy εu + [( +1)u] + u . (3)
X F Z −2 2 ∇ 4
achieved through resonant triad interactions, similar to (cid:8) (cid:9)
r
a the interactions we shall introduce below. The first term in the Lyapunov functional (3) favors the
We propose a simple rotationally-invariant model- growth of the instability whereas the quartic term is re-
equation,governingthedynamicsofarealfieldu(x,y,t), sponsible for its saturation by providing a lower bound
which describes the amplitude of the standing-wavepat- for . The growth rate ε of the instability is propor-
tern. Ourapproachisdifferentinthatitsearchesforthe tionFaltothe reduceddrivingamplitude (a a )/a . The
c c
minimal requirements for reproducing the steady states, positive-definitegradienttermissmallonly−nearthecrit-
which are observed in the experiments of Edwards and ical wave number k = 1, and thus inhibits the growth
c
Fauve [3]. We incorporate into our model only the two of any instabilities with wave numbers away from this
most essential aspects of the system: value.
Iftheparametricforcingissuchthattheu usym-
1. The dynamics is damped at frequencies away from →−
metryisbroken,thenthe Swift-Hohenberg“freeenergy”
1
is modified by the addition of a cubic term, αu3/3.
−
Such a term allows triad interactions of standing plane
waves to lower the value of and form hexagonal pat-
3.0
terns. The analysis of the SFwift-Hohenberg equation in ε/α2=0.08776
the presence of this term is summarized, for example, in ε/α2=1.91313
the review by Cross and Hohenberg [13]. With single-
frequency forcing one cannot break the u u sym- 12−fold
→ −
metry, but with certain combinations of two frequencies
2.0
the up-down symmetry is broken and triad interactions
become important.
We model the two-frequency parametric excitation of α Hexagons
a fluid by replacing the wavelength-selectingterm in the
Swift-Hohenberg equation (1) by a similar term which
dampsoutallmodesexceptthosenearoneoftwo critical 1.0
wavelengths:
2 2 2 2 2 2 3
∂ u=εu c( +1) ( +q ) u+αu u . (4)
t
− ∇ ∇ − Stripes
Theparameterccanbescaledout,butweincludeithere
0.0
because it is used in the numerical simulations, shown 0.0 0.5 1.0
later. Other model equations with similar wavelength- ε
selection properties are possible. We choose this equa-
tion because it is the simplest one that incorporates the
physics we are interested in — it allows two unstable FIG. 1. Phase diagram of the lowest-energy steady-state
length scales and contains triad interactions among the solutions of the model equation (4) for q =2cos(π/12). The
∗ 2
differentmodes. Since(4)canbeappliedtoanypattern- phase boundaries are lines of constant ε =ε/α .
forming system satisfying these requirements, it is not
our intention to provide a detailed derivation of it from
To study the formation of dodecagonal patterns we
any specific underlying microscopic dynamics.
choose q = 2cos(π/12), which is the magnitude of the
Let us turn now to an analytic investigation of the
vector sum of two unit vectors separated by an angle
model equation (4). When both ε and α are sufficiently
of 30 degrees. We minimize the Lyapunov functional (5)
small(orc sufficiently large)the wavelengthselectionby
the gradient term is nearly perfect and the Lyapunov withrespecttotheFouriercoefficientsuk describingfour
different pattern candidates: (a) a striped pattern with
functional may be written in Fourier space as
space group P2mm, whose Fourier spectrum contains
1 1 two opposite wave vectors of equal length; (b) a pat-
F =−2ε uku−k− 3α uk1uk2u−k1−k2 ternofperfecthexagonswithspacegroupP6mm,whose
|kX|=1,q |kiX|=1,q Fourier spectrum contains a single 6-fold star of wave
1
+4|kiX|=1,quk1uk2uk3u−k1−k2−k3, (5) vgreoctuoprsP;2(cm)ma,pwathtoesrenFoofucroimerpsrpeescsetrduhmexcaognotnaisnwsiftohurspvaecce-
torsononeringandtwovectorsontheotherring;and(d)
where the summations are restricted to wave vectors a dodecagonalpattern with space group P12mm, whose
whose magnitude is either 1 or q, lying on two rings in Fourier spectrum contains two 12-fold stars of wave vec-
Fourier space. The set of Fourier coefficients uk, that tors, one on each ring.
give rise to the lowest value of for a given choice of We use standard methods [14] to calculate for each
F F
the parameters ε and α, determines the most favorable of the cases. Because all the candidate patterns have
steady state solution of the model equation (4). We are symmorphic space groups [15] which are also centro-
only interestedin finding the globalminimum of ,thus symmetricwemayalwaystakealltheFouriercoefficients
F
establishingthatourmodelindeedpredictstheexistence on a given ring to be equal and their phases may all be
ofthepatternsobservedinthetwo-frequencyparametric chosen such that they are either 0 or π. The minimiza-
forcing experiments. Of course, this approach may over- tion of the Lyapunov functional is therefore always with
look meta-stable states or local minima of the free en- respect to no more than two real variables. We find the
ergy. Note that with the omission of the gradient term, values of the Lyapunov functional for the different pat-
one may perform a rescaling of the field u αu. The terns to be
rescaled free energy α−4 is then controlled→by a single
control parameter F 2 = 1ε∗2, (7a)
F −6
ε∗ =ε/α2. (6)
2
(a) (b)
(c) (d)
(e) (f)
FIG. 2. Numerical solutions of the model equation (4) showing real-space patterns along with their Fourier spectra for
∗ 2
differentvaluesofthecontrolparameterε =ε/α . Thereal-spaceimagesofu(x,y,t )showonequarterofthesimulation
→∞
cell with darker shades corresponding to larger values of the field. All figures are drawn to the same scale. In cases (a)-(d)
q = 2cos(π/12) = (2+√3)1/2 : (a) A 2-fold pattern of stripes for ε∗ = 2; (b) A 2-fold pattern of compressed hexagons for
ε∗ = 0.1; (c) A 6-fold pattern of perfect hexagons for ε∗ = 1.8; (d) A 12-fold pattern for ε∗ = 0.015. In (e) q =2, ε∗ =0.04,
yielding a 2-fold superstructure of stripes. In (f) q=√2, ε∗ =0.04, giving rise to a 4-fold pattern of squares.
4
F6 =F4−2 =−153(1+√1+15ε∗) dtheepipctheadseinditahgerapmhadseepidcitasgoranmly othfeFbigouurned1a.rieNsobteetwtheeant
2 (3+2√1+15ε∗)ε∗ 1 ε∗2, (7b) global minima; in certain regions of the phase diagram
− 152 − 10 additional states may be locally stable.
10 3 The model equation (4), supplemented with periodic
12 = (1+ 1+67ε∗/75) boundaryconditions,wassolvednumericallyonasquare
F −(cid:18)67(cid:19)
p domain using a pseudo-spectral method. The unit cell
20 (1+ 2 1+67ε∗/75)ε∗ 9 ε∗2. (7c) was typically chosen so that the simulation region held
− 672 3 − 67
p about 30 wavelengths. The simulation was performed
For ε∗ > 1.91313 the striped pattern has the lowest free on a 256x256 grid, with Adams-Bashforth second-order
energy. For 1.91313 > ε∗ > 0.08776 the 6-fold pat- time-stepping. The value of c was taken to be be-
tern of perfect hexagons and the 2-fold pattern of com- tween 10 to 100. Figures 2(a)-2(d) show the real-space
pressedhexagons(denotedby4-2),whicharedegenerate, and Fourier-space results of the simulations with q =
are most favorable. For ε∗ < 0.08776 the dodecagonal 2cos(π/12) for varying values of the control parame-
pattern is the most stable. These analytical results are
3
ter ε∗. The results are consistent with the Lyapunov- tion.
functional analysis and the phase diagram of Figure 1. WearegratefultoMichaelCross,JonathanMiller,and
Eight-foldandten-foldsymmetricpatternsarenotob- Peter Weichman for many helpful discussions. This re-
served in our model for any choice of q. An analytic searchwassupportedbytheCaliforniaInstituteofTech-
calculation of the Lyapunovfunctional (5) for these pat- nologythroughitsDivisionResearchFellowshipsinThe-
terns shows that it is greater than the free energy 6 oretical Physics.
F
(7b) ofthe six-foldstate, for anyvalue ofthe controlpa-
rameter ε∗. This is in accord with the experiments of
Edwards and Fauve [3], where such patterns are not ob-
served. Thisdoes notrule outthe possibility thatoctag-
onal and decagonal patterns are locally stable but only
that within the limits of our model they are not glob-
allystable. Twoadditionalpatternsthatareobservedin [1] M.Faraday,Phil.Trans.Roy.Soc.Lond.121(1831)299.
our model are a superposition of stripes of periodicities [2] B. Christiansen, P. Alstrøm, and M. T. Levinsen, Phys.
2π and π (shown in Figure 2(e)) and a square pattern Rev. Lett. 68 (1992) 2157.
for q = √2 (shown in Figure 2(f)). The latter has been [3] W. S. Edwards and S. Fauve, Phys. Rev. E 47 (1993)
R788; J. Fluid Mech. 278 (1994) 123.
reported by Edwards and Fauve. If one examines the
[4] J.P.Gollub,Proc. Natl.Acad. Sci.USA92(1995)6705.
Lyapunov functional (5) in its full generality by allow-
[5] M. Torres, G. Pastor, I. Jim´enez, and F. Montero De
ing the value of q and all the amplitudes and phases to
Espinoza, Chaos, Solitons, and Fractals, 5(1995) 2089.
vary independently, other patterns might be discovered.
[6] T. Besson, W. S. Edwards, and L. S. Tuckerman, Phys.
Wehaveonlyexaminedthesymmetricpatternsdiscussed Rev. E 54 (1996) 507.
here.
[7] B. A. Malomed, A. A. Nepomnyashchii, and M. I. Tri-
The simplicity of our model shows that for continu- belskii, Sov. Phys. JETP69 (1989) 388.
ous media very little is required to stabilize structures [8] W.ZhangandJ.Vin˜als, Phys. Rev. E53(1996) R4283.
with quasiperiodic long range order: two length scales [9] P.ChenandJ.Vin˜als,“Amplitudeequationsandpattern
and triad interactions. The reason that 12-fold patterns selection in Faraday waves,” preprint.
are stable and 8- and 10-fold patterns are not is purely [10] H. W. Mu¨ller, Phys. Rev. E 49 (1994) 1273.
geometrical. In view of the Lyapunovfunctional (5), the [11] A.C.NewellandY.Pomeau,J.Phys.A,26(1993)L429.
crucial issue is the competition between the number of [12] J.B.SwiftandP.C.Hohenberg,Phys.Rev.A15(1977)
modes, which tends to increase the value of , and the 319.
numberoftriadinteractions,whichtendstodFecreasethe [13] M. C. Cross and P. C. Hohenberg, Rev. Mod. Phys. 65
value of . The dodecagonalpattern ofFigure2(d) con- (1993) 851.
F
tains24non-zeroFouriermodesand32distincttriangles. [14] See, for example, P. M. Chaikin and T. C. Lubensky,
The octagonal and decagonal patterns do not contain a Principlesofcondensed matterphysics,(CambridgeUni-
sufficient number of triangles to compete with the 6-fold versityPress,Cambridge,1995), ch.4.7;or,formorede-
pattern of Figure 2(c). Our model confirms the conclu- tail, see L. Gronlund and N. D. Mermin, Phys. Rev. B
38 (1988) 3699.
sion of Edwards and Fauve that “12-fold patterns are
[15] Foradefinitionof“symmorphicspacegroup”inthecase
more common than previously supposed.”
ofquasiperiodicstructuresandofthenotation12mmsee,
Oursimplisticmodelisclearlynotadequateforstudy-
forexample,D.S.Rokhsar,D.C.Wright,andN.D.Mer-
ingthestructuralstabilityquasicrystalsinthesolidstate,
min, Acta Cryst. A 44 (1988) 197.
yet it may offer a very simple system in which to study
[16] H. W. Mu¨ller, Phys. Rev. Lett. 71 (1993) 3287.
general questions regarding quasiperiodic order. These
may include such questions as the formation and prop-
agation of defects and phase boundaries as well as the
dynamics of phason modes [16]. Moreover, we note that
(4) may apply to situations other than Faraday waves.
Any physical system that can be tuned such that two
wavelengths undergo a simultaneous supercritical bifur-
cation can be described by an equation similar to (4).
An equation similar to (4) could be used to study
multiple-frequency forcing of Faraday waves with more
thanjusttwofrequencies,assuggestedbythetitleofthis
letter. We may speculate that with three or four forcing
frequenciesitmightbepossibletostabilizequasiperiodic
patternswithevenhigherordersofsymmetry,suchas18
or 24. We leave the stability of higher-order symmetric
patterns as an open theoretical and experimental ques-
4