recent advances in physics of fluid parametric sloshing...

Raouf A. IbrahimDepartment of Mechanical Engineering,

Wayne State University,

Detroit, MI 48098

e-mail: [email protected]

Recent Advances in Physicsof Fluid Parametric Sloshingand Related ProblemsLiquid parametric sloshing, known also as Faraday waves, has been a long standing sub-ject of interest. The development of the theory of Faraday waves has witnessed a numberof controversies regarding the analytical treatment of sloshing modal equations andmodes competition. One of the significant contributions is that the energy is transferredfrom lower to higher harmonics and the nonlinear coupling generated static componentsin the temporal Fourier spectrum, leading to a contribution of a nonoscillating perma-nent sinusoidal deformed surface state. This article presents an overview of differentproblems of Faraday waves. These include the boundary value problem of liquid para-metric sloshing, the influence of damping and surfactants on the stability and response ofthe free surface, the weakly nonlinear parametric and autoparametric sloshing dynamics,and breaking waves under high parametric excitation level. An overview of the physics ofFaraday wave competition together with pattern formation under single-, two-, three-,and multifrequency parametric excitation will be presented. Significant effort was madein order to understand and predict the pattern selection using analytical and numericaltools. Mechanisms for selecting the main frequency responses that are different from thefirst subharmonic one were identified in the literature. Nontraditional sources of para-metric excitation and Faraday waves of ferromagnetic films and ferrofluids will be brieflydiscussed. Under random parametric excitation and g-jitter, the behavior of Faradaywaves is described in terms of stochastic stability modes and spectral density function.[DOI: 10.1115/1.4029544]

1 Introduction

Faraday waves refer to nonlinear standing waves, which appearon liquids enclosed by a container excited vertically with a fre-quency, X, close to twice the natural frequency xn, of the free sur-face, i.e., X ¼ 2xn. This condition is referred to as parametricresonance, since the motion of the liquid free surface is due to anexcitation perpendicular to the plane of the undisturbed free sur-face, and thus the generated waves are also known as parametricsloshing or Faraday waves. Faraday waves are named after Fara-day [1,2] who observed the fluid inside a glass container oscillatesat one-half of the vertical excitation frequency. Another similarseries of experiments, conducted by Mathiessen [3,4], showed thatthe fluid oscillations are synchronous, i.e., X ¼ xn. The contradic-tion of the two observations led Rayleigh [5,6] to make a furtherseries of experiments with improved equipment and his observationssupported Faraday’s results. During that time, Mathieu [7] formu-lated his equations, which helped Rayleigh to explain this phenom-enon mathematically. The problem was investigated again by others[8–10] who explained mathematically the discrepancy between Fara-day’s and Rayleigh’s observations and Matthiessen’s findings.

Depending on the fluid and excitation parameters, the solutionof Mathieu equation can be stable or unstable. The boundaries ofstability are usually given in a chart known as Ince–Strutt dia-gram. Analytical expressions for these boundaries are well docu-mented [11,12]. These boundaries are emanated at excitationfrequencies corrersponding to subharmonic (twice the sloshingnatural frequecy) or harmonic (same as the sloshing modal fre-quency) and they enclose regions of instability. Outside theseregions, the fluid free surface is stable. Benjamin and Ursell [10]showed that if the plane free surface is unstable, the resultingmotion could have frequency (N/2) times the excitation frequency,where N is an integer. Since the motion might be half-frequency

subharmonic, harmonic, or superharmonic, both Fraday [1,2] andMatthiessen [3,4] could be correct. However, the experimentalresults of Benjamin and Ursell [10] only showed that dampingresults in a threshold excitation amplitude below which the fluidfree surface is stable. Woodward [13] suggested that in most realfluids there is sufficient damping such that the unstable regions,except the first several unstable ones, will be located completelyabove the threshold excitation amplitude level and considerationsneed only be given to those modes in the lower frequency range.The fundamentals of parametric sloshing are well documented inRef. [14]. The purpose of this paper is to present the recent advan-ces of Faraday waves and related problems.

Generally, engineers have treated the hydrodynamic of para-metric sloshing of fluids with filling depths greater or smaller thanthe critical fluid depth. The critical fluid depth is the depth abovewhich the free-surface oscillations behave like a soft nonlinearspring. In the neighborhood of internal resonance (or autoparamet-ric resonance) among sloshing modes, Faraday waves may experi-ence complex behavior in the form of energy exchange and modescompetition. In particular, physicists dealt with the physics ofsmall layers of fluid and studied the competition of modes and themechanisms and generation of selected wave patterns. The pur-pose of this review article is to bring the recent developments ofFaraday waves as treated by Engineers, Physicists, and Mathema-ticians. Some controversies reported in the literature among dif-ferent researchers regarding the formulations of nonlinearsloshing modal interactions will be assessed.

This article is organized as follows: Section 2 provides a briefdiscussion of Faraday waves and cross waves generated by wave-maker. Section 3 presents the boundary value problem of paramet-ric sloshing of viscous fluids including surface tension. The basicidea of the quasi-potential approximation is introduced. This sec-tion enumerates the main techniques employed in the literature foranalyzing the boundary value problem. The damping effect onparametric sloshing is addressed in Sec. 4. The effects of insolublesurfactants on the damping rates, natural frequencies, and ampli-tudes of the fundamental sloshing mode are considered in Sec. 5.

Contributed by the Fluids Engineering Division of ASME for publication in theJOURNAL OF FLUIDS ENGINEERING. Manuscript received August 14, 2014; finalmanuscript received December 24, 2014; published online May 25, 2015. Assoc.Editor: Shizhi Qian.

Journal of Fluids Engineering SEPTEMBER 2015, Vol. 137 / 090801-1Copyright VC 2015 by ASME

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Section 5 also considers the parametric excitation of stratified flu-ids. Section 6 is devoted to the problem of nonlinear parametricsloshing including weakly nonlinear waves and breaking surfacewaves. The case of parametric excitation in the presence of inter-nal resonance among gravity waves is addressed in Sec. 7. Whenthe detuning between two or more adjacent modes is very small,these modes can enter in competition which may lead to chaoticbehavior. The problem of modes competition is treated in Sec. 8for fluids of small layers. The pattern formation and selection dueto fast time scales are considered in Sec. 9. This problem has beentreated mainly by Physicists who considered Faraday patterns ofthin fluid layers under single-, two-, three-, and multifrequencyexcitations. Section 10 addresses Faraday waves of other mediasuch as dielectric liquids, magnetic liquids, ferrofluids, smecticand nematic liquid crystal layers. The problem of random para-metric excitation under gravitational field and g-jitter of liquidfree surface is treated in Sec. 11. Section 12 provides some clos-ing remarks, conclusions, and need for further research.

2 Between Faraday Waves and Cross Waves

Faraday waves oscillate subharmonically with the externalexcitation frequency. They are excited as the forcing amplitude israised above a critical value. The surface waves could also be har-monic (or synchronous) with the external excitation in the pres-ence of large dissipation. This usually occurs in thin layers ofviscous fluids at low excitation frequencies. This leads to the pos-sibility of a bicritical point at the instability threshold point,where both subharmonic and harmonic (synchronous) surfacewaves can be excited for the same value of excitation amplitudeacceleration. The physical interpretation of the resonance at halfthe excitation frequency was given in Ref. [15] and is demon-strated in Fig. 1. When a fluid layer is excited parametrically athalf the excitation frequency, then when the vessel goes down, thefluid inertia tends to create a surface deformation, as in the Ray-leigh–Taylor instability1. This deformation disappears when thevessel comes backup, in a time equal to a quarter-period of the

corresponding wave. The decay of this deformation creates a flowwhich induces, for the following excitation period, the exchangeof the maxima and the minima. Thus, one obtains the period ofone fluid cycle to be twice the period of excitation.

As mentioned in the Introduction, the fluid-free surface motionunder parametric excitation may be described by a system ofMathieu equations in the form

d2amn

ds2þ Pmn � 2qmn cos 2sð Þamn ¼ 0 (1)

where amn is the nondimensional wave height of mode mn,Pmn ¼ 4x2

mn=X2, qmn ¼ 2 nmn=Rð ÞZ0 tanhðnmnh=RÞ, s ¼ xmnt, xmn

is the mn sloshing natural frequency, t is the time, Z0 and X arethe amplitude and frequency of parametric excitation, respec-tively, nmn are the roots of the first derivative of the Bessel func-tion of the first kind, i.e., d=dr JmðnmnÞ½ � ¼ 0, R is the tank radius(for circular tank), and h is the fluid depth. The domains of insta-bility could be reduced somewhat if linear damping due to fluidviscosity be added in Mathieu equation as [16,17]:

d2a

dt2þ 2c

da

dtþ x2

mn � X2Z0k cos Xt� �

a ¼ 0 (2)

where c ¼ 2�k2 is the damping coefficient as assumed in Refs.[17,18] k is the wave number, and � is the kinematic viscosity.The regions of instability according to Sorokin [17] are deter-mined by the boundaries

1�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2Z0kð Þ2�4

2c

X

� �2s

<2xmn

X

� �2

< 1þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2Z0kð Þ2�4

2c

X

� �2s

(3a)

Comparing with the undamped case [19]

1� 2Z0k <2xmn

X

� �2

< 1þ 2Z0k (3b)

The stability boundaries described by Eq. (3a) determine the criti-cal excitation amplitude, Zc, below which the fluid free surfacewould remain stable. This amplitude is determined by equatingthe expression under the radical sign to zero at X= 2xmnð Þ ¼ 1 togive

Fig. 1 The physical interpretation of the resonance at half the excitation frequency asprovided in Ref. [15]

1The Rayleigh–Taylor instability occurs at the interface between two plane-parallel layers of immiscible fluids, in which the more dense fluid is on top of theless dense one. The equilibrium is unstable to any perturbations or disturbances ofthe interface. This occurs if a parcel of heavier fluid is displaced downward with anequal volume of lighter fluid displaced upwards. The potential energy of theconfiguration is lower than the initial state and consequently the disturbance willgrow and lead to a further release of potential energy, as the more dense materialmoves down under the gravitational field, and the less dense material is furtherdisplaced upward.

090801-2 / Vol. 137, SEPTEMBER 2015 Transactions of the ASME


Zc ¼2c

kX(4)

For Z0 > Zc, the free-surface amplitude increases at an exponen-tial rate. Brand and Nyborg [20] carried out an experimentalinvestigation to measure Zc, i.e., the minimum values of excitationamplitude required for exciting half-frequency surface waves. Themeasured values of Zc were found to be much greater than thosepredicted analytically. They attributed this difference to the lackof development in the previous theories of the free-surface damp-ing coefficient. This will be discussed in Sec. 4.

Regarding the interface instability during vertical excitation ofthe free surface, Wright et al. [21] indicated that during one cycleof fluid oscillations the interface experiences stabilizing and desta-bilizing acceleration directed from the light to the heavy fluid andvice versa. The interplay between the Rayleigh–Taylor instabilityprevails during the first half-cycle of the vibration. The stabilizinginfluence of the fluid acceleration directed from the heavy to thelight fluid, prevailing in the second half-cycle of the vibration, isresponsible for temporal symmetry of the wave profiles during acomplete cycle. This asymmetry should be contrasted with thesymmetry of the unforced gravity or capillary waves. If the fre-quency of oscillations is small, the interface is subjected to desta-bilizing acceleration for a sufficiently long period. This allows theRayleigh–Taylor instability to proceed into its nonlinear stages,causing the development of spikes and bubbles.

Faraday waves are distinct from other waves with crests normalto a moving boundary (wave-maker) as cross waves [22–26]. Themechanism of creating the transverse waves observed in wavetanks was discussed in Ref. [27]. It was shown that regular trainsof progressive waves created by a wave-maker are unstable to dis-turbances along their crests, and that the disturbance has a fre-quency exactly half that of the progressive waves themselves. Thepossible relevance of this nonlinear subharmonic resonant interac-tion to the problem of edge wave generation was also discussed.Miles and Henderson [28] provided an account of the history ofFaraday waves. They designate those waves associated with anoscillation of the effective gravitational acceleration as Faradaywaves and those waves with crests normal to a wave-maker ascross waves or edge waves originally discovered by Stokes [29],which may be subharmonically excited by either an incomingwave or a disturbance moving parallel to the shore. The nonlineardynamics of nonlinear modulated cross waves of resonant fre-quency x1 and carrier frequency X � x1 was investigated by

Friedel et al. [30]. In a long horizontal container with finite depth,which is subjected to a vertical oscillation of frequency X ¼ 2x1,the wave can appear in solitary form. It is known that the solitarywave is only stable in a certain parameter regime, depending ondamping and driving amplitudes. It was shown how instabilitiesare saturated following generic routes to chaos in time with spa-tially coherent structures.

Figure 2 shows different waveforms, in which cross waves arestrictly 2:1 subharmonic waves produced by a wave-maker con-sisting in a partially submerged horizontally vibrating plate. Inter-action between both end-walls was found to produce sloshingmodes that exhibit new dynamics promoted by wave reflection atthe end-walls. Parametric excitation is triggered by the harmonicwave field produced by the vibrating end-walls, which exhibitstwo distinguished components. On the one hand, two counterpro-pagating harmonic wave-trains (see Fig. 2(d)) aligned along thevibrating end-walls are present, which propagate (and decay byviscous dissipation) inward from the end-walls. Perez-Graciaet al. [31] examined capillary–gravity, modulated waves of nearlyinviscid fluid parametrically excited by monochromatic horizontalvibrations in liquid containers whose width and depth are bothlarge compared with the wavelength of the excited waves. A gen-eral linear amplitude equation is derived with appropriate bound-ary conditions that provides the threshold acceleration andassociated spatiotemporal patterns, which compare very well withthe experimental measurements and visualizations. The resulting(quasi-periodic) waves are generally oblique, not perpendicular tothe vibrating end-walls. This article will not address this type ofwaves.

It should be mentioned that the mathematical formulations ofstability boundaries and nonlinear response under parametric ex-citation depend on the fluid boundary value problem. The mathe-matical formulation of the liquid boundary problem in closedcontainer depends on some assumptions pertaining to the fluidproperties and the container geometry. The general boundaryvalue statement of liquid sloshing is described in Sec. 3 forincompressible viscous fluids with the inclusion of its surfacetension.

3 Boundary Value Problem of Parametric Sloshing

In order to describe the fluid dynamics in a moving container,one should begin with the incompressible Navier–Stokes equa-tions, which are written in a moving Cartesian frame attached to

Fig. 2 Waveforms of (a) cross waves, (b) sloshing modes, (c) oblique subharmonic waves,and (d) harmonic wave-trains [31]

Journal of Fluids Engineering SEPTEMBER 2015, Vol. 137 / 090801-3


the container, with the plane of x- and y-axes coincide with theunperturbed free surface and the z-axis pointing vertically. Thegoverning equations in the moving frame attached to the containerare:

(1) The continuity equation for incompressible flow (diver-gence of velocity field is zero everywhere)

r � v ¼ 0 (5)

(2) Conservation of linear momentum (Navier–Stokesequation)

@v

@t� v � rv ¼ � 1

qrpþ �r2vþG (6)

where v ¼uiþ vjþ wk is the flow velocity, q is the fluiddensity, p is the fluid pressure, � is the fluid kinematic vis-cosity, and G ¼ �gðtÞk represents body forces per unitvolume acting on the fluid and in the present case is due tothe vertical acceleration (including gravitational accelera-tion). r ¼ ið@=@xÞ þ jð@=@yÞ þ kð@=@zÞ, i, j, and k areunit vectors along x, y, and z-axes, respectively. Note thatthe convective term, �v � rv, on the left-hand side, is theonly quadratic nonlinear term and any small amplitudeexcitation will result in mean fields that evolve on a slowertime scale. Such mean fields can either be a simple biprod-uct of the nonlinear vibration or couple to the primaryoscillating field, affecting the dynamics of the system.At low viscosity, the primary oscillating flow satisfies thelinearized momentum equation, @v=@t ¼ �rp=q, imply-ing that the flow is potential and satisfies the conditionr� v ¼ 0. Note that this condition applies to the bulk andnot to the oscillatory boundary layers attached to the solidwalls and the free surface. In the region of boundary layers,the mean flow is produced by the quadratic term in Eq. (6).Higuera et al. [32] indicated that the boundary layer meanflow does not vanish at the outer edge of the boundarylayers, but provides a nonzero velocity and a nonzerostress at the edges of the boundary layers near the solidwalls and the free surface. And it is these finite boundaryvalues that induce the viscous mean flow in the bulk.These new forcing terms are independent of the viscosityand quadratic in the amplitude of the primary waves, indi-cating that the effects of the mean flow cannot beneglected, even in the limit of vanishing viscosity, a strik-ing result considering that these terms are entirely due toviscous effects.

(3) At the free surface, z ¼ gðx; y; tÞ, the vertical velocity of afluid particle located on the free surface should equal thevertical velocity of the free surface itself. This is expressedby the kinematic boundary condition

@g@t¼ w� u

@g@x� v

@g@y

��z¼g

(7)

(4) At the free surface, z ¼ gðx; y; tÞ, the pressure of the freesurface should equal the capillary pressure due to surfacetension, i.e., Dp ¼ c ð1=r1Þ þ ð1=r2Þð Þ, where c is the sur-face tension,2 and r1 and r2 are the principal radii of curva-ture. In this case, the dynamic free-surface boundarycondition takes the form

p

q� 1

2vj j2þ gðtÞgþ cr � rgffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ rgj j2q264

375 ¼ �r2v (8)

(5) No slip condition at the walls

v ¼ 0 at the boundary of the free surface and z ¼ �h

(9)

According to Ruvinsky et al. [33], one can assume, for low viscos-ity fluids, that the oscillating part of the flow in a surface waveconsists of a potential component given by rU and a small vortexpart v1 excited by the potential component. In the linear analysis,v1j j= rUj j � k=d ¼ 2p=kð Þ

ffiffiffiffiffiffiffiffiffiffiffi2�=x

p, where k ¼ 2p=k and

d ¼ffiffiffiffiffiffiffiffiffiffiffi2�=x

p. x and k are characteristic time and space scale of

gravity–capillary waves, respectively. U is the velocity potentialfunction. When the thickness of the viscous boundary layer issmall compared to the typical wavelength of the free surface,d < k, then weak effects due to viscosity can be taken intoaccount by introducing effective boundary conditions for the oth-erwise potential bulk flow. This is the basic idea of the quasi-potential approximation developed in Refs. [33] and [34]. Thisquasi-potential formulation was extended by Zhang and Vi~nals[35] to three-dimensional (3D) flow for quasi-potential fluid.

For an incompressible fluid, the field equations may be writtenin terms of the potential function as

Continuity equation:

r2Uðx; y; z; tÞ ¼ 0 (10)

The boundary conditions at the free surface z ¼ gðx; y; tÞ

@g@tþrU � rg ¼ @U

@zþ wðx; y; tÞ (11)

@U@tþ 1

2rUð Þ2� gðtÞgþ 2�

@2U@z2¼ cj (12)

@wðx; y; tÞ@t

¼ 2�@

@x2þ @

@y2

� �@U@z

(13)

@U@z

��z!1¼ 0 (14)

where wðx; y; tÞ is the z-component of the rotational part of the ve-locity field at the free surface, gðtÞ ¼ g0 þ gzðtÞ, and g0 is thegravitational acceleration, and j is the mean curvature of the freesurface given by the expression

j ¼ r �� @g@x;� @g

@y; 1

� �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ ðrgÞ2

q (15)

Combining Eqs. (13) and (11) gives

@wðx; y; tÞ@t

¼ 2�@2

@x2þ @2

@y2

� �@g@tþrU � rg� w

� (16)

Note that 2�rU � rg is a nonlinear viscous term and can beignored. Furthermore, w � Oð�Þ and �w � Oð�x2Þ, which canalso be ignored. With these approximations, Eq. (16) takes theform

@

@tw� 2�r2g� �

¼ 0 (17)2The dimension of the surface tension is dyn=cm ¼ 0:001 N=m ¼ 0:001 J=m

2.



Equation (17) is solved to give

wðx; y; tÞ ¼ 2�r2gðx; y; tÞ þ w0 � 2�r2g0 (18)

where g0 and w0 are the initial conditions of the wave height andthe rotational velocity at the free surface. If the motion starts fromrest, then w0 ¼ 0 and also in view of the small value of 2�r2g0 itcan be ignored for nonlinear finite-amplitude states. In this case,the free-surface boundary conditions (11) and (12) take the form

@g@t¼ @U@zþ 2�r2g�rU � rg (19)

@U@t¼ gðtÞg� 2�

@2U@z2� 1

2rUð Þ2þ cj (20)

The boundary values problem of liquid in closed containers com-prised the continuity equation represented by Laplace’s equation(for incompressible) together with dynamic and kinematic condi-tions of the free surface and conditions at the wetted walls andbottom where the velocity component normal to the boundariesmust vanish. Large liquid-free surface oscillations in closed con-tainers have been treated in the literature using a number ofapproximations. The following are selected number of theories:

(a) Moiseev [36] constructed normal mode functions and char-acteristic numbers by integral equations in terms of Green’sfunction of the second kind (Neumann function). Chu [37]generalized Moiseev’s method by employing a perturbationtechnique using the characteristic functions to determinethe subharmonic response to an axial excitation.

(b) Penney and Price [38] carried out a successive approxima-tion approach where the potential function is expressed as aFourier series in space with coefficients that are functionsof time. These coefficients are approximated by Fouriertime series using the method of perturbation. The resultingsolution is expressed in terms of a double Fourier series inspace and time. This method was applied by others[19,38,39].

(c) Hutton [40] expanded the dynamic and kinematic free-surface equations in Taylor series about a stationary surfaceposition. This method was modified and used in manysloshing problems by Woodward [13,41].

(d) Time-average Lagrangian developed by Miles [42]: Miles[42–44] solved the kinematical boundary value problem fornonlinear gravity waves in a cylindrical basin using a varia-tional formulation together with the truncation and inver-sion of an infinite matrix. The results were applied toweakly coupled oscillations, using the time-averagedLagrangian, and to resonantly coupled oscillations, usingPoincar�e’s action-angle formulation. Miles [44] demon-strated that a Lagrangian formulation, in which the general-ized coordinates are the coefficients in a normal-modeexpansion of the free-surface displacement and are slowlymodulated sinusoids, leads to a set of evolution equationsfor 2N slowly varying amplitudes, where N is the numberof modes retained in the truncated modal expansion. TheLagrangian formulation becomes a very powerful tool indeveloping the fluid field equations in Lagrangian coordi-nates and boundary conditions [45–50]. This approachbecomes very attractive in developing numerical algorithmssuch as finite-difference and finite element, in particular,for cases of large amplitude motions at resonance sloshingand Faraday waves.

(e) Center-manifold and normal form formulation developedby Meron and Procaccia [51,52]: A detailed formulation tosolve the boundary value problem using the center-manifold theory and normal form averaging was presentedin Refs. [51,52]. Their approach was found in differencewith the Lagrangian formulation of Miles [43,44]. The

difference was in the lack of symmetry of the nonlinearterms of modal amplitude equations. However, the twoapproaches confirmed the essential features of experimentalresults.

(f) A spectral technique developed by Horsley and Forbes [53]to solve the nonlinear boundary value problem of Faradaywaves and reducing the system to a set of nonlinear ordinarydifferential equations for a set of Fourier coefficients: Time-periodic solutions of the main subharmonic resonance wereobtained in both the full and weakly nonlinear theories.These solutions were found to undergo several bifurcations,which give rise to chaos for appropriate parameter values.

As mentioned earlier, the damping plays an important role inthe dynamic behavior of Faraday waves. The influence of damp-ing on the stability–instability boundaries together with thedynamic response of the free surface has received extensiveresearch activities. It is important to realize that the damping fac-tor involves uncertainties, since the measured value is not alwaysfixed and depends on temperature and oscillations of the free sur-face. This is in addition to the condition of the container wettedwalls. This problem is considered in Secs. 4 and 5.

4 Damping Effect

Considerable efforts were given to determine the damping coef-ficient of liquid-free surface motion and its influence on the fluidnatural frequencies and its response. The kinematic viscosity, �, isa fundamental parameter for measuring the rate of vorticity andmomentum by molecular transport. In fact, it is known that thevorticity is present at a distance, d, outside the boundary of thebody, whereas the time required for diffusing the vorticity ormomentum through this distance is of order ðd2=�Þ, which isreferred to as the diffusion time. Boussinesq [54] introduced theinfluence of viscous damping to study progressive and standingwaves in closed containers. Boussinesq’s work was extended byKeulegan [55] to calculate the attenuation of solitary waves. Caseand Parkinson [56] determined the damping of surface waves ofsmall amplitude in partially filled cylindrical tank. They consid-ered viscous dissipation in an assumed laminar boundary layer asthe primary cause of damping. Van Dorn [57] conducted experi-mental investigations and attributed the differences between thepredicted and measured results to a surface film produced byspontaneous contamination. It was stated: “while the observedattenuation agreed with that computed for solid boundaries whenthe water was fresh, the former tended to increase with time tosome higher limiting value, usually within an hour.”

Miles [58] and Mei and Liu [59] indicated that the damping ofsurface waves in closed basins appears to be due to several sour-ces. These include viscous dissipation at the boundary of the sur-rounding basin, viscous dissipation at the surface in consequenceof surface contamination, and capillary hysteresis associated withthe meniscus surrounding the free surface. The theoretical resultsfor the logarithmic decrements of gravity waves in circular andrectangular cylinders were compared with the decay ratesobserved by Case and Parkinson [56] and by Keulegan [55],which typically exceeded the theoretical value based on walldamping alone by a factor of 2–3. It was concluded that both sur-face films and capillary hysteresis can account for these observeddiscrepancies. Despite of the elegant formulations of sloshingdamping reported in the literature [10,14,55,60], there is alwayssignificant discrepancy between predicted and measured dampingfactors. The measurements become more difficult for highermodes as confirmed in Ref. [61].

For two-dimensional (2D) linear flow, outside the boundary-layers, in a rectangular tank of length Lx in the x-direction, widthLy in the y-direction, and for antisymmetric sloshing modes, thedamping due to boundary-layer flow expressed by the damping ra-tio based on the theory of Keulegan [55] was given by theexpression



fn ¼1

2

ffiffiffiffiffiffiffiffiffiffiffiffiffi�

2xnL2x

rsinhð2knhÞ � knhð Þ þ knLx

sinhðknhÞcoshðknhÞ

� þ

ffiffiffiffiffiffiffiffiffiffiffiffiffi�

2xnL2y

s

¼ffiffiffiffiffiffiffiffiffiffip�Tn

p

2pLy

Ly

Lx1þ ðknl=2Þ � knh

sinhðknhÞcoshðknhÞ

� �þ 1

� (21)

where kn ¼ np=l, n ¼ 1; 2; :::, and Tn is the period of nth sloshingmode.

Although the measured natural frequencies were found inexcellent agreement with the theoretical values, Ikeda et al. [62]found significant differences between the measured values (usinglogarithmic decrement) and the theoretical values based on Keule-gan. For example, based on Keulegan theory, the values of thedamping ratios of the three sloshing modes, in a square container,were f1;0 ¼ f0;1 ¼ 0:003142, f2;0 ¼ f0;2 ¼ 0:0028274 andf3;0 ¼ f0;3 ¼ 0:0025638. Curve fitting was used to match the theo-retical results to the experimental data, and the damping ratioswere identified as fm;n ¼ 0:015. These ratios were similar to boththe measured value of f10 ¼ 0:018 obtained by the logarithmicdecrement and to the theoretical values based on Keulegan theory.These values were obtained for a square tank of cross dimensionof 100 mm� 100 mm and liquid depth of h ¼ 60 mm. Note thatthe measured values involved significant uncertainties due to tem-perature differences, contamination of the free surface, etc. Ingeneral, damping is an inherent parameter that involves the mostsignificant level of uncertainty, and for this reason, structuraldynamicists adopted a probabilistic description of the damping pa-rameter, which is usually represented by a random variable with agiven probability distribution [63].

The frequency and damping rate of surface capillary–gravitywaves in a bounded region depend on the conditions imposedwhere the free surface makes contact with the boundary. An edgecondition that models the dynamics associated with moving con-tact lines, but not contact angle hysteresis, was obtained by mak-ing the slope of the free surface at contact proportional to itsvelocity as proposed by Hocking [64]. This model was used toobtain the frequency and damping rate of a standing wavebetween two parallel vertical walls. The effect of viscosity in theboundary layers on the walls was included and it was shown thatthe dissipation associated with the surface forces can exceed thatproduced by viscosity. Henderson [65] and Henderson and Miles[66] calculated the natural frequencies and damping ratios for sur-face waves in a circular cylinder based on the assumptions of afixed contact line, Stokes boundary layers, and either a clean or afully contaminated surface. The differences between the predictedand observed frequencies were found to be less than 0.5% for allbut the fundamental axisymmetric mode with a clean surface. Thedifference between the predicted and observed damping ratio forthe dominant mode with a clean surface was 20%.

Experiments on single-mode Faraday waves in small rectangu-lar and circular cylinders in which both capillary and viscouseffects are significant were reported by Henderson and Miles [67].Theoretical predictions of the resonant frequency of a single modeand of the threshold amplitude for its excitation based on the hy-pothesis of linear boundary-layer damping were found in agree-ment with the measured data. In their theoretical analysis, theyused the measured damping rate to predict these quantities forwaves in the rectangular cylinder. Later, Henderson et al. [68]measured the damping rates and natural frequencies of the funda-mental axisymmetric mode in circular cylinders when the contactangle between the water and the side walls was acute, obtuse, andabout p=2. It was found that damping rates decrease with increas-ing contact angle, while the natural frequencies increase withincreasing contact angle. Pritchett and Kim [69] described an ap-paratus for studying the Faraday instability in a viscous fluid.

A nonlinear model of Faraday waves was developed by Decent[70] and Decent and Craik [71] to investigate the hysteresis,which occurs when both finite-amplitude solutions and the flatsurface solution co-exist. The lower hysteresis boundary in

forcing frequency space was defined by the lower boundary abovewhich nontrivial stationary points exist. Single-mode limit cycleswere found as solutions of a nonlinear evolution equation for para-metrically excited standing surface waves in a rectangular con-tainer. Later, Decent and Craik [72] examined their structureanalytically and numerically, and described local and global bifur-cations. To calculate the linear and nonlinear damping coeffi-cients, Decent [73] indicated that it is necessary to determine thedissipation in the main body of the liquid, the dissipation in theboundary layers at the sidewalls and at the surface, and the dissi-pation due to capillary hysteresis. Decent [74] extended Miles’calculations [58] to obtain the cubic damping coefficient. Decentand Craik [71] estimated experimentally the linear and cubicdamping coefficients in a rectangular tank. Both the theoreticalmethod of Decent [74] and the experimental method of Decentand Craik [71] revealed that the cubic damping coefficient fordeep water is positive for water depths greater than approximately1.2 cm. Decent [70] showed that the free-surface amplitude equa-tion exhibits a Hopf bifurcation when the cubic damping coeffi-cient is positive, which is absent for negative cubic dampingcoefficient. This Hopf bifurcation was found to give rise to a sta-ble limit cycle solution, which corresponds to a time-modulatedstanding wave, where the maximum amplitude of the standingwave varies with the slow time scale.

Martel et al. [75] estimated the natural frequencies and dampingrates of surface waves in a circular cylinder with pinned-endboundary conditions in terms of the gravitational Reynolds andBond numbers, Rg ¼

ffiffiffiffiffiffiffiffigR3

p=� and B ¼ qgR2=c, respectively, to-

gether with the fluid depth ratio, h=R. Higher-order approxima-tions that include the effect of viscous dissipation in the Stokesboundary layers and the bulk were considered. A comparison withclean surface experimental results [66] showed a satisfactoryagreement except for the first axisymmetric mode, which exhibitsa 26% discrepancy. The boundary-layer calculation was supple-mented by a calculation of interior damping based on Lamb’s dis-sipation integral for an irrotational flow [76]. The analysis yieldedresults of comparable accuracy within the parametric domain ofexperiments. The corresponding calculations for a fully contami-nated surface were found to reduce the discrepancy between cal-culation and experiment but, in contrast to the results for a cleansurface, leave a significant residual discrepancy.

The vertical oscillation of a plate partially immersed in a non-wetting fluid was found to cause a radiated wave-train when thecontact line between the plate and the free surface of the fluid can-not move freely along the plate [64]. Realistic conditions to applyat the contact line when capillarity is not negligible were found toinclude the dynamic variation of the contact angle and contactangle hysteresis. The amplitude of the radiated waves and theenergy dissipation at the contact line were calculated. Jiang et al.[77] found a complicated nonlinear relationship between wavefrequency and amplitude near contact lines. The relative dampingrate was found to be dependent on the wave amplitude, increasingsignificantly at smaller wave amplitude. These results were dis-cussed in relation to different formulations of contact line condi-tions for oscillatory motions and free-surface flows.

Faraday waves can be modeled by a damped Mathieu equation[81]

d2gdt2þ 4�k2 dg

dtþ gk þ c

qk3 � ak cosðXtÞ

� g ¼ 0 (22)

where g is the surface wave amplitude, c is the surface tension, ais the excitation amplitude acceleration, and k is the instabilitywave number. Benjamin and Ursell [10] noted that standingwaves can be created even if a=g < 1. When the water becomesshallow and viscous effects remain small, i.e., kh � 1; kd� 1,where d �

ffiffiffiffiffiffiffiffiffi�=X

pis the boundary layer thickness at the free sur-

face, an equation which is nonlocal in time is needed to describethe behavior of g [78,79]. The nonlocality arises because of



viscous dissipation at the bottom boundary. The surface waveheight obeys a nonlocal equation when the fluid is deep and vis-cous effects are strong kh� 1; kd � 1ð Þ. However, Cerda andTirapegui [80,81] indicated that a local equation can quantita-tively describe g when the fluid becomes shallow. This equation isagain a Mathieu equation, but the dependence of the damping onthe wave number is much different than when viscous effects areweak. In this regime, kh � 1; kd � 1ð Þ, diffusion of momentumoccurs so rapidly relative to the surface oscillations that the damp-ing no longer depends on the history of the motion. When viscouseffects are strong, the critical acceleration amplitude, ac, requiredto excite the standing waves is typically larger than the accelera-tion of gravity. In the deep water limit (wavelength � liquiddepth), Kumar [82] indicated that the preferred wave number inthe Faraday instability is primarily determined through Rayleigh–Taylor instability. In the case of shallow water (wavelength � liq-uid depth), the agreement between the Rayleigh–Taylor and Fara-day wave numbers does not appear to be as good, probably due tothe interaction between the oscillatory motion of the standingwaves and the bottom boundary.

Lapuerta et al. [83] considered a horizontal heavy fluid layersupported by a large aspect ratio container, which is subjected toparametric excitation. When the density and viscosity of the fluidare small compared to their counterparts in the heavier fluid, anonlinear analysis was found to yield a generalized Cahn–Hilliardequation for the evolution of the fluid interface. This equationrevealed that the stabilizing effect of vibration is like that of sur-face tension and is used to analyze the linear stability of the flatstate, the local bifurcation at the instability threshold, and someglobal existence and stability properties concerning the steadystates without dry spots. A set of equations was derived for thecoupled evolution of the left- and right-traveling surface wavesand the associated mean flow. The viscous mean flow was foundto drastically affect the dynamics of the system and the resultingsurface wave patterns. Funada et al. [84] extended the work ofBenjamin and Ursell [10] to purely irrotational waves on a viscousfluid. Two irrotational theories were presented. The first theorywas based on viscous potential flow in which the effects of viscos-ity enter only through the viscous normal stress term evaluated onthe potential. In the second irrotational theory, a viscous contribu-tion was added to the Bernoulli pressure. The second theory givesrise to the same damped Mathieu equation as the dissipationmethod. The damping term in the second theory was found to betwice the damping rate of the first theory. The growth rates ofunstable disturbances computed by viscous potential flow werefound to be uniformly larger than those computed by the secondtheory. Comparisons with the exact solution and the Rayleigh–Taylor instability revealed that thresholds and growth rates forviscously damped waves are better described by the viscouspotential flow.

Numerical algorithms were proposed in the literature for calcu-lating the damping of liquid sloshing with small amplitudes.These algorithms are based on finite element method [85,86] andthe finite volume schemed based on the volume-of-fluid (VOF)method [87–89]. The simulated results showed that the equivalentdamping coefficients are directly proportional to the liquid viscos-ity. Surfactants, such as oil on water, can cause an increase in thedamping of surface waves [58]. Miles [58] concluded that bothsurface contamination and capillary hysteresis might have contri-bution to the damping surface waves in closed basins. For detailedaccount of damping associated with liquid sloshing, the readermay refer to Chap. 3 in Ref. [14].

5 Influence of Surfactants and Stratified Fluids

Surfactants are compounds, which may lower the interfacialtension between two liquids or between a liquid and a solid. Sur-face contamination has been modeled by phenomenological for-mulae [90–92] based on Marangoni elasticity with insolublesurfactant. It has the effect of dramatically increase the damping

rate of gravity capillary waves as first shown by Dorrestein [90].The effects of insoluble surfactants on the damping rates, naturalfrequencies, and amplitudes of the fundamental axisymmetricFaraday waves in circular cylinders were experimentally and theo-retically studied by Henderson [92]. In particular, Henderson [92]determined the effects of elastic, insoluble films on Faraday wavedynamics and examined the effects of films on the damping ratesof large amplitude waves with low frequency. The water surfacewas contaminated with varying concentrations of surfactants suchas oleyl alcohol, lecithin, diolein, arachidyl alcohol, and choles-terol. It was found that when the surface is saturated with surfac-tant, the damping rates are greatly in excess of both the cleansurface model and inextensible surface model predictions, indicat-ing that, although the quiescent surface had zero elasticity duringwave passage, the surface had a finite elasticity. Measured damp-ing rates were found unaffected by arachidyl alcohol and choles-terol surfactants when the wave amplitudes are large. An analysiswas presented by Nicol�as and Vega [93] who anticipated that sur-face contamination would enhance the generation of the streamingflow produced by the surface wave.

The nonlinear subharmonic resonant wave heights in rectangu-lar tank under parametric excitation were measured by Virniget al. [94] for liquids without and with additives. They comparedthe measured results with those predicted analytically by Gu et al.[95] and Fig. 3(a) shows the dependence the response amplitudeon the excitation frequency for the sloshing mod (1,1) for two ex-citation amplitudes. The results were taken for a rectangular con-tainer of width of 17.78 cm, breadth of 22.86 cm, and liquid depthof 11.43 cm. Only the stable branches of the analytical solutionare shown by solid and dashed–dotted curves. It is interesting toobserve that both experimental and predicted results have goodagreement for higher values of wave amplitudes. At lower waveamplitudes, the experimental results reveal a “tailing” which isrepeated for both cases of increasing and decreasing the excitationfrequency at constant amplitude. The observed tailing effect isattributed to the influence of surface tension. Figure 3(b) shows

Fig. 3 Amplitude–frequency response in a rectangular con-tainer (dimension of 17.78 cm 3 22.86 cm) for mode (1,1) of fluiddepth of 11.43 cm under two different excitation amplitudes:0.9 mm (___ predicted branch, � measured) and 1.55 mm (____ - ____

predicted,�measured). (a) Without additive and (b) with additive[94].



similar set of results after applying a surfactant (Kodak Photo Flo200 solution). It is seen that as the surface tension is reduced the tail-ing effect is eliminated. The inclusion of surface tension in the analy-sis of Gu et al. [95] revealed that the natural frequency of mode (1,1)is reduced when the surface tension is reduced. It is observed thatthere is a critical fluid depth, which separates two nonlinear regimesof the fluid free surface referred to as soft and hard spring character-istics. This depth was originally predicted by Tadjbakhsh and Keller[96] and verified experimentally [97,98] for waves produced by exci-tations in a direction parallel to the free surface.

The dependence of the liquid wave amplitude on the excitationamplitude for constant excitation frequency was obtained by Vir-nig et al. [94] and the results are shown in Fig. 4(a) for four differ-ent excitation frequencies after applying a surfactant. Theanalytical results are seen to be higher than those measured exper-imentally and convergence of both results appears when the exci-tation frequency is close to twice the natural frequency of the firstmode (2f1 ¼ 4:692 Hz). For excitation frequencies less than twicethe natural frequency, the analytical results reveal the occurrenceof a saddle node bifurcation at critical excitation amplitude. Theunstable branch is shown by a dashed curve. The saddle node sig-nifies the occurrence of a jump in the response amplitude and thispoint is governed by the system damping ratio and excitation fre-quency. The amplitude–frequency response is shown in Fig. 4(b)for different values of liquid depth. It is known that below the crit-ical depth, the liquid free surface behaves like a “hard” oscillator,i.e., the amplitude increases with the excitation frequency. Abovethe critical depth, the “softening” characteristic takes place. Forthis case, the predicted critical depth is 7.7 cm, while the experi-mental measurements suggest that it is between 6.6 cm and7.47 cm.

The role of insoluble surfactants on the stability of parametri-cally driven surface waves was studied by Kumar and Matar [99].It was found that in order to obtain time–periodic solutions, whichinvolve Marangoni3 forces, it is necessary to consider the high-P�eclet4 number limit of the surfactant transport equation. Theresults showed that the presence of surfactants raises or lowers thecritical amplitude and wave number depending on the spatialphase shift between the surfactant-concentration variations andsurface deflections. If the concentration variations are in phasewith the surface deflections (maximum concentration at wavecrests), they will drive a Marangoni flow that pulls fluid awayfrom the wave crests, and this will produce larger critical ampli-tude. Similarly, if the concentration variations are out of phasewith the surface deflections (minimum concentration at wavecrests), they will drive a Marangoni flow that pulls fluid towardthe wave crests, and this will produce smaller critical amplitude.For nonzero diffusivities, it was found that disturbances in the sur-face concentration of the surfactant simply decay exponentiallyon a time scale, which is inversely proportional to the surface dif-fusivity of the surfactant. Kumar and Matar [100] considered thethickness of the liquid layer to be much smaller than the wave-length of the interfacial disturbance. It was found that the liquidlayer is unstable to long-wavelength disturbances if it is coveredby surfactants, while it is stable to such disturbances if the surfac-tants are absent. These results were found valid for nonzero sur-factant diffusivities and represent standing wave solutions inwhich Marangoni flows are present. Later, Kumar and Mater[101] found that surfactants can potentially lower the value of thecritical amplitude relative to its value for an uncontaminated freesurface. The critical wave number was found to be an increasingfunction of the Marangoni number.

The effect of surface contamination, modeled by Marangonielasticity with insoluble surfactant and surface viscosity, on driftinstabilities in spatially uniform standing Faraday waves was stud-ied by Mart�ın and Vega [102,103]. The order of magnitude of theMarangoni elasticity was taken as that already obtained byNicol�as and Vega [93] to fit the experimentally measured dampingrate for contaminated water. Surface viscosity is expected to besmall in contaminated water, but it can also be large in other sys-tems. It was found that contamination enhances drift instabilitiesthat lead to various steadily propagating and oscillatory patterns.The elastic effects of an insoluble surfactant on the formation andevolution of 2D Faraday waves were studied numerically by Ubalet al. [104–106]. The numerical results revealed that the interfaceis always subharmonically excited at the onset and that the pres-ence of the surfactant requires a higher external force to inducestanding waves. The magnitude of the external amplitude wasfound to be related to the temporal phase shift that exists betweenthe evolution of the surfactant concentration and the free-surfaceshape.

Giavedoni and Ubal [107] studied the formation of Faradaywaves on the free surface of a liquid layer covered by an insolublesurfactant. The linear analysis included the effects of both surfaceelasticity and surface viscosity. The critical force needed to formthe waves, as well as the critical wave number were determinedwithin a large range of values of the dimensionless parameters

Fig. 4 Dependence of fluid amplitude on excitation (a) ampli-tude and (b) frequency in a rectangular container (dimension of17.78 cm 3 22.86 cm): Solid curves are predicted points andsolid and hollow symbols are measured points [94]. (a)Response amplitude–excitation amplitude for different excita-tion frequencies for mode (1,1) and fluid depth of 11.43 cm and(b) response amplitude-frequency for different values of fluiddepth (h) and excitation amplitude (Z0) for mode (0,1).

3The Marangoni effect is the mass transfer along an interface between two fluidsdue to surface tension gradient. The Marangoni number may be regarded asproportional to (thermal-) surface tension forces divided by viscous forces and isgiven by the expression Ma ¼ ðð�dc=dTÞðLDT=laÞÞ, where L is a characteristiclength (m), l is the dynamic viscosity (kg/(s �m)), DT (in Kelvin K), a ¼ �k=ðqcpÞ isthe thermal diffusivity (m2/s), �k is the thermal conductivity (Watts/m K), and cp isthe specific heat at constant pressure (J/kg K).

4The P�eclet number (Pe) is defined as the ratio of the rate of advection (atransport mechanism of a substance or conserved property by a fluid due to thefluid’s bulk motion) of a physical quantity by the flow to the rate of diffusion ofthe same quantity driven by an appropriate gradient. It is equivalent to the product ofthe Reynolds number and the Prandtl number, i.e., Pe ¼ LU=a ¼ Re � Pr. ThePrandtl number Pr is the ratio of momentum diffusivity (kinematic viscosity) tothermal diffusivity, i.e., Pr ¼ �=a ¼ cpl=�k, where U is the velocity (m/s), D is themass diffusion coefficient (m2/s), and � ¼ l=q is the kinematic viscosity (m2/s).



representing the physicochemical properties of the surfactant.When Marangoni number is small, the largest concentration gradi-ent was formed when the convective transport is largest, i.e.,when the free surface is a horizontal plane. Then, at this instant,the Marangoni traction produces its maximum effect on the inter-facial velocity, slowing the motion of the liquid from the troughto the crest of the wave; therefore, the tangential velocity turns tozero before the free surface attains its maximum deformation. Fortypical values of P�eclet number (Pe), the capillary number5 (Ca),Bond number6 (Bo), and the fluid depth ratio, it was shown thatthe presence of a surfactant always increases the force required todevelop a wavy interface. However, the dependence of the forceon Marangoni number was found nonmonotonic when the surfaceviscosity is negligible and Pe> 1.

Faraday waves in a thin sheet of a viscous fluid subjected to auniform and slow rotation about the vertical axis were studied byMondal and Kumar [108]. The rotation created a Coriolis force,which breaks the mirror symmetry of the flow due to parametri-cally excited surface waves. The Coriolis force was found to delaythe Faraday waves on the free surface. However, the Faraday(subharmonic) waves can always be excited at the onset of surfaceinstability if the excitation frequency is much greater than fourtimes the angular frequency of the rotating body. The surfacewaves were found to be synchronous with the vertical vibration ina thin sheet of viscous fluid. Instability phenomena in a thin layerof a slowly rotating viscous fluid (whose viscosity is approxi-mately ten times the viscosity of water) were observed. The syn-chronous surface waves developed additional local maxima, whenthe excitation acceleration amplitude was raised above certain

value. This was manifested when any point on the free surfacemoves up and attains two different local maxima in one period ofexternal forcing. The wave number of these waves is double thatof the synchronous waves. Superharmonic waves at the instabilityonset were found possible in parametrically forced viscous fluids.Furthermore, different responses, such as subharmonic, harmonic,and superharmonic with different wave numbers, may co-exist forthe same forcing amplitude. This leads to the interesting possibil-ity of a tricritical point as the primary instability.

Figure 5 shows instability regions on the plane of excitation/Galileo ratio, A=G (where A ¼ ah3=�2 and G ¼ gh3=�2, a is theexcitation acceleration amplitude) versus wave number, k. Thefigure shows a tricritical point and superharmonic waves in vis-cous sheet of water–glycerol mixture with Galileo number,G ¼ gh3=� ¼ 2:7� 103, Capillary number, Ca ¼ q�2=ðchÞ ¼ 4:7� 10�4, and excitation frequency ratio - ¼ Xh2=� ¼ 7:5. Theshaded zones and zones bounded by dots are for subharmonic andharmonic regions, respectively. Note that harmonic response goesto a tricritical point for 2xrh

2=�¼ 2.9, plot (c) of Fig. 5, wheresurface waves with wave numbers k1; 1:5k1 and 2k1 with k1 as thewave number of the first harmonic response co-exist at the insta-bility onset. Further increase in rotational rate would lead tosuperharmonic waves as primary instability. Stability of the freesurface of thin sheets of a metallic liquid on a vertically vibratinghot plate, in the presence of a uniform and small rigid body rota-tion about the vertical axis, was studied by Mondal and Kumar[109]. It was found that the inhomogeneity in the surface tensiondue to a uniform thermal gradient across the liquid sheet preferssubharmonic response. On the other hand, it was indicated that therigid body rotation prefers harmonic response at the fluid surface.The competition results in Marangoni and Coriolis forces actingas fine-tuning parameters in the selection of wave numbers corre-sponding to different instability tongues for subharmonic and har-monic responses of the fluid surface. Bicritical points were foundto involve both the solutions oscillating subharmonically,

Fig. 5 Stability zones on the plane of excitation/Galileo ratio A=G and wave number kshowing a tricritical point and superharmonic waves in viscous sheet of water–glycerolmixture with Galileo number G 5 2:73103, capillary number Ca 5 4:7310�4, and excitationfrequency ratio - 5 7:5. The shaded zones and zones bounded by dots are for subharmonicand harmonic, respectively, and for different rotational rates (a) 2xr h2=m 5 0.0, (b) 2.3, (c)2.9, and (d) 4.0 [108].

5The capillary number (Ca) represents the relative effect of viscous forces versussurface tension acting across an interface between two immiscible liquids and isdefined by the expression Ca ¼ lV=c, where V is a characteristic velocity.

6Bond number measures the relative magnitudes of gravitational and capillaryforces and is defined by the expression Bo ¼ qgL2=c.



harmonically, or one oscillating subharmonically and the otherharmonically with respect to the vertical forcing frequency. Theeffect of small Marangoni and Coriolis forces on the onset ofstanding surface waves and bicritical points in a verticallyvibrated sheet of mercury was studied by Mondal and Kumar[109]. When both subharmonic and harmonic (synchronous) sur-face waves can be excited for the same value of forcing excitationamplitude, the corresponding point is referred to as bicritical. Themetallic fluid was subjected to a small thermal gradient and rigidbody forces (Coriolis), which resulted in a variety of bicriticalpoints and the possibility of two different kinds of tricriticalpoints. In the absence of the Marangoni force, a tricritical pointinvolving two harmonic solutions with different wave numbersand a subharmonic solution was reported. On the other hand, inthe presence of both Marangoni and Coriolis forces, it was foundthat a tricritical point involves two subharmonic solutions withdifferent wave numbers and a harmonic solution is possible. Itwas concluded that one may use a Marangoni number and a Corio-lis force as tuning parameters to influence the nature of the multi-critical point at the onset of parametrically forced surface waves.Galileo number, G ¼ gh3=�, was found to have an influence on thethreshold acceleration and critical wave number in molten sodium.

Faraday waves were observed on the free surface of helium-4layer when excited vertically at low temperature ðT ¼ 700 mK,where TðKÞ ¼ TðCÞ þ 273:15) [110]. Standing wave patternswere observed to appear on the surface. Threshold excitationamplitudes for the instability were clearly manifested. The differ-ence in the threshold amplitude between the superfluid and thenormal fluid is that the threshold amplitude is larger for the nor-mal fluid than for the superfluid, and the difference was attributedto the wall damping in the cell geometry. Hysteretic behavior spe-cific to the nonlinear waves was also observed in the superfluidphase. Higuera et al. [32] presented an overview on the small vis-cosity limit and the effects of symmetries and nonlinearity on theresponse of fluid systems. The role of viscous mean flow wasfound to couple the primary surface waves. Under low frequencyexperiments of surface rheology, the importance of fluid dynamicsprovided an explanation of the irreversible character of surfacepressure versus surface area. An experimental study of the Fara-day instability of viscoelastic fluid was presented by Cabeza andRosen [111] who used a shear thinning polymer solution in whichthe elastic effects are predominant. Depending on the fluid layerdepth and the driving frequency, harmonic or subharmonic regimeswere developed. In addition, the onset acceleration was used to esti-mate the rheological properties of the fluid. Ezersky et al. [112]showed experimentally that parametric excitation of capillarywaves in a liquid polymer may give rise to spatially periodic distri-bution of microparticles. It was found that photopolymerizationmakes it possible to fix position of microparticles and producematerials with controllable spatially periodic inhomogeneities.

The parametric excitation of internal 2D waves of a viscouscontinuously stratified fluid, completely filling a rectangular ves-sel under vertical oscillations, was considered in the literature[113–117]. Approximate formulae were obtained for the thresholdamplitude of the vessel excitation and the boundaries of the reso-nance zones. The dynamics of internal gravity waves excited byparametric instability in a stable stratified medium, either at theinterface between water and a kerosene layer, or in brine with auniform gradient of salinity was studied by Benielli and Sommeria[117]. Each internal wave mode was amplified for an excitationfrequency close to twice its natural frequency, when the excitationamplitude is sufficient to overcome viscous damping. In the interfa-cial case, each mode was made well separated from the others in fre-quency and was found to behave like a simple pendulum. The caseof a continuous stratification is more complex as different modeshave overlapping instability tongues. Foster and Craik [118] consid-ered capillary–gravity waves of an inviscid liquid, which exhibit sec-ond- or sub-harmonic resonance at precise frequencies. Equationsdescribing this situation were derived incorporating slight detuningfrom two-wave and Faraday resonances.

The case of two incompressible viscous fluids with differentdensities meet at a planar interface subjected to oscillating accel-eration directed normal to the interface was considered by Jacq-min and Duval [119]. General viscosities and densities for the twofluids were considered but a Boussinesq equal-viscosity approxi-mation was adopted. It was shown that the linear evolution of aperturbation to the interface subjected to an arbitrary oscillatingacceleration is governed by a single integrodifferential equation.Parameter regions of subharmonic, harmonic, and untuned modeswere delineated. The critical 7Stokes–Reynolds number was givenin terms of the surface tension and the difference in density andviscosity between the two fluids. The critical Stokes–Reynoldsnumber and the most unstable perturbation wavelengths werefound to be insensitive to the degree of density and viscosity dif-ferences between the two fluids. The parametric excitation of in-ternal 2D waves of a viscous continuously stratified fluidcompletely filling a rectangular vessel was studied by Kravtsovand Sekerzhi-Zenkovich [120]. The fluid was assumed to have asmall viscosity. Approximate formulae were obtained for thethreshold amplitude of the oscillations of the vessel and the boun-daries of the resonance zones.

The Faraday resonance of interfacial waves in a two-layerweakly viscous system in a rectangular domain was considered inRefs. [121,122]. The scaling of the viscosity results in boundarylayer corrections at the solid walls and at the interface. The damp-ing in the meniscus region, where the interface contacts the sidewalls, was included. As a result of the presence of both destabiliz-ing effects, due to vertical oscillation, and stabilizing effects, dueto viscosity, a threshold condition for instability was derived. Far-aday waves formed on the interface between two immiscibleliquids in a cylindrical cell were studied in Ref. [123]. The effectsof the volume filling ratio on the bifurcation set associated withthe onset of the fundamental axisymmetric mode were examined.In particular, the study considered the subharmonic regime of thecontrol parameter space where the response was significant. Bothsuper- and subcritical bifurcations were uncovered, with hystere-sis in the latter case. The extent of the hysteresis was observed tobe strongly dependent on volume filling ratio, suggesting that non-linear damping effects are influenced by this parameter. At largeexcitation amplitudes, a precessional periodic motion was foundto develop via a Hopf bifurcation. This mode was observed to dis-appear catastrophically at an excitation frequency equal to1.853 6 0.006 times the natural frequency of the resonant mode.The problem of capillary–gravitational Faraday waves on theinterface between fluids was studied in Ref. [124].

Sections 2 through 5 provided an overview of the pertinent fea-tures of parametric instability boundaries of liquid free surfaceand the boundary value problem of incompressible viscous fluidswith surface tension. These sections were very useful to assess thedamping associated with Faraday waves and the influence of con-taminations and surfactants. However, they did not address thenonlinear response characteristics of the free surface under para-metric excitation, which is only predicted by including the nonlin-ear effects of the free-surface boundary conditions as will bedemonstrated in Sec. 6.

6 Nonlinear Parametric Sloshing

In the neighborhood of parametric resonance, the amplitude ofthe free surface grows without limit. As the amplitude increases,geometric nonlinearities (nonlinear inertia) emerge and the motion

7The Stokes number (Stk) is a dimensionless number corresponding to thebehavior of particles suspended in a fluid flow. It is defined as the ratio of thecharacteristic time of a droplet to a characteristic time of the flow and may be givenby the ratio Stk ¼ ~tU0=dc where ~t is the relaxation time of the particle (the timeconstant in the exponential decay of the particle velocity due to drag), U0 is the fluidvelocity of the flow, and dc is the characteristic dimension of the obstacle. Particleswith low Stokes number follow fluid streamlines (perfect advection) whereas forlarge Stokes number, the particle’s inertia dominates so that the particle willcontinue along its initial trajectory.



achieves bounded amplitude. For the case of parametric excitationof liquid free surface, the response exhibits other nonlinear phe-nomena such as nonplanar motion, rotational motion, chaoticmotion, and free-surface disintegration. Kalinichenko andSekerzhi-Zenkovich [125] subdivided Faraday waves into threecategories, namely, regular, irregular, and breaking waves. Theregular Faraday waves include those whose profiles possess eithertime-periodic or symmetrical about vertical planes passingthrough the wave antinodes. The limiting angle at the crest ofsuch waves was estimated to be 80 deg. The waves, which havedisturbed temporal and spatial symmetries but retain the connec-tivity of the oscillating fluid volume, were considered to be irregu-lar. Finally, waves with separate droplets shed from the freesurface of the fluid or with jet launches were assigned to the classof breaking Faraday waves. It was noted that in the case of irregu-lar and breaking waves their height can be estimated only in thestatistical sense, that is, using a quantitative characteristic, such asthe wave steepness, is permissible only for individual wave pro-files. Among the irregular waves are those with a small depressionin the wave crest and periodic triplets. In the case of breakingwaves, the mechanism of jet launch formation on the wave crestwas considered by Kalinichenko [126] who used the experimentalsetup reported by Kalinichenko et al. [127,128]. It was experimen-tally demonstrated that the breaking of standing waves in a rectan-gular reservoir starts with cavity collapse on the wave crest inprocess of formation. It was shown that jet launch from the wavecrest is preceded by the initiation, development, and collapse of acavity. Based on a comparison of the experimental data with ananalytical model, it was suggested that cavity initiation is due tothe nonlinearity of the waves themselves, namely, the presence oftwo small disturbances of the free-surface traveling counter to oneanother and forming a cavity.

The nonlinear behavior of liquid free surface under parametricexcitation for filling depths greater than the free-surface wavelength may be classified into weakly nonlinear and strongly non-linear characterized by breaking waves. These two classes will beaddressed in Secs. 6.1 and 6.2.

6.1 Weakly Nonlinear Parametric Sloshing. The weaklynonlinear dynamic behavior of the nonlinear theory can predictthe steady-state response and uncover complex free-surfacedynamic behavior such as quasi-periodic and chaotic motions.Some controversies were reported in the literature regarding theanalytical description of the sloshing modal equations. For exam-ple, Dodge et al. [19] developed a finite-amplitude analysis for acircular cylindrical container, but their equations of motion for themodal amplitudes were found by Miles [43] to violate reciprocityconditions. Miles [43] rectified this problem by performing somealgebraic manipulation that resulted in nonlinear inertia terms inthe first antisymmetric mode. The amplitude modal equations ofDodge et al. [19] were developed using Laplace’s equation for thevelocity potential function together with the kinematic boundaryconditions on the tank walls and the nonlinear kinematic anddynamic boundary conditions at the free surface. They retainedterms of first-, second-, and third-orders in the amplitude of theprimary mode. The sloshing amplitude equations of the first anti-symmetric, a11; first symmetric, a01; and second symmetric, a21;modes were obtained in the form

€a11þ 1� e�2 cos�s� �

a11 1þK11a211þK01a01�K21a21

� �þ0:034780k2

11 €a11a211þ k11a11 _a2

11þ0:165a11 €a01

�0:198686a11 €a21þ k01 _a01 _a11�k21 _a21 _a11¼ 0 (23a)

€a01 þ k01 tanh k01h 1� e�2 cos �s� �

a01

� a11 €a11 0:12158k01 tanh k01h� 0:198686k211

� �þ a2

11 k01 tanh k01h 0:070796k211 � 0:060741

� �þ 0:263074k2

11� ¼ 0 (23b)

€a21 þ k21 tanh k21h 1� e�2 cos �s� �

a21

þ a11 €a11 0:350807k21 tanh k21h� 0:48267k211

� �þ a2

11 k21 tanh k21h 0:175403� 0:065931k211

� �� 0:48267k2

11� ¼ 0 (23c)

where the modal amplitudes are nondimensionalized with respectto the length 1=ðkmn tanh kmnhÞ, � ¼ X=x11, X is the parametricexcitation frequency, x11 is the natural frequency of the first anti-symmetric sloshing mode, s ¼ x11t is the nondimensional time, his the fluid depth, e ¼ x2

11Z0=g, kmn are the roots of the first deriv-ative of the Bessel function of the first kind, i.e.,d=dr JmðkmnrÞ½ �jr¼R ¼ 0, R is the tank radius, and Kmn and kmn areconstants, which depend on the the fluid depth and kmn and aredocumented in Refs. [19,129].

Miles [43] noted that these equations differ from those derivedusing Lagrangian formulation. The main difference is in the pres-ence of the expression K11a2

11 þ K01a01 � K21a21 in Eq. (23a).The two formulations can be reconciled if the entire Eq. (23a) isdivided throughout by the expression ð1þ K11a2

11 þ K01a01

� K21a21Þ, which is approximated by one for all nonlinear terms,while the first term €a11 is multiplied by the inverse of that expres-sion, which takes the form ð1� K11a2

11 � K01a01 þ K21a21Þ. Thisprocesses results in identical coefficients of the terms €a11a2

11 anda11 _a2

11, and likewise the coefficients of the terms €a11a01, _a11 _a01,€a11a21, and _a11 _a21. Figure 6 shows the experimental measure-ments [19] of the dependence of the liquid-free surface height onexcitation frequency for three different values of excitation ampli-tudes. The measured amplitude was taken at r=R ¼ 0:837 and thewave height was estimated as the one-half the difference betweenthe maximum and minimum liquid-free surface excursions forfluid depth ratio h=R ¼ 2. The figure shows the points at whichthe response amplitude jumps and collapses depending on whetherthe excitation frequency increases or decreases. The characteris-tics of the response are belonging to soft spring nonlinearity.

Another finite-amplitude analysis for a circular cylindrical tankwas developed in Ref. [130]. The free-surface boundary condi-tions were applied at the equilibrium position of the free surfaceand a correction to the resonant frequency was obtained in termsof first-order in the amplitude, which was noted by Miles [43] to

Fig. 6 Experimental measurement of liquid response ampli-tude of sloshing mode (1,1) as a function of excitation fre-quency for 1/2 subharmonic response for different values ofexcitation amplitude: D : Z0 5 0:65 mm, �: Z0 5 1:31 mm, and w:Z0 5 2:18 mm [19]



be of second-order. Parametric nonlinear excitation vibration ofthe liquid surface in a partially filled circular cylindrical tank wasstudied in Refs. [131] and [132]. The analysis was extended byTakahara and Kimura [133] to the case of 3D nonlinear liquidmotion in a rigid rectangular tank partially filled with liquid underpitching excitation. It was noted that the pitching excitation alsocauses the parametric excitation when the pitching axis does notintersect the symmetrical axis of the circular cylindrical tank. Thetime histories of the liquid surface displacement to the harmonicvertical excitations were calculated for the case of axisymmetricmode possessing nodal radii. Dynamical behavior of parametri-cally excited solitary waves in Faraday’s water trough was studiedby Wang et al. [134]. The endless collision of two solitary wavesof like polarity, and the periodical reflection and attraction of asolitary wave by the boundary (one end of the water trough) wereanalyzed.

The fluid free surface changes dramatically when the shape ofthe container is deviated from circular to ellipsoidal as pointed outby Higuera et al. [135]. In this case, the mean flow is coupled withthe amplitudes of surface waves as well as the spatial phases ofthe resulting pattern. The ellipticity was found to break the rota-tional symmetry of the primary waves and would select two stand-ing oscillations with nodes along either the major or the minoraxis of the ellipse. In Faraday waves, coupled amplitude meanflow equations were derived for patterns consisting of multimodeplane waves as reported by Higuera et al. [136] who predictedcoupling between the mean flows and patterns when the patternssatisfy some symmetry requirements. In a fluid with a free surfacebeing vibrated vertically, surface waves are excited when thevibration amplitude exceeds a frequency-dependent critical value.Subharmonic standing waves are known to have the lowest thresh-old driving amplitudes, i.e., the wave frequency being half of thedriving frequency. By using suitable cell geometry, Chen [137]generated waves with curved wave fronts. It was argued that thesepredictions should also be true in Faraday waves. Chen [137]measured the mean flow velocities driven by curved rolls in a pat-tern formation system. Curved rolls in Faraday waves were gener-ated in experimental cells consisting of channels with varyingwidths. The mean flow magnitudes were found to scale linearlywith roll curvatures and squares of wave amplitudes, agreeingwith the prediction from the analysis of phase dynamics expan-sion. The effects of the mean flows on reducing roll curvatureswere also reported.

Experimental investigation of Faraday crispation was reportedby Cabeza et al. [138] who measured the fluid surface displace-ment. The results revealed the period-doubling cascade with atleast two bifurcations and an incoherent signal. In other studies,Cabeza et al. [139,140] obtained the numerical reconstruction ofthe patterns appearing on the surface and the dynamical evolutionto chaos of a localized point of the surface. The classic evolutionscenario of the Faraday instability system was modified throughcontrolling the height of the fluid layer, in a regime of highly dis-sipative fluid. It was possible to create a periodicity window,yielding within it a global pattern, roll-type structures, very stable,and with a subharmonic temporal behavior. In addition, Cabezaet al. [140] observed the generation of strongly localized solitarywaves, which propagate over these structures that are destroyedonly when they reach the border of the trough.

Parametric excitation of a narrow rectangular container (crosssection of 600 mm� 60 mm and fluid depth of 300 mm) was con-sidered by Jiang et al. [141] who observed mild to steep standingwaves of the fundamental mode over an excitation frequencyrange of 3.15 Hz to 3.34 Hz. These standing waves were alsosimulated by a 2D spectral Cauchy integral code. The experimen-tal results showed that contact line effects increase the viscousnatural frequency and alter the neutral stability curves. The addi-tion of a wetting agent, such as Photo Flo, was found to signifi-cantly change the stability curve and the amplitude–frequencyresponse of the free surface. Figure 7 shows the free-surfaceamplitude–frequency responses for three different values of

excitation amplitude: 2.5 mm, 3.0 mm, and 3.5 mm. The shownarrows indicate that the response of amplitude was obtained whenthe excitation frequency is changed according to the arrow direc-tion. Solid/hollow symbols represent the experiments withincreasing/ decreasing frequency. For increasing forcing fre-quency, the response curves (dashed lines) exhibit jump from zeroto bounded amplitude. The frequency ratio, X=2xn, where thejump occurs is seen to be larger for smaller forcing amplitude:1.002 for 2.5 mm, 0.992 for 3.0 mm, and 0.982 for 3.5 mm excita-tion amplitude.

Figure 8(a) shows the amplitude–frequency response plots fortreated water, while Fig. 8(b) is for treated water with Photo Flo.Figure 8(a) includes the analytical response based on Hendersonand Miles [67] theory with excitation amplitude of 2.5 mm. It isseen that the addition of Photo Flo resulted in a significant changein the jump position as shown in Fig. 8(b). Strong modulation inthe wave amplitude for some forcing frequencies higher than 3.30Hz was observed experimentally. Over the frequency ratio rangeof 1.02–1.05, large modulations were observed in the time historyrecord of the free-surface wave amplitude as shown in Fig. 9 fortwo different excitation frequencies and amplitudes. It is seen thatthe magnitude of the modulation for higher excitation frequencyof 3.32 Hz is more significant with a modulation frequency of0.03 Hz. No modulation was predicted [141] by using nonlinearnumerical simulations, even with higher harmonics in the sinusoi-dal excitation. However, when the experimental forcing signalwas used as the numerical input, modulation was predicted, albeitwith less magnitude. Since no contact line effect was simulated, itwas concluded that the modulation may be caused by contact lineeffect and sideband noise in the excitation signal.

The Faraday waves were revisited by Craik [142,143] and Craikand Armitage [144] who were able to describe the wavelengthselection and hysteresis observed experimentally. Their experi-mental investigation dealt with surface gravity–capillary waves ina rectangular tank undergoing small vertical oscillations. Thelarge aspect ratio of the tank enabled them to study the behaviorof several neighboring 2D wave modes; their onset, hysteresis andinstability. Two water depths, close to one and two centimeters,were investigated in detail. Hysteresis below the minimum forcingfor linear wave-onset was found to imply that the excitation andnonlinear damping are both significant as confirmed by Miles[145]. Jiang et al. [141] performed boundary-integral simulations

Fig. 7 Experimental Faraday wave amplitudes dependence onexcitation frequency. Solid symbols represent frequencies var-ied in small steps from low to high frequency at constant forc-ing amplitude. Hollow symbols represent frequency variationsin the opposite direction. Three forcing amplitudes are used:2.5 mm (� and w), 3.0 mm (� and D), 3.5 mm (� and �), and3.5 mm. Solid/dashed lines with arrows represent excited Fara-day waves with decreasing/increasing frequency [141].



of 2D motion to determine the highly nonlinear motion of aninterface with finite- and large-amplitude deformations. Their ex-perimental investigation revealed that contact line effects increasethe viscous natural frequency and alter the neutral stability curves.Furthermore, strong modulations in the wave amplitude for someforcing frequencies higher than 3.30 Hz were observed. Reducingcontact line effects by Photo Flo addition suppresses thesemodulations.

One or more surface wave solitons8 with polar-on-like behaviorwere reported by Wu et al. [146] in partially filled container para-metrically driven at an appropriate frequency and amplitude cre-ated. An amplitude equation in the form of a perturbed nonlinearSchr€odinger equation was derived for parametric excitation of sur-face waves in an extended system by Elphick and Meron [147].The existence of a stable nonpropagating kink solution was pre-dicted. In addition, a stable nonpropagating soliton solution was

found for subcritical excitation. Lioubashevski et al. [148] pre-sented an experimental study of the onset of the Faraday instabil-ity in highly dissipative fluids. It was found that the criticalacceleration for the transition to parametrically excited surfacewaves scales as a function of two dimensionless parameters corre-sponding to the ratios of the critical driving amplitude height andviscous boundary layer depth to the fluid depth. This scaling,which exists over a wide range of fluid parameters, was found toidentify the proper characteristic scales and indicates that aRayleigh–Taylor type mechanism drives the instability in this re-gime. An exact equation, which is nonlocal in time for the linearevolution of the surface of a viscous fluid, was derived by Cerdaand Tirapegui [81]. It was shown that this equation becomes localand of second-order was used to study Faraday’s instability in astrongly dissipative regime.

The mechanism responsible for the presence of complex dy-namics in the damped nonlocal parametrically forced nonlinearSchr€odinger equation was examined by Higuera et al. [149]. Theevolution equations take the form of a pair of damped parametri-cally driven nonlinear Schr€odinger equations9 with nonlocal cou-pling [150,151]. As the strength of the applied excitationincreases, this equation undergoes a sequence of transitions tochaotic dynamics. The origin of these transitions is linked to thepresence of heteroclinic connections between the trivial state andspatially periodic standing waves. These connections are associ-ated with cascades of gluing and symmetry-switching bifurca-tions. The dynamics near the minima of the resulting resonancetongues were described by a system of coupled nonlocalSchr€odinger equations with damping and parametric forcing byVega and Knobloch [152]. Near the bicritical points where twoadjacent resonance tongues intersect, a pair of coupled dampedcomplex Duffing equations captures the properties of both pureand mixed modes, and of the periodic solutions resulting from aHopf bifurcation on the branch of mixed modes. The coupled sys-tem of equations describing fast surface oscillations and slowlyevolving mean flows in three dimensions was developed by Vega

Fig. 9 Time history records of two cases of steadily modulatedFaraday waves [141]. (a) Excitation frequency of 3.30 Hz andexcitation amplitude of 2.90 mm. (b) Excitation frequency of3.32 Hz and excitation amplitude of 2.65 mm. Measurementswere taken at the horizontal center of the tank [141].

Fig. 8 Comparison of prediction and experimental measurements of dependence of Far-aday wave amplitude on the excitation frequency ratio at an excitation amplitude of2.5 mm: ...... predicted using Henderson and Miles [67] theory; _____ numerical results.Symbols are experimental results [141]. (a) The fluid is treated water: � and w measuredresults with increasing and decreasing frequencies, respectively. (b) The fluid is treatedwater with Photo Flo in a ratio of 100:1. • and � represent measured results with increas-ing and decreasing frequencies, respectively.

8Soliton is a wave that maintains its shape while it travels at constant speed.Solitons are caused by a cancellation of nonlinear and dispersive effects in which thespeed of the waves varies according to frequency. Dispersion and nonlinearity caninteract to produce permanent and localized wave forms. Solitions are of permanentform and are localized within a region.

9In quantum mechanics, the analogue of Newton’s law is Schr€odinger’s equationfor a quantum system (usually atoms, molecules, and subatomic particles whetherfree, bound, or localized). In general, it is a linear partial differential equation, whichmay describe the wave function of the system, also called the quantum state or statevector.



et al. [153]. A phenomenological model of parametric surfacewaves was introduced in the limit of small viscous dissipation thataccounts for the coupling between surface motion and slowlyvarying streaming and large-scale flows. The analysis was limitedto the simplest regular pattern consisting of stripes. Mean flowswere induced by perturbation of the stripes, and their coupling tothe order parameter equation affected the stability of stripe solu-tions. The results for the secondary instabilities of the primarywave revealed that the mean flow would lead to a weak destabili-zation of the base state and introduced a longitudinal oscillatoryinstability.

The dynamics of parametrically driven counterpropagatingwaves in a one-dimensional extended nearly conservative annularsystem were examined by Martel et al. [154]. The waves weredescribed by two coupled damped parametrically driven nonlinearSchr€odinger equations with opposite transport terms due to thegroup velocity and small dispersion. The system was character-ized by two length scales defined by a balance between forcingand dispersion (the dispersive scale), and forcing and advection atthe group velocity (the transport scale). The weakly nonlinearevolution of Faraday waves of a vertically excited annular con-tainer was studied by Mart�ın and Vega [155]. In the small viscos-ity limit, the evolution of the surface waves was coupled to anonoscillatory mean flow that develops in the bulk of the con-tainer. A system of equations for the coupled slow evolution ofthe spatial phase of the surface wave and the streaming flow wasnumerically integrated to show that the simplest reflection sym-metric steady-state becomes unstable for realistic values of the pa-rameters. The spatiotemporal evolution of the surface in theasymptotic regime was obtained by Rojas et al. [156]. The ampli-tude of the surface deformation was saturated to a finite value, dueto the nonlinear terms. At very low Reynolds number, the tempo-ral dynamics were found to be strongly nonlinear. The bifurcationwas found to be supercritical.

The standing Faraday wave-trains that appear near threshold ina nearly conservative, parametrically excited system were studiedby Mancebo and Vega [157]. Sufficiently close to threshold, therelevant equation whose cubic coefficient is extremely sensitive towave number shifts, which can only be understood in the contextof a more general quintic equation that also includes two cubicterms involving the spatial derivative. A weakly nonlinear analy-sis of one-dimensional viscous Faraday waves in 2D large aspectratio containers was presented by Mancebo and Vega [158]. Thesurface wave was found to be coupled with a viscous long-wavemean flow.

Under parametric excitation, whose frequency is a slowly time-dependent “chirped,” i.e., XðtÞ ¼ X� lt, where l is the constantchirp rate, Assaf and Meerson [159] showed that, when passingthrough resonance, resonance always occurs when the chirp rate issufficiently small. The critical chirp rate, above which breakdownof autoresonance occurs, was found for different initial conditions.The theory of parametric autoresonance predicts that a downwardchirp of the vibration frequency should cause persistent wavegrowth, which is only expected to terminate at large amplitudes,when an underlying constant frequency system ceases to exhibit anontrivial stable fixed point [159]. Ben-David et al. [160] pre-sented experimental verification of parametric autoresonance ex-citation of a nonlinear wave. They demonstrated thatautoresonance is not hindered by moderate dissipation and thepredicted (negative) frequency chirp indeed drives persistentwave growth, via the parametric autoresonance mechanism, toamplitudes that surpass the theory’s region of validity.

6.2 Breaking Surface Waves. A surface wave whose ampli-tude reaches a critical level at which some process can suddenlystart to cause large amount of wave energy to be transformed intoturbulent kinetic energy is known as a breaking wave. Duringbreaking, a deformation in the form of a bulge forms at the wavecrest. High-frequency detail was found to be present in a breaking

wave and to cause crest deformation and destabilization. After thetip of the wave overturns and the jet collapses, it creates a verycoherent and defined horizontal vertex. The main vortex along thefront of the wave diffuses rapidly into the interior of the wave af-ter breaking, as eddies on the surface become more viscous.Advection (transport mechanism of a substance or property by afluid due to the fluid’s bulk motion) and molecular diffusion playa part in stretching the vortex and redistributing the vorticity, aswell as the formation turbulence cascades.

Under relatively high-frequency parametric excitation, largeamplitude surface waves were observed in Refs. [161–164]. Thismotion usually occurs in the form of typical spray-formed waves.Sometimes the surface motion becomes very violent at larger ex-citation levels with small vapor bubbles entrained in the liquid.The bubbles can become negatively buoyant and sink to the tankbottom. Analytical and experimental studies to examine the occur-rence of surface disintegration with liquid particles spray and thewave characteristics of spray-excited low-frequency waves werereported in Refs. [129,165–167]. The surface disintegration andbubble formation in a vertically excited liquid column were theo-retically and experimentally investigated by Buchanon et al.[168–180]. These studies showed that the smaller fluid height thelarger is the minimum excitation amplitude required for the sur-face disintegration. It was also shown that the frequency of aspray-excited low-frequency wave is independent of the liquidheight-to-diameter ratio.

Steep standing waves were found to undergo a surprising trans-formation at even large parametric excitation amplitude [181].Small plunging breakers first appear to each side of the dimpledcrest. A further increase in excitation amplitude would lead even-tually to period tripling with breaking every two out of threewaves. The existence regime for period tripling is shown in Fig.10. It is seen that the measured neutral-stability curve is shifteddownward due to contact line effects [141]. All experimentsincluding those with period-tripled breaking were found to be lim-ited to within the neutral-stability curve. Jiang et al. [181,182]showed experimentally that increasing the excitation amplitudefurther leads to breaking waves in three recurrent modes (periodtripling): sharp crest with breaking, dimpled, or flat crest withbreaking, and round crest without breaking. Interesting steepwaveforms and period-tripled breaking were found to be relateddirectly to the nonlinear interaction between the fundamentalmode and the second temporal harmonic. The period-tripledbreaking consists of three distinct modes: A, B, and C, as illus-trated in Fig. 11. The maximum wave profile in mode A is seen tobe characterized by its high elevation, sharp crest angle, violentbreaking, and drop formation. Mode B follows with a dimpled orflat crest and double plungers to the sides of the crest. Mode C hasa round (nonbreaking) crest similar to Penney and Price’s [38] so-lution. The sharp-crested mode A reappears after mode C, forminga recurrent cycle with a three-wave period. The mode A wave wasnever realized experimentally on steep nonbreaking waves.

Fig. 10 Stability boundary shows regions of various wave-forms in Faraday wave experiments [181]



Although the first sharp crest forms an upward jet in mode A, itcan also develop into a large plunger with its crest listing to oneside. The appearance of an asymmetric plunger is random anddoes not affect the triple periodicity. A leftward plunging breakershown in Fig. 12(a) was captured experimentally with both seedparticles and dye in the water so that both the free surface and theunderlying water are illuminated. Jiang et al. [181] demonstratedmode-B breaking in Fig. 12(b). Double plungers were formed ateach side of the flat crest and create local bores (0.02 and 0.04 s).These postbreaking plungers slide down the wave crest at 0.06and 0.08 s, creating irregular motions near the surface. The bul-bous centre at 0.04 s to 0.08 s was found to be caused by therebounding jet initiated in the previous part of the wave cycle.The bright spots beneath the wave crest (the last four frames)were entrained air bubbles. Mode C has the least breaking andusually no irregular surface motions were observed as shown inFig. 12(c).

The method of multiple scales to study the nonlinear Faradaywaves in a circular container partially filled with inviscid fluidwas employed in Refs. [183,184]. It was shown that different free-surface standing wave patterns can be formed at different valuesof excitation frequency and amplitude. At low excitation fre-quency, the effect of surface tension on mode selection of surfacewave was found to be not significant. However, at high excitationfrequency, the influence of surface tension is significant. Based onweakly viscous fluids assumption, the fluid field was divided intoan outer potential flow region and an inner boundary layer region.A linear amplitude equation of slowly varying complex ampli-tude, which incorporates damping term and external excitation,was derived from linearized Navier–Stokes equation. The resultsrevealed that when the forcing frequency is low, the viscosity ofthe fluid is prominent for the mode selection. However, when theforcing frequency is high, the surface tension of the fluid isprominent.

Steep forced waves generated by moving a tank containingwater were studied experimentally and numerically by Bredmoseet al. [185]. The most unusual feature was type-A table-topbreaker in the form of a flat-topped wave crest with almost verti-cal sides was observed. A sequence of video snapshots shown inFig. 13 reveal the generation of sharp-crested waves followed byflat-topped waves in a water tank of water depth of 400 mm. Theprocess of steepening of successive wave crests can be observedby comparing the first two frames of Fig. 13. Table-top crestswere predicted numerically by Topliss [186] using a Cauchyboundary-integral method for steep, unsteady 2D waves [187,188]with high-energy initial conditions. Figure 14 shows a set of

surface wave profiles at the time of maximum elevation for a setof systematically differing initial conditions. These energeticwaves are characterized by sustaining a flat top through the riseand fall of the crest. For some of the table-top breakers, a thinningof the profile during the downward motion was observed.

Under parametrically excited surface waves, the breaking stateejects droplets from wave peaks, when the applied forcingexceeds an acceleration threshold. The rate of breaking eventsapproaches zero gradually with decreasing acceleration. Twoproperties of these ejections were studied around the ejectionthreshold [189]. Analysis of the ejection rate dependence onacceleration allowed the determination of the ejection thresholdand an inference about the wave height distribution. A Poissondistribution was found for the times between ejections. Cabezaet al. [190] studied parametrically excited surface waves when fin-ger structures are generated on the free surface. The intermediatestates between regular patterns and droplet ejection in Faradayinstability were also analyzed. It was found that the surface breaksand ejects droplets from wave peaks when the applied force

Fig. 11 (a) Profile of three different modes during period-tripled breaking. T is the temporal wave period before period tri-pling (twice the forcing period). Each crest feature appears atthe end-walls of the tank 1:5 T after it appears at the centerline[181]. (b) Images showing the wave-amplitude modulationsperiod tripling scenario [197].

Fig. 12 Experimental snapshots of three breaking modes withexcitation amplitude F 5 4:60 mm and f 5 1.60 Hz. Time (unit: s)is shown in each frame. (a) Mode A with “leftward” plungingbreaker, (b) double plunger to each side of the dimpled crest inmode B, and (c) Mode C with maximum elevation at t 5 0:04 s[181].



exceeds an acceleration threshold. Both chaotic and turbulentbifurcation behaviors of Faraday surface waves were studied andmany different transitions were reported. The spatial and temporalfinger distribution was described in terms of external acceleration.Kahan et al. [191] used proper orthogonal decomposition to char-acterize the evolution of fingers regime in Faraday instability. Thestructural transition was analyzed for varying accelerationamplitude.

On a vertically vibrating fluid interface, a droplet can be indefi-nitely bouncing. When approaching the Faraday instability onset,the droplet was found to couple with the wave and starts to propa-gate horizontally. The resulting wave–particle association(referred as a walker) was shown to have remarkable dynamical

properties, reminiscent of quantum behaviors [192,193]. The na-ture of a walker’s wave field was investigated experimentally,numerically, and theoretically by Eddi et al. [194] who showedthat the walker field results from the superposition of waves emit-ted by the droplet collisions with the interface. It was shown thateach shock emits a radial traveling wave, leaving behind a local-ized mode of slowly decaying Faraday standing waves. As itmoves, the walker keeps generating waves and the global struc-ture of the wave field results from the linear superposition of thewaves generated along the trajectory.

Surface singularities produced by the collapsing depressions ofstanding waves were examined in Ref. [195]. This phenomenonwas first observed by Longuet-Higgins [196]. Under parametricexcitation of a cylindrical tank partially filled with a fluid, it wasobserved that below a critical standing wave height gc the fluidsurface topology remains smooth and simply connected. Abovegc, the collapsing wave entrains an air bubble beneath the surfaceand changes its topology from simply to multiple-connected.Thus, the critical height represents the threshold of the surface to-pology change. The resulting inertial collapse creates a singularityon the fluid surface at which the velocity and surface curvaturediverge. This singularity was found to localize the kinetic energyof the fluid along the central axis and produces a narrow, high-speed vertical jet as shown in Fig. 15. In the model of Zeff et al.[195], the jet velocity was found to be proportional to the squareroot of the kinematic surface tension. Their model was reconsid-ered by Das and Hopfinger [197] who presented some results per-taining to parametrically forced gravity waves in a circularcylinder in the limit of large fluid-depth approximation. The insta-bility boundary in terms of the excitation amplitude ratio Z0=Rand excitation frequency ratio X=ð2x01Þ, where Z0 is the excita-tion amplitude, R is the tank radius, X is the excitation frequency,and x01 is the first damping axisymmetric sloshing mode fre-quency is shown in Fig. 16 for Fluorocarbon (FC-72) fluid. It isseen that the axisymmetric mode exists in the range

Fig. 13 Time sequence of snapshots taken video camera (25 frames per second) for freesurface under vertical excitation shaking [185]

Fig. 14 Surface profiles of “standing waves” at maximum ele-vation for a sequence of different initial amplitudes for a cosinesurface velocity distribution and initial uniform water depth ofh 5 1 and L 5 2p (Refs. [185,186])



0:94 < X=2x01 < 1:02. The solid curve corresponds to the theo-retical prediction with damping ratio f ¼ 0:0022. This curve isgiven by the expression

Z0

R¼ 1

3:832 tanhð3:832h=RÞ f2 þðX=2Þ2 � x2

01

h i2

ð2x201Þ

2

264

375 (24)

where h is fluid depth in the tank, and the detuning parameter

is b ¼ ðX=2Þ2 � x201

h i=ð2ex2

01Þ, and forcing parameter

�e ¼ Z0k01 tanhðk01hÞ. At X=2x01 ¼ 1:02, b4 ¼ 0:875, the wavemotion was found to bifurcate to the wave mode (3,1) of dimen-sionless wave number k31R ¼ 4:201. In the range0:94 < X=2x01 < 0:974, b1 b > b2, where b1 ¼ �0:962 andb2 ¼ �1:015; the wave motion is unstable when the parametricinstability threshold is crossed with exponential growth in waveamplitude up to breaking (inset image 1 in Fig. 16) with possiblejet formation. Over the interval 0:974 � X=2x01 � 0:987 andb2 � b � b3, where b3 ¼ �0:753, the wave motion is unstableabove the instability threshold but the amplitude remains finite(inset image 2 in Fig. 16). In the range 0:987 < X=2x01 � 1:02,the instability is subcritical for X=2x01 < 1 and supercritical atX=2x01 ¼ 1 and above.

The period tripling event at excitation amplitude ratioZ0=R ¼ 0:008 is shown in Fig. 11(b). It is seen that the wave creststeepens, is flat-topped in the next period, and then takes interme-diate amplitude before the cycle begins again. For three differentliquids, Fig. 17 shows typical shapes of the last stable wave crestsas well as the cavities, which form half a period later at the wavetrough. Clearly, surface tension and viscosity affect the shape ofthe wave crests and consequently the cavity size by inhibiting par-asitic capillary waves and by preventing Taylor instability fromdeveloping. Figure 17(a) reveals that the large viscosity of glycer-in–water solution inhibits parasitic capillary waves and Taylorinstability. The cavity, which is formed (lower part of image)when the wave column is accelerated downward, and impinges onthe wave trough, is deeper and of smaller size than in water andFC-72. Its aspect ratio (cavity depth, ‘, to cavity radius, rca) is‘=rca 1. The cavity size shown in the images corresponds to theinstant when the cavity just begins to contract. When the kine-matic surface tension is small as in FC-72, shown in Figs. 17(d)and 17(e), a cylindrical precursor fluid column is projectedupward (Taylor instability) that is then partially or completelytaken over by the following wave crest; drop pinch-off may occurin some cases, as shown in Fig. 17(d), or the wave crest may beflat-topped, as shown in Fig. 17(e). Figures 17(b) and 17(c) belongto water in which upward projection of a cylindrical fluid columnalso occurred, but is less pronounced because of higher surfacetension. Parasitic capillary waves are clearly present but becauseof the high surface tension, the wave crest remains axisymmetric.The wave shape shown in Fig. 17(f) is for FC-72 and containerradius R¼ 2.5 cm resulting in a lower Bond number and, hence, asmoother wave crest.

The Faraday waves’ instability was studied in domains withflexible boundaries implemented by triggering this instability infloating fluid drops in Ref. [198]. An interaction of Faraday waveswith the shape of the drop was observed, the radiation pressure ofthe waves exerted a force on the surface tension held boundaries.Two regimes were observed. These are co-adaptation of the wavepattern with the shape of the domain so that a steady configurationis reached, and the radiation pressure dominates and no steadyregime is reached. The drop stretches and ultimately breaks intosmaller domains that have a complex dynamics including sponta-neous propagation. For the second regime, Fig. 18 shows the dis-placement of the boundary due to the waves did not lead toequilibrium. The formation of parallel standing waves results intoa constantly increasing elongation of the drop into a snakelikestructure which breaks into fragments having a large variety ofdynamical behaviors. Figure 18 was obtained with ethanol of

Fig. 15 Surface wave collapses and resulting jets in a cylindri-cal tank with an inner diameter of 12.7 cm filled to a depth of6.5 cm under vertical excitation: (a) Snapshot showing the col-lapse of a surface wave depression and the subsequent upwardjet caused by self-focusing of the kinetic energy associatedwith a near singularity under excitation frequency of 7.84 Hzand excitation amplitude of 2.39 m/s2, which was increased to3.23 m/s2. Once the standing wave is sufficiently tall, it pro-duces a deep depression which collapses to a singularity andjet. (b) Snapshot of fluid of viscosity of 0.26 cm2/s. The drivingfrequency and acceleration amplitude are 8.00 Hz and 4.64 m/s2,respectively [195].

Fig. 16 Instability boundary for Fluorocarbon (FC-72) fluid.The solid curve corresponds to the theoretical curve withdamping ratio f ¼ 0:0022. Bifurcation points are bifurcation tomode (2,1), detuning parameter, b1 5 20:962; a excitation fre-quency ratio, X=2x01 5 0:94; wave-breaking bifurcation,b2 5 21:015,X=2x01 5 0:974; unsteady wave motion,b3 5 20:753, X=2x01 5 0:987; bifurcation to mode (3,1),b4 5 0:875, X=2x01 5 1:02. - - -: experimental wave breakingthreshold and � � �: onset of unstable wave motion [197].



volume of 1:060:02 ml, density q2 ¼ 789 kg=m3; dynamic vis-

cosity l2 ¼ 0:9� 10�3 Pa � s; and surface tension c2 ¼ 23 mN=mfloating on silicon oil of corresponding properties q1

¼ 965 kg=m3; l1 ¼ 100� 10�3 Pa � s; and c1 ¼ 20 mN=m. The

measured interfacial tension was c1;2 ¼ 0:7 mN=m. The maincharacteristic is that ethanol is wetted by silicon oil. At rest, an oilfilm was observed to cover the upper surface of the drop as shownin Fig. 18(a). The instability in ethanol appears in a subcriticalway; i.e., large amplitude waves form at threshold. These wavesstretch the drop as shown in Fig. 18(c) but there is no convergencetoward a final stable shape. The elongation is followed by buck-ling as shown Fig. 18(d) then breaking into several fragments asdemonstrated in Fig. 18(e). Kiyashko et al. [199] experimentallyinvestigated the dynamics of defects in the domain walls that ariseat the interface of two domains under a parametric excitation ofcapillary waves on the fluid free surface. It was shown that in asymmetric domain wall defects move strictly along it. It wasfound that a single defect located near the domain wall can attractdefects of domain wall and embedded in it.

This section has addressed both weakly nonlinear and breakingFaraday waves. The occurrence of each type depends on the fluidand excitation parameters. However, when the natural frequenciesof different modes are related to each other through nonlinear cou-pling of the modal equations of motion, the problem becomescomplicated as the sloshing modes are sharing energy and the free

surface becomes unsteady. This problem will be considered inSec. 7.

7 Parametric Excitation Under Internal Resonance

Weak resonant coupling among gravity waves on the surface ofdeep water can cause significant energy transfer among wave-trains. The nonlinear coupling may give rise to the occurrence ofinternal resonance conditions among the interacting modes. Inter-nal resonance implies the presence of a linear algebraic relation-ship among the natural frequencies of the sloshing modes in theform

Pnj¼1 sjxj ¼ 0, where sj are integers, and xj are the natural

frequencies of the coupled modes, the number S ¼Pn

j¼1 j sj j isknown as the order of internal resonance. The problem of internalresonances in nonlinearly coupled oscillators is of interest in con-nection with redistribution of energy among the various naturalmodes. The coupling among these modes plays a crucial role insuch interactions. In a straightforward perturbation theory, inter-nal resonances lead to the problem of small divisors.

Experimental investigations conducted by McGoldrick [27] andLonguet-Higgins and Smith [200] demonstrated the presence ofresonant interaction among surface waves. McGoldrick [27]showed that both second-harmonic and triadic resonances are pos-sible for deep-water gravity–capillary waves. Hammack andHenderson [201] presented an overview of resonant interaction

Fig. 17 Snapshots of free surface ((a) and (c)–(e)) obtained in container R 5 5 cm andimage (f) obtained in container R 5 2:5 cm, showing the last stable wave amplitude andthe following (half a period later) wave depression (cavity) below, for glycerin–water (a),water (c), and FC-72 ((d) and (e)). (b) The wave crest for water, with parasitic capillarywaves, taken at 0.186 wave period before the maximum wave amplitude is reached,shown in (c); (a) X=2p 5 8:85 Hz, excitation amplitude ratio Z0=R 5 0:0368; (c)X=2p 5 8:86 Hz, Z0=R 5 0:0154; (d) X=2p 5 8:77 Hz, Z0=R 5 0:0179; (e) X=2p 5 8:35 Hz,Z0=R 5 0:0143, (f) R 5 2.5 cm, FC-72, X=2p 5 11:65 Hz, Z0=R 5 0:0175 [197].



theories and experiments for waves on the surface of a deep layerof water. Pattern formation in parametric surface waves was stud-ied in the limit of weak viscous dissipation in Refs. [202,203]. Amultiscale expansion of the quasi-potential equations revealed theimportance of triad resonant interactions, and the saturating effectof the driving force leading to a gradient amplitude equation. Min-imization of the associated Lyapunov function was found to yieldstanding wave patterns of square symmetry for capillary waves,hexagonal patterns, and a sequence of quasi-patterns (QPs) formixed capillary–gravity waves.

Nonlinear modal interaction was also studied for the case ofspatiotemporal resonant triads in a horizontally unbounded fluiddomain. The interaction takes place such that each critical wavenumber from linear analysis actually corresponds to a circle ofcritical wave-vectors. It has been argued that resonant triads mayplay a central role in the Faraday’s wave pattern selection problemas indicated in Refs. [204–208]. Resonant triads comprised threecritical wave-vectors that sum to zero, i.e., k3 ¼ k16k2, wherek1j j ¼ k2j j is the wave number of one critical mode and k3j j is the

wave number of the other critical mode. The bifurcation problemwas formulated using a stroboscopic map and estimation of Flo-quet multipliers [209–214]. It was shown that there is a fundamen-tal difference in the pattern selection problems for subharmonicand harmonic instabilities near the 10codimension-two point.Many experimental [204,215] and theoretical [208,216] studiesattributed the formation of exotic patterns near the codimension-two (or “bicritical”) point to resonant triad interactions involvingthe critical or near-critical modes with different spatial wave num-bers. The approximate (broken) symmetries of time translation,time reversal, and Hamiltonian structure were utilized by Porter

and Sliber [217] who obtained general scaling laws governing theprocess of pattern formation in weakly damped Faraday waves.For the case of two-frequency forcing, it was found that thestrength of observed three-wave interactions depends on the fre-quency ratio and on the relative phase of the two driving terms.Porter and Sliber [218] and Porter et al. [219] considered two-frequency forcing with an emphasis on the resonant triad occur-ring near the bicritical point where two pattern-forming modeswith distinct wave numbers onset simultaneously. This triad wasobserved in the form of rhomboids11 and was involved in the for-mation of QPs and superlattices.

The period-doubling bifurcation of parametrically excitedstanding waves was analyzed for square and circular containersby Crawford [210,211] using reduced maps for the critical modeamplitudes. Experimental and theoretical investigations for sur-face waves in square containers suggested that the effective sym-metry may be larger than the geometric symmetry of the containercross section. The difference between the geometric symmetryand effective symmetry is demonstrated in Fig. 19 by consideringthe cross section P of a square of nondimensional length p. Forsquare containers, the geometric symmetry is generated by reflec-tion about x ¼ p=2 and reflection across the diagonalðx; yÞ ! ðp� x; yÞ and ðx; yÞ ! ðy; xÞ. A standing wave patterncan be smoothly extended by reflection in x and y to give a solu-tion to the corresponding free-boundary problem posed on thelarger domain ~P with periodic boundary conditions. The extrasymmetries complicate the construction of normal forms andappear to stabilize effects in the experiments. These symmetriesmay be directly observed, if the sidewalls are deformed to a non-square cross section that retains square symmetry. Crawford et al.[212] indicated that there are hidden translational and rotational

Fig. 18 Behavior of ethanol drop deposited on silicon oil: (a)sketch of the vertical section of the floating drop in the absenceof oscillations and (b) It is circular at rest, and (c)–(e) three suc-cessive images showing the temporal evolution of the dropunder parametric excitation frequency of 130 Hz [198]

Fig. 19 Demonstration of effective and geometric symmetries:(a) the extension by reflection of a Neumann boundary condi-tion solution on O; xð Þ leads to periodic boundary condition onð2p; pÞ. (b) The relation between the physical domain P and theextended domain ~P [211].

10A bifurcation that requires at least m control parameters to occur is called acodimension-m bifurcation.

11The rhomboid is a parallelepiped, a solid figure with six faces in which eachface is a parallelogram and pairs of opposite faces lie in parallel planes. Somecrystals are formed in 3D rhomboids. This solid is also sometimes called a rhombicprism.



symmetries that further constrain the linear and nonlinear behav-ior of the fluid free surface. As a result, unexpected degeneracyamong the linear wave frequencies and unexpected branches ofnonlinear solutions in the bifurcation equations for the surfacewaves can occur. These additional symmetries are not obvious,since they are not symmetries of the square container and conse-quently do not preserve the boundary conditions of the problem.

The interaction between surface wave modes leads to a varietyof interesting nonlinear phenomena, including chaotic dynamics.When two or more spatial modes are simultaneously excited, theamplitude equations contain the independent dynamics of eachmode and also their coupling. Symmetry considerations greatlysimplify the analysis by reducing the number of allowed couplingterms to any given order in the perturbation expansion. Internalresonance arises from the coupling of two sloshing modes whosefrequencies are either equal or one modal frequency is n times thefrequency of another mode. The case of equal frequencies occursnaturally in a circular cylinder for which the nonaxisymmetricmodes occur in degenerate pairs. These modes differ only by anazimuth rotation of p=2 and have the same natural frequency x1.They are uncoupled (orthogonal) in the linear approximation, butare nonlinearly coupled both through direct, third-order interac-tions and through secondary modes that are excited at second-order and affects indirect, third-order interactions. However, thecoupled motion of such a pair necessarily comprises angularmomentum, which cannot be generated by parametric excitation.It is possible to have approximately coincidental modal frequen-cies in the doubly infinite, discrete spectrum for any liquid con-tainer [220].

A more common internal resonance may occur between two or-thogonal modes with natural frequencies in the approximate ratioof 2:1. As indicated by Miles [221], the direct coupling is quad-ratic while the indirect coupling, through secondary modes, is ofhigher order and is therefore negligible. Internal resonance withx2 ¼ 2x1 for two gravity-wave modes with wave numbers k2 andk1 in cylindrical tank of depth h requires

k2 tanh k2h ¼ 4k1 tanh k1h (25)

An example with a much larger coupling coefficient is resonancebetween the dominant antisymmetric and axisymmetric modes,for which k1R ¼ 1:8412; k2R ¼ 3:8317, and h=R ¼ 0:1523,where R is the tank radius [221].

For a rectangular tank, the internal resonance of liquid modesin a rectangular tank under parametric excitation was studied inRefs. [95,222–225]. The liquid free surface under parametric exci-tation may be represented by the expression

gmnðx; y; tÞ ¼ AmnðtÞ cos Xt=2ð Þ þ BmnðtÞ sin Xt=2ð Þ½ �� cos mpx=Lxð Þ cos mpy=Ly

� � �(26)

where Amn and Bmn are mode amplitudes which vary with time,in the weakly nonlinear regime, on a timescale much larger than1=X, m and n are integers giving the number of half-wavelengthsin the x- and y-directions, and Lx and Ly are the width andbreadth dimensions of the tank. The complex amplitude,CmnðtÞ ¼ AmnðtÞ � iBmnðtÞ, was governed by the first-orderequation:

_CmnðtÞ ¼ aCmnðtÞ þ b �CmnðtÞ þ c CmnðtÞj j2CmnðtÞ ¼ S Cmn; �Cmnð Þ(27)

where a, b, and c are constant coefficients. In rectangular andsquare tanks, the evolution equations of modes (3,2) and (2,3)were obtained in the form [225]

_C32ðtÞ ¼ S32 C32; �C32ð Þ þ l32C32 C23j j2þ�32�C32C2

23 (28a)

_C23ðtÞ ¼ S23 C23; �C23ð Þ þ l23C23 C32j j2þ�23�C23C2

32 (28b)

where the function S is defined in Eq. (27) for a single mode. Thecoefficients lmn and �mn are constants. For a square tank, thesecoefficients are identical and the two equations are interchange-able. This satisfies the symmetry requirements and internal reso-nance condition. For a square tank of 6-cm side and fluid depth of2.5-cm, Simonelli and Gollub [225,226] measured the slowlyvarying amplitudes Amn and Bmn that contribute to the fluid freesurface wave height, gðx; y; tÞ, using two problems under paramet-ric excitation. Figure 20(a) shows the image of the free surface ofmode (3,2), while Fig. 20(b) shows the superposition of the twomodes (3,2) and (2,3). Figure 21(a) shows three regions of stabil-ity boundaries of three modes and their degenerate modes. Figure21(b) shows the detailed structure of stability boundaries near the(3,2) and (2,3) resonance including additional stability region B inwhich the stable states are superposition of the two modes, withequal amplitudes C32j j2¼ C23j j2. In view of the symmetry proper-ties of these modes, four equivalent mixed states are expected andwere observed experimentally. Inside region D only pure stateswere found. Due to the symmetry, either pure state can be founddepending on the initial conditions. Regions A and C were foundto be hysteretic: in A the flat surface state co-exists with mixedstates, while in C mixed states co-exist with pure states. Thegeneral bifurcation problem of Faraday resonance in a square con-tainer was also studied by Silber and Knobloch [227] using a 2Dmap. They suggested the necessity of fifth-order nonlinearity intheir model in order to reproduce Simonelli and Gollub [225] dia-gram shown in Fig. 21.

The case of parametric excitation when the modal frequenciesare in the ratio of 1:2 was studied in Refs. [95,223,228]. Holmes[228] qualitatively showed the existence of chaotic motions forcertain parameter ranges close to 2:1 subharmonic resonances. Guand Sethna [223] studied periodic, almost periodic, and chaoticwave motions in a rectangular tank subjected to vertical sinusoidalexcitation. The internal resonance condition 1:2 requires the fluidheight to be relatively small, which causes excessive energy dissi-pation. Such energy dissipation suppresses nonlinear phenomenamaking it impossible to verify the analytical results experimen-tally. When the frequencies are nearly equal, the free oscillationin a nearly square container was discussed by Bridges [229] forthe case of standing waves. For the case of a nearly square con-tainer, all nonsymmetric modes have nearly equal natural frequen-cies independent of the fluid depth. The case of 1:1 internalresonance was considered in Refs. [227,230–232] and it wasshown that the system is capable of exhibiting periodic and quasi-periodic standing and traveling waves. They were able to identifyparameter values at which chaotic behavior can occur. Faradaywaves in a circular cylinder, which are internally resonant with ei-ther the subharmonic mode (with frequency one-fourth that of theforcing) or the superharmonic mode (with frequency equal to that

Fig. 20 Liquid surface images in a square tank produced by (a)pure (3,2) mode and (b) superposition of the (3,2) and (2,3)modes with equal amplitudes. The images were averaged overone period of the forcing period [225].



of the forcing) were investigated experimentally by Hendersonand Miles [233]. For subharmonic resonance, both modesachieved comparable amplitudes that were steady or were modu-lated with one or two periods, or exhibited quasi-periodic or cha-otic motions. For low modes, an energy exchange occurred duringthe initial period of growth, and precession instability was devel-oped. For high modes for which both frequencies and wave num-bers are in a 2:1 ratio, superharmonic resonance occurredirreproducibly and it was found to be prevailed by 1:1 interactionsamong the possible Faraday wave modes.

Nayfeh [234] presented weakly nonlinear analysis and derivedthe general fluid field equations. The free-surface conditionswere expanded into Taylor series in terms of linear mode shapes.The governing equation of the liquid-free surface elevation wasin full agreement with Miles Lagrangian formulation. In the pres-ence of week damping, the free oscillation component of anymode that is not directly excited by the parametric resonance orindirectly excited by internal resonance will decay with time. Ifthe condition of perfect internal resonance of two modes,x2 ¼ 2x1, was relaxed, then Hopf bifurcations are possible.Nayfeh and Nayfeh [235] considered the case of 2:1 internal res-onance condition when the higher mode, rather than the lowermode, is excited by a principal parametric resonance. The solu-tions of the modulation equations were found to be fixed-point,limit-cycle, or chaotic solutions. Multiple limit cycles with dif-ferent amplitudes and periods were detected and shown to co-exist over some ranges of the external parametric resonancedetuning parameter of the second mode taken as the bifurcationparameter. Some limit cycles were found to experience pitchforkbifurcation while others undergo cyclic fold bifurcation. Thepitchfork bifurcation produces symmetry-breaking bifurcationand the cyclic fold bifurcation results in cyclic jumps. In thecase of principal parametric resonance of the lower mode, theroute to chaos was found to be a period-doubling sequence ofbifurcations.

The role of weakly damped modes in the selection of Faradaywave patterns forced with rationally related frequency compo-nents mx and nx was examined by Topaz and Silber [236]. Sym-metry considerations were used for the special importance of theweakly damped modes oscillating with twice the frequency of thecritical mode, and those oscillating primarily with the “difference

frequency” j n� m jx and the “sum frequency” j nþ m jx. Theresonance effects predicted by symmetry were emerged in the per-turbation results for one spatial dimension and agree with the nu-merical results for two dimensions. The difference frequencyresonance was found to play a key role in stabilizing superlatticepatterns observed by Kudrolli et al. [237]. Later, Topaz et al.[238] showed how pattern formation in Faraday waves may bemanipulated by varying the harmonic content of the periodic forc-ing function. Their approach was based on the crucial influence ofresonant triad interactions coupling pairs of critical standing wavemodes with damped, spatiotemporally resonant modes. For forc-ing functions with arbitrarily many frequency components, it wasfound that there are at most five frequencies that affect each of theimportant triad interactions at leading order. The relative phasesof those forcing components were found to make the differencebetween an enhancing and suppressing effect. Their approach wasapplied to one-dimensional periodic patterns obtained with impul-sive forcing and to 2D superlattice patterns and QPs obtained withmultifrequency forcing.

Moehlis et al. [239] extended the work of Feng and Sethna[232] to study periodic orbits associated with heteroclinic bifurca-tions in a model of the Faraday system for containers with squarecross section. These periodic orbits were found to correspond toquasi-periodic surface waves in the physical system. The periodicorbits were associated with heteroclinic bifurcations. Chaoticattractors were also found in the model equations, which corre-spond to chaotic surface waves. The case when the sloshing modalfrequencies are in the ratio of 3:1 was considered in Refs.[240,241]. For the standing gravity waves, 3:1 internal resonancebetween (1,1) mode and (5,2) mode was found to be the strongestfor the Faraday resonance in the 2D standing gravity waves offinite depth. Wada and Okamura [241] derived the nonlinearevolution equations of the dominant free-surface modes up tofifth-order in wave amplitude. It was found that the third-orderevolution equations are integrable while the fifth-order evolutionequations are chaotic. This means that the fifth-order nonlinearitybreaks the integrability of the third-order integrable system. The3:1 resonance was found possible between two normal modes for2D standing gravity waves with the dimensionless depthhk � 0:6232, where k is the wave number of the standing gravitywave. Related to the problem of internal resonance is the problem

Fig. 21 Stability boundaries of liquid free surface in a square tank (6.17 3 6.17 cm): (a)regions of three modes and their degenerate modes. The resonances are asymmetric: sub-critical on the left and supercritical on the right, (b) expanded view near the (3,2)–(2,3) reso-nance. Three primary regions are shown: the flat surface, mixed states in region B, and purestates in region D. The intermediate regions A and C are characterized by co-existence of dif-ferent types of fixed points (flat or mixed in A, mixed or pure in C) which are realized fordifferent initial conditions [225].



of sloshing modal competition and associated free-surface pat-terns. These issues will be discussed in Secs. 8 and 9.

8 Mode Competition of Faraday Waves

It is known that under parametric excitation of frequency X thefirst excited mode is the one near to the parametric resonance atX=2. However, in some cases, the detuning may be of the sameorder for two or more adjacent modes, so that these modes canenter in competition even close to threshold. The competitionbetween nearly degenerate modes was shown to lead to chaoticbehavior in a circular container [220] and to complex selectionrules in a square cell [242]. Depending on the value of the forcingfrequency, the system may either display a subcritical transitionfrom one mode to the other or a competition between the twomodes, leading to the onset of a mixed mode.

At a certain critical depth-radius ratio of liquid in a circularcontainer, the first-mode oscillation may be coupled with a highermode motion at an integral multiple of the basic frequency. Cha-otic behavior may arise due to competition between two differentspatial modes or patterns [220,243,244]. An axisymmetric modeand two completely degenerate antisymmetric modes of gravitywaves in a circular cylindrical container were, respectively, stud-ied by Mack [245] and Miles [221]. A region of mode competitionemerges in which the fluid surface can be described as a superpo-sition of two modes with amplitudes having slowly varying enve-lopes. These slow variations can be either periodic or chaotic.Ciliberto and Gollub [243] developed a phenomenological modelin the form of the nonlinear Mathieu equation

€amn þ fmn _amn þ x2mn � wmnZ0 cos Xt

� �amn ¼ 1mna3

mn (29)

where 1mn is the coefficient of the cubic term that limits thegrowth of the mode, and wmn is the gain coefficient of the para-metric excitation term. The subscripts mn indicate the number ofangular maxima and the number of nodal circles, respectively.The cubic term on the right-hand of Eq. (29) was added to limitthe growth of the response amplitude and was not based on theactual nonlinear modeling as in the case of the boundary valueproblem of liquid sloshing dynamics. This cubic term should beregarded as equivalent to all nonlinear inertia terms of mode mn.Figure 22 shows the surface mode profile of the two modes (4,3)and (7,2), which were the subject of many studies in the literaturefor mode competition [246].

Figure 23 shows the digitized optical intensity fields formed bystable patterns of modes (4,3) and (7,2), whose natural frequenciesare f4;3 ¼ 7:892 Hz and f7;2 ¼ 8:1042 Hz, respectively. The

crosses are experimentally determined points on the stabilityboundaries [243,244]. At the intersection of the two stabilityboundaries, both modes oscillate simultaneously. Above the sta-bility boundaries, the liquid surface oscillates at half the drivingfrequency in a single stable mode. The shaded areas are regions ofmode competition, in which the surface can be described as asuperposition of the (4,3) and (7,2) modes with amplitudes havinga slowly varying envelope in addition to the fast oscillation at halfthe excitation frequency, f0=2. These slow variations can be eitherperiodic or chaotic. At driving amplitudes higher than thoseshown in Fig. 23, the surface can become chaotic even if the driv-ing frequency is resonant, so that a single mode is dominant.

As one crosses from the region of slow periodic oscillationsinto the chaotic region, one finds a period-doubling bifurcationfollowed by a transition to chaos. A typical example is shown inFig. 24, where time history records are shown for three differentdriving amplitudes but fixed driving frequency of 16.05 Hz. Thefigure also shows the corresponding power spectra of the timeseries. It is seen that subharmonic bifurcation occurs at relativelyhigher excitation amplitude (Z0 ¼ 149 lm and Z0 ¼ 190 lm).This is accompanied by a slight broadening of the peaks. AboveZ0 ¼ 180 lm, the time history record is chaotic. Ciliberto andGollub [243,244] indicated that at amplitudes higher thanZ0 ¼ 200 lm, mode competition disappears and only the fourfoldsymmetric mode dominates. An experimental study was per-formed by Karatsu [247] to study nearly degenerate (4,1) and

Fig. 22 Surface mode contours: (a) (m,n) 5 (4,3) and (b) (m,n) 5 (7,2) [246]

Fig. 23 Stability boundaries of modes (4,3) and (7,2) in termsof the excitation amplitude Z0 and frequency f0 of a circular con-tainer of radius 6.35 cm filled with 1 cm layer of water. Shadedregions belong to slow periodic and chaotic oscillations involv-ing competition between these modes [220].



(1,2) surface wave modes in a circular cylinder and observed peri-odic and chaotic mode competition.

The work of Ciliberto and Gollub [243,244] attracted the inter-est of some researchers in an effort to understand the appearanceof few-dimensional chaos systems with an infinite number ofdegrees-of-freedom. For example, Meron and Procaccia [51,52]employed the center-manifold and normal-form theories to deriveevolution dynamical equations in cylindrical container in terms oftwo time scales, t and the slow time s ¼ et, wheree ¼ 2

X ðx4;3 � x7;2Þ is a small parameter, in the form

da4;3

ds¼ �f4;3 þ ir4;3

� �a4;3 þ i C1�a4;3 þ C2 a4;3

�� 2a4;3

hþC3 a7;2

�� 2a4;3 þ C4�a4;3a27;2� (30a)

da7;2

ds¼ �f7;2 þ ir7;2

� �a7;2 þ i D1�a7;2 þ D2 a7;2

�� 2a7;2

hþ D3 a4;3

�� 2a4;3 þ D4 �a7;2a24;3� (30b)

where rmn ¼ 2xmn � Xð Þ=ð4XÞ is a detuning parameter, fmn arephenomenological damping parameters, and the coefficients Ci

and Di depend on the excitation amplitude and system parameters.Meron and Procaccia [51,52] were able to rationalize essentiallyall the major experimental observations such as the appearance ofregular and chaotic mode competitions, the existence of asymme-try between (4,3) and (7,2) modes, and the qualitative structure ofthe stability diagram. However, Miles [248] and Miles andHenderson [28] disputed the analytical results of Meron and Pro-caccia [51,52] on the basis that Eqs. (45) and (46) violate symme-try conditions and thus do not lead to the canonical formulations.Miles [248] claimed that the formulation of Meron and Procaccia[51,52] would preserve the canonical form if the coefficientsC3 ¼ D3 and C4 ¼ D4. On the other hand, Meron and Procaccia[51] neglected nonlinear inertia terms containing the time deriva-tive on the ground that these terms are not expected to contributesignificantly to the long-time behavior of the amplitude equations.In their reply, Meron and Procaccia [249] confirmed that thesecoefficients should be different, in principle. They also claimedthat the Hamiltonian formulation of Miles [43] yields disagree-ment with the linearized hydrodynamics treatment with C1 ¼ D1.The differences between C3 and D3, also between C4 and D4, are

of the same order as that between C1 and D1. It was affirmed byMeron and Procaccia that the difference between these coeffi-cients is small for almost degenerate models of the orderðx4;3 � x7;2Þ=X and can be considered as a high-order correction.Recall

x2mn ¼ kmn tanhðhkmnÞ

ck2mn

qþ g

� �(31)

where c is the surface tension and g is the gravitational accelera-tion. Note that when the term gkmn dominates the mode is a grav-ity wave, while when ck3

mn=q dominates the waves are capillary.The nonlinear dynamical equations of two nearly degenerate

subharmonic modes of Faraday waves was developed by Umeki[250] and Umeki and Kambe [251]. The system equations werereduced into a two degrees-of-freedom system because each non-axisymmetric mode has two completely degenerate componentsand the associated angular momentum of each mode tends tovanish owing to the damping and the circular symmetry. Period-doubling bifurcation and chaotic solutions with one positive Lya-punov characteristic exponent were obtained numerically. It wasshown that some of the period-doubling bifurcations are related tothe symmetry. Using the results of Umeki and Kambe [251] andUmeki [252], the controversy between the results of Meron andProcaccia [249] and Miles [248] concerning the existence ofcanonical formulations was resolved. A two-parameter analysis ofthe interaction between two period-doubling modes with azi-muthal wave numbers k and lðl > k 1Þ was developed by Mileset al. [253]. It was found that when one mode bifurcates subcriti-cally and the other supercritically the pure mode branches losestability to a branch of reflection-symmetric mixed modes. Thesystem was found to undergo a Hopf bifurcation then to a quasi-periodic reflection-symmetric pattern. The results explain a num-ber of observations reported by Ciliberto and Gollub [243,244] onparametrically excited surface waves in circular container.

Parametric excitation of surface waves in a container under ver-tical forcing was investigated by Kambe and Umeki [209]. A sys-tem of evolution equations of third-order nonlinearity was derivedfor the case of excitation frequency, which is close to twice thefrequencies of two nearly degenerate free modes. It was foundthat this dynamical system yields not only excitation of a

Fig. 24 (A) Time history records showing the transition from periodic to chaotic oscillationsand (B) corresponding power spectra under excitation frequency of 16.05 Hz and three differ-ent excitation amplitudes [220]



single-mode state but also interaction between two modes inwhich each mode oscillates either periodically or chaotically.These results were found in good agreement with the experimentalobservations, except for the case of strong nonlinearity. Homo-clinic chaos in the Hamiltonian system of two degrees-of-freedomwithout damping was studied numerically. It was suggested thatthe chaotic mode competition observed experimentally is differentfrom the homoclinic chaos. The analytical results for a pair of(4,3) and (7,2) modes in a circular cell were found in agreementwith the experiment by Ciliberto and Gollub [220,244], and theresults for the (2,3) and (3,2) mode pair in a rectangular cell arealso in good agreement with the experiment by Simonelli andGollub [225], except for the presence/absence of mixed-mode ex-citation. Umeki [252] showed that the periodic mode competitionoccurs from a Hopf bifurcation, which is supercritical for the pre-dicted nonlinear coefficients of gravity waves but may becomesubcritical for slightly modified values of the coefficients, of amixed rotating wave state for the slightly rectangular case. If thedamping coefficient is sufficiently small, the periodic orbits in thephase space bifurcate into complicated orbits.

Miles [253] extended the formulations reported in Refs. [225],[230–232], and [252] by incorporating capillarity, cubic forcing,and cubic damping. Weakly nonlinear capillary–gravity waves offrequency x and wave number k in a square container subjectedto the vertical displacement Z0 cos 2xt were studied based on theassumptions that 0< f< kZ0; where f is the linear damping ratio.The fixed points of the evolution equations comprised four solu-tions. These are: (1) the null solution; (2) an orthogonal pair ofrolls described by either cos kx or cos ky; (3) an orthogonal pair ofsquares described by either cos kxþ cos ky or cos kx� cos ky; and(4) coupled-mode solutions for which both modes are activeand neither in phase nor in antiphase. The fixed points for rollsand squares lie on separate loci in an energy-frequency plane thatintersect the null solution at a pair of pitchfork bifurcations, oneof which is definitely supercritical and the other of which may beeither subcritical or supercritical. Crawford et al. [254] formulatedthe modal interaction between two period-doubling in the pres-ence of symmetry as a map in four dimensions, and classify theprimary, secondary, and tertiary bifurcations. The observed ampli-tude oscillations were found to arise as a tertiary Hopf bifurcationfrom a mixed mode pattern provided exactly one of the primaryinstabilities is subcritical. Riecke [255] analyzed the stable wavenumber kinks in parametrically excited standing waves in largeaspect ratio. It was found that for small group velocity of theunderlying traveling waves the stable band of wave numbers cansplit up into two sub-bands, which are separated by a region ofunstable wave numbers. This gives rise to solutions with stablewave number kinks which bridge the unstable regime between thesub-bands.

Modal competition between three neighboring 2D modes in anarrow rectangular tank was experimentally and theoreticallystudied [256]. In particular, the stability of a standing wave toperturbations from the two neighboring sideband modes wasstudied. Cubic damping, cubic forcing, and fifth-order conserva-tive terms were retained in the analytical model for deep water.Retaining these higher order nonlinearities gave rise to fairlygood agreement with experimental results. Faraday instability inan elongated rectangular cell was studied by Residori et al. [257]for a range of the forcing frequency for which two nearly degen-erate modes enter in competition. The stability properties of thetwo modes were found to either co-exist in a bistable way orgive rise to a mixed mode. It was indicated that different bifurca-tion scenarios exist depending on the relative detuning betweenthe two modes. Slow oscillations accompanying the mixed modewere explained by the frequency beating of the two competingmodes.

In the presence of small viscosity, i.e., Cg � � gh3½þ ðch=qÞ��1=2 � 1, where Cg is the capillary–gravity number, itwas shown that the waves couple to a streaming flow driven inoscillatory viscous boundary layers at rigid walls and the free

surface [150]. This flow in turn affects the waves responsible forthe oscillatory boundary layers. It was indicated by Mart�ın et al.[258] that this coupling is responsible for different types of driftinstabilities of the waves, instabilities that were observed inexperiments in annular containers [259] but are absent from thetheory when the coupling to the streaming flow is neglected.These instabilities were found to arise not only in annular contain-ers but in cylindrical containers as well, and are driven by a cou-pling between the streaming flow and the spatial phase of thewaves. Richer dynamic behavior was reported when the containeris deformed into an elliptical one as shown in Higuera et al. [135].In this case, the streaming flow was found to couple with theamplitudes of the standing waves, resulting in a much strongercoupling between the waves and the streaming flow. Higuera et al.[260] explored the consequences of this coupling in model equa-tions derived originally by Higuera et al. [135] under the assump-tion that the Reynolds number of the streaming flow is small. Thesystem was described by a five-dimensional system of ordinarydifferential equations. The eccentricity of the container, althoughsmall, was found of crucial importance since it results in a cou-pling of the streaming flow to the complex amplitudes of the twonearly degenerate modes.

The Floquet theory was used by Mancebo and Vega [261] todetermine the threshold acceleration for the appearance of Fara-day waves in large aspect ratio containers. Different distinguishedlimits were considered in the analysis. In particular, the case ofnearly inviscid limits, i.e., when the capillary–gravity numberCg � 1 then the most dangerous mode at threshold is potential,except in two thin boundary layers near the bottom wall and thefree surface, and an approximation of the critical excitation accel-eration amplitude, ac can be found in closed form. If that limit ofthe capillary–gravity number does not hold, then the most danger-ous mode at threshold exhibits nonlocalized vorticity due to vis-cous effects. The other case is the basic limit and highly viscoussublimit, which is captured as Cg > 1. In this case, viscous effectsdominate gravity and surface tension, which can both be ignoredin the analysis. The resulting simplified problems either admitclosed-form solutions or are solved numerically by the well-known method introduced by Kumar and Tuckerman [262]. Thecritical excitation acceleration amplitude ac was found to dependon excitation frequency, the capillary–gravity number, Cg, and thegravity–capillary balance parameter, S ¼ c= cþ qgh2ð Þ, whichmeasures the ratio of surface tension to its combined effect withgravity.

Knobloch et al. [263,264] considered nearly inviscid parametri-cally excited surface gravity–capillary waves in 2D periodicdomains of finite depth with small and large aspect ratios.Coupled equations describing the evolution of the amplitudes ofresonant left- and right-traveling waves and their interaction witha mean flow in the bulk were derived. The mean flow consisted ofan inviscid part together with a viscous streaming flow driven bya tangential stress due to an oscillating viscous boundary layernear the free surface and a tangential velocity due to a bottomboundary layer. Through nonlinear rectification, Reynolds stresseswere generated, which drive a streaming flow in the nominallyinviscid bulk. This flow in turn was found to advect the wavesresponsible for the boundary layers. The resulting system wasdescribed by amplitude equations coupled to a Navier–Stokes-likeequation for the bulk streaming flow, with boundary conditionsobtained by matching to the boundary layers. The coupling to thestreaming flow was found to be responsible for various types ofdrift instabilities of standing waves and in appropriate regimescan lead to the presence of remarkable relaxations oscillations.

A sequence of symmetry-breaking instabilities leading to a cha-otic state was observed in the surface deformations of a fluid layersubjected to a vertical oscillation by Gollub and Meyer [265]. Fordriving amplitudes above a critical value, a primary instabilitywas observed in the form of circularly symmetric standing wavesat half the driving frequency. A second instability at a higherthreshold was found to break the circular symmetry and lead to a



slow precession of the pattern, so that the overall motion is quasi-periodic. Keolian et al. [266] and Keolian and Rudnick [267] usedliquid helium and water in thin annular troughs and observed bothperiod-doubling and quasi-periodic motions apparently involvingthree modes. Nonlinear evolution equations for the amplitudes ofresonant capillary–gravity waves with different directions of wavenumber vector were derived by Yoshimatsu and Funakoshi [268].These equations include cubic nonlinearity and the effects of vis-cous damping and parametric forcing obtained from the energyequation. The center-manifold was used to obtain quintic ampli-tude equations for unstable modes in which cubic damping andforcing were included. It was concluded that the squares are stablefor sufficiently short waves, the hexagons and the eightfold QPsare stable when the wavelength is within two intermediateregions, and the stripes are always unstable. The disordered struc-ture of the free-surface flow under relatively large harmonic exci-tation amplitude was also experimentally observed by Gollub andMeyer [265]. Their measurements showed a sequence ofsymmetry-breaking instabilities leading to chaotic state.

In the weakly inviscid regime, parametrically driven surfacegravity–capillary waves generate oscillatory viscous boundarylayers along the container walls and the free surface. Throughnonlinear rectification these generate Reynolds stresses, whichdrive a streaming flow in the nominally inviscid bulk. This flow inturn advects the waves responsible for the boundary layers. Theresulting system was described by amplitude equations coupled toa Navier–Stokes-like equation for the bulk streaming flow.Higuera et al. [260,269,270] examined two model systems, whichinclude an elliptically distorted cylinder while in the second it isan almost square rectangle. The forced symmetry breaking wasfound to result in a nonlinear competition between two nearlydegenerate oscillatory modes. This interaction was found to desta-bilize standing waves at small amplitudes and amplifies the roleplayed by the streaming flow. In both systems, the coupling to thestreaming flow triggered by these instabilities was found to lead toslow drifts along slow manifolds of fixed points or periodic orbitsof the fast system, and to generate behavior that resembles burst-ing in excitable systems. The new dynamical behavior includedrelaxation oscillations involving abrupt transitions between stand-ing and quasi-periodic oscillations, and exhibiting “canards.”Here, relaxation oscillations are nonlinear oscillations, which arenonsinusoidal repetitive such as those shown in Fig. 25. Theselimit cycles are evidently relaxation oscillations, but of anunusual type, involving slow drifts along branches of both equili-bria and of periodic orbits, with fast jumps between them. More-over, these oscillations may be symmetric or asymmetric, with thesymmetry alternately present and broken in successive periodicwindows. The presence of relaxation oscillations in this systemcan be attributed to the disparity between the decay times of free

surface gravity–capillary waves and the streaming flow that pre-vails in the nearly inviscid regime.

In the case of a slightly elliptical container, two types of stand-ing waves, oriented along the major and minor axes of the ellipse,were found to come in close succession as the amplitude of theparametric excitation increases, and these may interact at smallamplitude, producing mixed modes which are much more efficientat driving a streaming flow. In this case, the fluid flow couples tothe two amplitudes, as well as to the spatial phase of the resultingpattern. Higuera et al. [260,269,270] demonstrated that this inter-action can lead to relaxation oscillations characterized by switch-ing from single frequency standing waves to two-frequency wavesand back. Under appropriate conditions these relaxation oscilla-tions can exhibit the so-called canard phenomenon in which thesystem follows nominally unstable solutions in the slow phase.Various global bifurcations are located as well, of which perhapsthe most interesting is responsible for the appearance of chaoticdynamics right at threshold of the primary instability. In an ellipti-cal container, Faraday waves were described by a third-order sys-tem of ordinary differential equations with characteristicslow–fast structure [271]. These equations describe the interactionof standing waves with a weakly damped streaming flow drivenby Reynolds stresses in boundary layers at the free surface and therigid walls.

9 Patterns of Faraday Waves

Faraday waves provide a convenient experimental system forstudying pattern formation due to fast time scales and large aspectratio. The theoretical starting point of pattern formation is usuallya set of deterministic equations of motion, typically in the form ofnonlinear partial differential equations. The study of pattern for-mation in fluids has greatly been benefited from the careful andcontrolled experiments as well as the development of new analyticand numerical tools. Cross and Hohenberg [272] and M€uller [273]presented two different review accounts on the spatiotemporalpattern formation of Faraday wave patterns. Knobloch [274] pre-sented an overview of the theory of pattern formation in manysystems of interest in physics, which exhibit spontaneoussymmetry-breaking instabilities that form structures of patterns.

Early experimental and analytical investigations dealt withsmall systems, in which the pattern is determined by the shape ofthe boundary and few eigen-modes are excited. Parametric excita-tion experiments were performed to excite low-order modes ofsmall liquid layers [225,242,244]. Other experiments were per-formed on liquid cells of thickness in the order of 100 wavelength[275–277]. These experiments displayed several remarkable fea-tures such as: (1) square patterns, even in cylindrical containers;(2) a secondary instability to a pattern with transverse modulationsof the amplitude of standing waves; and (3) at still higher excita-tion strength, a transition to spatiotemporal chaos. The transitionto chaos was studied experimentally in a system of capillarywaves parametrically excited in a thin layer of fluid by Ezersky[278,279]. In a certain range of fluid depths, when the amplitudeof oscillation is increased a regular wave system is replaced by achaotic field. Irrespective of the container shape used in theexperiments, the temporal evolution of the system from a regularto a chaotic state was found to occur through the excitation of lowfrequency.

Other experimental investigations were conducted on large sys-tems to examine the pattern selection and chaotic behavior[278,280–287]. Ezersky et al. [288] observed a grid of waves ofsquares on a vibrating thin layer of silicone, while three types ofgrids were reported by Levin and Turbnikov [276] and presenteda hexagonal grid, a grid of squares, and one-dimensional grid.Christiansen et al. [283] and Edwards and Fauve [289] observedan eightfold and 12-fold QPs analogous to a 2D quasi-crystal inparametric excitation of capillary waves experiment. Theobserved QP was found independent of the shape of the containerside walls. Regions with perfect crystal patterns separated by

Fig. 25 Time series of relaxation oscillations showing slowdrifts along branches of both equilibrium states with fast jumpsbetween them [260]



domain walls-chains of dislocations were found to exist in para-metrically excited capillary ripples [287]. The transition of ensem-bles of domains to perfect crystals was found to be due to collapseof individual domains when the domain wall is a closed contourof annihilating dislocations and merging of neighboring domainsdue to the dislocation climb to the walls of a cell. A stable fivefoldstanding wave pattern exhibiting a germinal quasi-crystalline struc-ture was observed in a symmetry breaking Faraday instabilityexperiment with a low aspect ratio and the liquid depth being lessthan the surface wavelength by Torres et al. [290].

When Faraday waves are excited well beyond the threshold forpattern formation, the ordered patterned structure is lost [291].Such spatial disorder, known also as “defect-mediatedturbulence,” was found to occur in a wide class of driven nonlin-ear systems [277,292,293]. It was indicated that defect-mediatedturbulence is well characterized. Above a critical value of theexcitation acceleration amplitude, defects in the form of disloca-tions of pattern lines were found to spontaneously nucleatethroughout the system and their nucleation rate increases with theexcitation. Furthermore, their motion was found to generate rapidexponential decay of temporal and spatial correlations. The transi-tion from an ordered pattern to disorder corresponding to defect-mediated turbulence was examined experimentally by Shani et al.[294]. It was found that this transition is mediated by a spatiallyincoherent oscillatory phase, which consists of highly dampedwaves that propagate through the effectively elastic lattice definedby the pattern. As these waves decay within a few lattice spaces,they are spatially and temporally uncorrelated at larger scales.Significant effort was made in order to understand and predict thepattern selection using analytical tools. Two mechanisms forselecting the main frequency responses that are different from thefirst subharmonic one were identified in the literature [295]. Thefirst mechanism occurs when two or more frequency componentsare introduced in the parametric excitation. Each component willtend to excite its own corresponding first subharmonic mode.Their relative amplitudes will determine which of these responseshas the lowest global excitation strength threshold, which estab-lishes the instability that is observed at onset. The second mecha-nism can only arise in the high viscosity regime. If the fluid layeris shallow enough, even a single component excitation with lowenough frequency can excite instability different from the firstsubharmonic one. As the viscous boundary layer reaches the bottomof the fluid container, the threshold of the lowest unstable modesrises allowing others with higher main frequency components tobecome unstable at onset. In Secs. 9.1 and 9.2, this paper will pro-vide an account of the formation and selection of Faraday patternsunder single-, two-, three-, and multifrequency excitations.

9.1 Single-Frequency Parametric Excitation. Under single-frequency parametric excitation of a fluid layer can displayFaraday waves of different types of patterns depending on the ex-citation level and frequency. Weak nonlinear effects can causeinteractions between surface waves with different wave modesand can be important in determining free-surface wave patterns.The liquid free surface may be regarded as a superposition ofinteracting waves propagating in different directions. Faradaywaves were observed to be especially versatile and exhibit thecommon patterns familiar in convection such as stripes, squares,hexagons, and spirals. These patterns include triangles [296], QPs[237,283], superlattice patterns [237,297], time-dependent rhom-bic patterns [298], and localized waves [299]. Christiansen [300]conducted experimental investigation to measure the dampingrates and the critical amplitude of the primary instability of highaspect ratio of Faraday waves. Above the primary instability, asequence of ordered crystalline states was observed. These statesinclude a quasi-crystalline pattern in the form of eightfoldsymmetry.

It was reported that if the dissipation is large, the preferred pat-tern consists of parallel stripes [284,301–303]. At higher driving

amplitudes, stationary patterns with a hexagonal shape in thecentre of the cell were observed experimentally in Ref. [242].Standing wave patterns with different symmetries were observedand understood as superposition of two linearly unstable modes ofthe fluid free surface. The pattern wave number was fixed by theexcitation frequency, whereas the shape was found to depend onthe nonlinear interaction. Douady et al. [259] presented a study ofsecondary instabilities of parametrically generated standing wavesin a horizontal layer of fluid subjected to vertical vibrations. Itwas found that when the driving frequency is increased (respec-tively, decreased), the system bifurcates abruptly to another stand-ing wave pattern by nucleation (respectively, annihilation) of onewavelength. An experimental study of surface waves parametri-cally excited was presented by Douady [15]. Stability boundaries,wave amplitude, and perturbation characteristic time of decaywere measured and found to be in agreement with an amplitudeequation derived by symmetry. The measurement of the amplitudeequation coefficients provided some interpretation of the observedtransition, which is supercritical, and showed the effect of theedge constraint on the dissipation and eigenvalues of the variousmodes. The fluid surface tension was obtained from the dispersionrelation measurement.

The threshold accelerations necessary to excite surface wavesin a vertically excited fluid square container was measured[304,305]. Figure 26 shows the stability boundaries according tothe numerical solution (solid curve), while the dashed curve repre-sents the threshold curve in the absence of mode quantization.Because the actual thresholds are larger than the predictions, itwas concluded that another source of energy dissipation was pres-ent. The small inset images in Fig. 26 are examples of patternsobtained at different frequencies. The Faraday instability wasexamined experimentally and theoretically by Barrio et al. [306]and found that for certain liquids, in which viscosity is very smalland with peculiar physical properties such as a high molecularweight and density, there exists an additional dissipation mecha-nism that introduces an important nonlinear term in the equationsof motion. This mechanism was found to produce the stabilizationof symmetric patterns. The model equation was solved numeri-cally and the patterns produced were analyzed and compared withexperimental images. QPs were observed in two different Faradaywave experiments, one with a low-viscosity deep layer of fluidwith single-frequency parametric excitation [204,205,283],and the other with a high-viscosity shallow layer of fluid and forc-ing with two commensurate temporal frequencies [301]. The pat-tern instability for a layer of a viscous fluid in a large aspect ratio

Fig. 26 Threshold measurements for oil at 55F in a squarecontainer of width of 11.4 cm and depth of 0.7 cm. The solid lineis the finite depth numerical calculation. The dashed line repre-sents the threshold curve in the absence of mode quantization(only the tongue minima are shown). The insets show wave pat-terns at onset for different frequencies [304].



container subject to vertically arbitrarily periodic excitation was stud-ied by Chen and Wei [307]. The instabilities for Faraday water wavesystem under excitations of the triangle and square waves wereanalyzed.

Under single-frequency excitation and depending on the fluidviscosity of glycerol/water mixture, the pattern of squares wasobserved for kinematic viscosity below � � 0:70 cm2=s. Forlarger viscosity � � 1:00 cm2=s, the pattern changes to parallellines as shown in Fig. 27. The lines preferred to be perpendicularto the sidewalls, especially at high excitation amplitude. For thecircular geometry of the container, the lines of the pattern arebowed. It is seen that the wavelength at the center is smaller thannear the circumference. As the excitation amplitude increases, aspontaneous defect pair generation was observed similar to thebehavior in convective rolls at low Prandtl number [308,309]. Itwas observed that these defects migrate outward until they vanishnear the boundary and then another defect pair is produced. If theexcitation amplitude is abruptly increased, patterns of circles andspirals were observed by Edwards and Fauve [301]. The spiralpattern was found to rotate with a period of about one minute.Two topological defects of the same sign belonging to the waves,traveling in opposite directions in one pair, may form a stablebound state in parametrically excited capillary ripples [310]. Notethat a perfect structure consists of two mutually orthogonal pairsof standing waves. Individual dislocations in the form of suchbound states may interact with each other. It was shown that thedislocations may either annihilate, if they have opposite topologi-cal charges, or are arranged into quasi-stable states in the form ofa linear chain, in the case of like topological charges. Kalini-chenko et al. [128,311] derived the conditions for harmonic insta-bility of the free surface of a liquid of low viscosity in arectangular vessel of finite horizontal dimensions.

Spirals and targets were studied by Kiyashko et al. [312] usingsilicon oil with viscosity � ¼ 1:0 cm2=s, density q ¼ 0:97 g=cm

3,

surface tension coefficient c ¼ 20:5 dyn=cm, and fluid depth of0.5 cm. It is known that plane standing waves are completely sta-tionary with nonmoving nodes and antinodes. On the other hand,standing waves forming targets and spirals drift slowly toward thecore [312]. The speed of the drift was found to depend on themagnitude of vertical excitation acceleration, as shown in Fig. 28.Wave drift phase velocity was found to sensitively depend on theprofile of the side walls of the fluid cell. It is seen that the driftvelocity is maximal for a straight vertical wall, and is reduced fora wall with a step. It was assumed that the wave drift is related toa shear flow produced near the wall by rapidly decaying surfacewaves at the fundamental frequency, generated by an oscillatingmeniscus. This shear flow was observed even in the subcritical

regime when the parametric instability is absent. Near the thresh-old, the magnitude of velocity of the shear flow was close to0.5 mm/s. Both contracting targets and rotating multi-armed spi-rals were found to persist for a long time (a few hours) in the cav-ity. The experimental images of the topology of the free surfacewere documented in Ref. [312] and it was shown that multi-armedspirals with different values of topological charge emerge as aresult of dislocations interacting with the background target pat-tern. Dislocation pairs were observed produced by perturbing rollsat the periphery of the target. One dislocation of the pair quicklydisappears at the wall, and another moves toward the center alongsome curved trajectory as shown in Fig. 29. The velocity of thedefect was found to increase as it approaches the core. When sev-eral defects were introduced on the target, stable multi-armed spi-rals were observed. The evolution of a target with four dislocations(two positive and two negative) is shown in Fig. 29(a). A spiral isformed when one of the defects is already at the center while theothers are still on the periphery as shown in Fig. 29(b), and later atarget is restored as shown in Fig. 29(c) when all the dislocationsannihilate at the center. In another experiment, a two armed spiralwas formed and persisted for a long time as shown in Fig. 29(d).

Faraday waves in a stadium shaped container were experimen-tally studied by Kudrolli et al. [313]. Figure 30 shows samples ofsurface wave patterns generated at different parametric excitationfrequencies. As pointed by Agam and Altshuler [314], the experi-mental findings are quite intriguing, since these patterns appear in90% of the cases, and their magnitudes is unexpectedly large.Agam and Altshuler [314] further examined the problem andshowed that, at sufficiently low frequencies, the wave patterns are“scars” selected by the instability of the corresponding periodicorbits, the dissipation at the container side walls, and interactioneffects which reflect the nonlinear nature of Faraday waves. Largeboundary dissipation was found to prevent the formation of ran-dom patterns as well as other modes such as the whispering gal-lery modes. Moreover, increasing the boundary dissipation, bylowering the fluid level, should eventually suppress all scarsexcept the horizontal one.

Two mechanisms were proposed for QP formation. The first isapplied to single-frequency forced Faraday waves and was testedexperimentally by Westra et al. [206]. The second was developedto explain the origin of the two length scales in superlattice pat-terns [219,236], which were observed in two-frequency experi-ments [237]. These two mechanisms were examined by Rucklidge

Fig. 27 Surface pattern of essentially parallel lines wasobserved for fluid mixture of 88% glycerol and 12% water andkinematic viscosity of m � 1:00 cm2=s under parametric excita-tion of single frequency of 80 Hz [301]

Fig. 28 Dependence of velocity of wave drift (measured bynode displacement) on the parametric excitation accelerationfor two different profiles of side walls. 1—Vertical profiles of theside wall and the structure of the shear flow and 2—vertical wallwith corner step near the bottom [312]



and Silber [315] with the purpose of explaining the selection ofQPs in single- and multifrequency forced Faraday wave experi-ments. Both mechanisms were used to generate stable QPs in aparametrically forced partial differential equation that shares

some characteristics of the Faraday wave experiment. The firstmechanism was found to be robust for single-frequency forcingand two different forcing strengths, for which 12-fold and 14-foldQPs were obtained. The second mechanism, which requires more

Fig. 29 Experimental images of Faraday ripples in laboratory experiment: (a) a targetwith four dislocations (two positive and two negative); (b) one dislocation is attracted tothe target core, spiral is formed; (c) all dislocations are attracted to the center and annihi-lated, perfect target reappeared (images (a)–(c) are separated by 2.0 s); and (d) asymptoticstate of another experiment where a three-armed spiral was formed and rotated for a longtime (one period of a standing wave corresponds to two white and two dark stripes onthe photos due to time averaging) [312]

Fig. 30 Surface wave patterns in a stadium shaped container at various external frequen-cies [313]



delicate tuning, was used to select particular angles betweenwave-vectors in the QP. Nonlinear three-wave interactionsbetween driven and weakly damped modes play a key role indetermining which patterns are favored. Rucklidge and Sliber[316] performed quantitative comparisons between the predictedpatterns and the solutions of the system partial differential equa-tion based on strong damping of all modes apart from the drivenpattern-forming modes. This is in conflict with the requirementfor weak damping if three-wave coupling is to influence patternselection effectively. Rucklidge and Sliber [316] distinguishedtwo different Faraday experiments that three-wave interactionscan be used to stabilize QPs, and examples of 12 -, 14 -, and20-fold approximate QPs were presented. The first is an experi-ment with a low-viscosity deep layer of fluid with single-frequency excitation [204,205,283]. The second is with ahigh-viscosity shallow layer of fluid and forcing with twocommensurate temporal frequencies [301].

Free-surface waves, which are synchronous with the excitation,were shown to occur in thin layers of fluid vibrated at low fre-quency in Refs. [78–80]. They also occur in certain viscoelasticfluids [297] and in fluids forced periodically, but with more thanone frequency component [301,317]. For each case, it is possibleto tune the forcing parameters in order to access the transitionbetween subharmonic and harmonic response. At codimension-two point, both instabilities set in simultaneously, but with differ-ent spatial wave numbers. Kumar [78] showed that harmonicinstability could only occur in a fairly shallow viscous liquid in avessel of infinitely large horizontal dimensions. Coupled evolutionequations can be written for the various wave amplitudes. Thecoupling coefficients depend on the angles between the wave-vectors, and these coupling functions depend in turn on theimposed parameters such as wave frequency.

A nonlinear analytical approach dealing with the wave patternselection for parametric surface waves, not restricted to fluids oflow viscosity was presented in Ref. [318]. A standing wave ampli-tude equation was derived from the Navier–Stokes for viscous flu-ids. The associated Lyapunov function was calculated fordifferent regular patterns to determine the selected pattern nearthreshold as a function of a damping parameter C ¼ 2�k2

0=x0,where x0 ¼ X=2. For C � 1, it was shown that a single wave (orstripe) pattern is selected. For C� 1, patterns of square symmetryin the capillary regime, a sequence of sixfold (hexagonal), eight-fold patterns in the mixed gravity–capillary regime, and stripe pat-terns in the gravity dominated regime were depicted. Figure 31shows the symmetry of the preferred patterns predicted by Chenand Vi~nals [318] in the parameter space defined by the viscosityof the fluid and excitation frequency for a fluid depth of 0.3 cm,

density of 0:95 g=cm3, and surface tension of 20.6 dyn/cm, and

the experimentally observed patterns are indicated by symbols (asreported in Ref. [319]). It is seen that stripe patterns are preferredat high viscosity, whereas at low viscosity, hexagons (at lowfrequency) and squares (at high frequency) were observed.The shallowness of the fluid layer was accounted for the observa-tion of a hexagonal pattern for viscosity of 1 cm2=s and low fre-quency, and not observing a quasi-periodic pattern for viscosity0:04 cm2=s and excitation frequency of 27 Hz. The experiments ofBinks and van de Water [204] detected this latter region in a deepfluid layer.

The parametric excitation of two mutually orthogonal pairs ofthe quasi-monochromatic (modulated) capillary waves on the sur-face of a liquid layer was studied by Reutov [320]. The numericalsolution predicted the formation of the modulation lattice with tet-ragonal cells with disorder. The disorder was manifested as localvariations of the period of the vertical and horizontal lines of thelattice. The weakly nonlinear modal interaction for a finite depthof fluid subjected to a vertical oscillation was analytically devel-oped from the full Navier–Stokes equations by Skeldon and Gui-doboni [321]. The coefficients of the amplitude equations and theconsequences for stability of different spatially periodic patternsin the infinite depth case were calculated. Although symmetryarguments provided a qualitative explanation for the selection ofsome of these patterns, quantitative analysis was found to be hin-dered by mathematical difficulties inherent in a time-dependent,free-boundary Navier–Stokes problem. Skeldon and Porter [322]reconsidered weakly nonlinear behavior and compared the scalingresults derived from symmetry arguments in the low viscositylimit with the computed coefficients of appropriate amplitudeequations using both the full Navier–Stokes equations and areduced set of partial differential equations due to Zhang andVin�als [35,207]. An optimal viscosity range was found for locat-ing superlattice patterns experimentally.

The phase relaxation of ideally ordered patterns of Faraday waveswas examined by Kityk et al. [323]. A combined frequency–amplitude modulation of the excitation signal a periodic expansionwas used and dilatation of a square wave pattern was generated. Itwas shown that the measured relaxation time allows a precise evalua-tion of the phase diffusion constant. Later, the spatiotemporal behav-ior of Faraday surface waves was studied by Kityk et al. [324]. Thebifurcation of a doubly hexagonal superstructure out of a simple hex-agonal pattern was presented. Some attempts were made to predictthe surface elevation profile of Faraday surface waves [325]. Localsurface deformation measurements using a focused laser beam werepresented by Westra et al. [206] who obtained quantitative values ofthe temporal phases. Kityk et al. [326] presented the results of experi-mental measurements of the complete spatiotemporal Fourier spec-trum of Faraday waves generated at the interface of two immiscibleliquids of different density. The container consists of an aluminumring of diameter of 18 cm with its bottom and top made of two paral-lel glass plates with a gap between them of 10 mm. The gap wasfilled by two immiscible liquids: silicone oil of viscosity of20 m � Pa � s, density of 949 kg/m3 and an aqueous solution (sugarand nickel sulfate) of viscosity of 7.2 m � Pa � s and density of1346 kg/m3. The liquid–liquid interfacial tension was measured to be35 6 2 dyn/cm yielding a capillary length of 3 mm. Kityk et al. [326]obtained the bifurcation scenario from the flat surface to the pat-terned state for each complex spatial and temporal Fourier compo-nent separately. For example, Fig. 32 shows the dependence of thespatial amplitude on the normalized acceleration ðe ¼ ða0 � acÞ=ac,where ac is the critical acceleration above which the interfacial sur-face in unstable) for modes 10 and 11. The amplitudes were meas-ured at subharmonic, harmonic, and superharmonic frequencies. Thefigure shows the regions of pure square (see Fig. 33) and hexagonal(see Fig. 34) patterns and a narrow region of mixed patterns. It wasshown that the energy is transferred from lower to higher harmonicsand the nonlinear coupling generated static components in the tempo-ral Fourier spectrum leading to a contribution of a nonoscillatingpermanent sinusoidal deformed surface state. A comparison of

Fig. 31 Regions of surface wave patterns in terms of fluid vis-cosity and excitation frequency, experimental results are shownby 3 for stripes pattern, w for square pattern, D for hexagonalpatter. Alternating 3 and w indicate mixed stripe–squarepatterns [318].



hexagonal and rectangular patterns reveals that spatial resonance cangive rise to a spectrum that violates the temporal resonance condi-tions given by the weakly nonlinear theory.

9.2 Two-Frequency Parametric Excitation. It is believedthat Edwards and Fauve [289,301,327] were the first to study the

two-frequency driven Faraday instability. The general form oftwo-frequency parametric excitation is

€ZðtÞ ¼ a cosðvÞ cosðmxtÞ þ sinðvÞ cosðnxtþ /Þ½ � (32a)

where the phase angle 0 deg � v � 90 deg represents the rela-tive mixing between the two modes and the angle / describestheir phase difference, 0 < / < 2 pm=n.

The importance of the correlation length, which determines theinfluence of the shape of the container on the wave pattern wasdiscussed by Edwards and Fauve [301]. The square pattern con-sisting of two perpendicular standing waves was found to occur athigh frequencies [275,277,281,283,296]. Edwards and Fauve[301] presented the experimental results obtained with single-frequency and two-frequency parametric excitation of fluid cell ofdiameter of 12 cm and depth of 0.29 cm. They adopted the tech-nique of Benjamin and Scott [328] using a precisely machinedcorner of a container filled to the prim. In this case, the fluid freesurface becomes pinned at discontinuity of the slope of sidewallof the cylindrical container and the VOF is adjusted so that thesurface is flat everywhere and thus has no meniscus. Thus, thebrim-full state provides a homogeneous Dirichlet condition on thesurface height in which the no-slip condition for viscous fluid

Fig. 32 Bifurcation diagrams showing the dependence of the spatial amplitudes on thenormalized acceleration at different excitation frequencies for (a) mode 10 and (b) mode 11,the excitation frequency is X 5 12ð2pÞ rad=s [326,364]

Fig. 33 Snapshots of the square Faraday patterns over a unitcell at excitation frequency of 12 Hz and acceleration of 30 m/s2:(a) maximum surface elevation and (b) minimum surfaceelevation [326]

Fig. 34 Snapshots of hexagonal Faraday patterns: (a)–(c) at excitation frequency of 12 Hz andacceleration of 39:3 m=s2 for three different temporal phases: (a) down hexagons; (b) patternnear the minimal surface elevation; and (c) up hexagons [326]



states at a solid boundary will have zero velocity relative to theboundary. Several other containers including square, hexagonal,and octagonal were used to verify that patterns do not depend onthe container geometry. The excitation was generated as a combi-nation of two sinusoidal components of frequencies X1 ¼ 4x andX1 ¼ 5x, with x=2p ¼ 14:60 Hz, such that the general form ofthe excitation acceleration is

€ZðtÞ ¼ a cosðvÞ cosð4xtÞ þ sinðvÞ cosð5xtþ /Þ½ � (32b)

where the phase of 4x component is zero by choice of time originand the phase angle / is associated with 5x component.

Under two-frequency excitation given by Eq. (32b) with fre-quency components 4x and 5x for x=2p ¼14:6 Hz, / ¼ 75 deg,and v ¼ 45 deg, the free surface was found to take hexagons pat-tern as shown in Fig. 35(a). Hexagons-to-lines transition wasfound to co-exist over a finite range of bifurcation parameterl � ða� acÞ=ac, where ac is the threshold excitation accelerationamplitude above which the free surface is unstable. Figure 35(b)shows a 12-fold QP observed under excitation frequencyx=2p ¼ 28:0 Hz, / ¼ 68:4 deg, and v ¼ 72 deg.

The way in which these different patterns become stable orunstable as the parameters are varied was analyzed in some detailand experimentally verified by Binks and van de Water [204]. Anexample of a quasi-crystalline pattern was presented by Golluband Langer [329] and is shown in Fig. 36. Binks and van de Water[204] conducted an experimental investigation on a circularcontainer of 44 cm diameter and fluid depth of 2 cm.They used fluid of kinematic viscosity of � � 0:03397 cm2=s,density q ¼ 0:8924 gm=cm

3, and surface tension c ¼ 1:83 J=m

2

¼ 1830 dyn=cm. Under parametric excitation, different fluid sur-face patterns were observed depending on the parametric excita-tion frequency. Figure 37 shows samples of fluid surface wavepatterns in a cylindrical container of 44 cm diameter and fluiddepth of 2 cm under different values of excitation frequency wherefourfold square ðn ¼ 2Þ, sixfold hexagonal ðn ¼ 3Þ, eightfoldquasi-periodic ðn ¼ 4Þ, and tenfold quasi-periodic ðn ¼ 5Þ emergeclose to the onset. It was found that the n¼ 2, 3, and 4 patternsreveal a clear long-range orientation order, with minor defectsmanifested by a slow bend in the case of the square pattern, andthe appearance of triangularlike structures in the case of ðn ¼ 3Þ.These apparent triangles are the result of a p-phase defect, as thealteration of hexagons and triangles can be reversed by shiftingthe phase at which the image is taken by p. For the eightfoldquasi-periodic pattern, there appears to be a point defect in the topcentral portion of the image. The ordering of the tenfold pattern is

only strong in the central region, although some tenfold orienta-tion order can be observed throughout the image.

Zhang and Vi�nals [35,207] developed a weakly nonlinearanalysis for the dynamics of small amplitude surface waves on asemi-infinite weakly inviscid fluid layer. Kudrolli et al. [237]experimentally observed that two-frequency parametricallyexcited waves produce an intriguing “superlattice” wave patternnear a codimension-two bifurcation point where both superhar-monic and harmonic waves occurred simultaneously, but with dif-ferent spatial wave numbers. The superlattice pattern issynchronous with forcing, spatially periodic on a large hexagonallattice, and exhibits small-scale triangular structure. Silber andProctor [330] showed that similar patterns can exist and may bestable if the nonlinear coefficients of the bifurcation problem sat-isfy certain inequalities. Silber et al. [214] used the spatial andtemporal symmetries to indicate that weakly damped harmonicwaves may be critical to understanding the stabilization of suchpattern in the Faraday system.

Fig. 35 Fluid free surface hexagons pattern produced by two-frequency parametricexcitation 4x and 5x components for (a) x=2p 5 14:6 Hz, / 5 75 deg, and v 5 45 deg and(b) x=2p 5 28:0 Hz, / 5 68:4 deg, and v 5 72 deg [301]

Fig. 36 Standing wave pattern takes the form of a quasi-crystalline wave with 12-fold rotational symmetry of a layer ofsilicone oil under two-frequency parametric excitation. Thebrightest regions are locally horizontal, whereas darker colorsindicate inclined regions [329].



Two-frequency-forced liquid layer systems were also studiedexperimentally [215,237,298,299]. These studies were conductedboth in the near vicinity and far from the phase angle correspond-ing to the codimension-two point. They revealed a number ofqualitatively superlattice type states. Silber and Skeldon [213]studied the two-frequency Faraday system in the vicinity of thecodimension-two point. They considered forcing ratios m=n to beeither odd/even or even/odd parities, where interactions betweenharmonic and subharmonic waves may occur. Arbell and Fineberg[331] presented an experimental investigation of superlattice pat-terns generated on the surface of a fluid under parametric excita-tion with two commensurate frequencies. Four qualitativelydifferent types of superlattice patterns were generated via a num-ber of different three-wave resonant interactions. They occureither as symmetry-breaking bifurcations of hexagonal patternscomposed of a single unstable mode or via nonlinear interactionsbetween the two primary unstable modes generated by the twoforcing frequencies. Besson et al. [332] showed that a transitionbetween two patterns with different linearly unstable wavelengthscan be obtained in various fluid regimes by changing the relativeamplitudes of a two-frequency parametric excitation function.This transition occurs through a bicritical point, where both modesare simultaneously neutrally stable.

At low excitation frequencies, a hexagonal pattern consisting ofthree standing waves spaced at 120 deg was reported inRefs. [319,333,334]. The preferred surface pattern near the onsetof instability is the one that minimizes a certain functional of the

wave amplitudes. Regular patterns can be formed that are notspatially periodic but do have rotational symmetry, i.e., quasi-crystalline patterns [283,301]. Transitions to spatially and tempo-rally disordered states occur were found to occur when the waveamplitudes are increasing. Some of these states, such as the hex-agonal lattice, were found to disappear, while the striped phasebreaks down in regions where the stripes are most stronglycurved. On the other hand, if the fluid is not too viscous, so thatthe correlation length of the pattern is relatively long, then thesymmetry imposed by the boundaries can be recovered by averag-ing over a large number of individually fluctuating patterns asreported by Gluckman et al. [335]. A case of strongly turbulentcapillary waves was studied experimentally by Wright et al.[336]. The dynamics of a fluid surface filled with high-amplituderipples were also studied using a diffusing light photography,which resolves the height at all locations instantaneously. Whennonlinearities are strong enough to generate a cascade from longwavelength to shorter wavelength, the resulting turbulent statecontains large coherent spatial structures.

Faraday instability of viscous fluids driven by an excitation oftwo frequencies exhibited nonlinear effects, which give rise to ahexagonal pattern, as well as an unusual “QP” with 12-fold orien-tational order. The development of complex states of fluid motionwas illustrated experimentally on film flows by Gollub [337]. Inaddition, surface waves and thermal convection were considered.In one-dimensional pattern, cellular patterns bifurcate to states ofspatiotemporal chaos. In 2D pattern, even ordered patterns can be

Fig. 37 Samples of fluid surface wave patterns in a cylindrical container of diameter of44 cm and fluid depth of 2 cm under parametric excitation showing (a) square symmetryunder excitation frequency f 5 45 Hz, (b) hexagonal symmetry at f 5 30 Hz, (c) eightfoldquasi-periodic at f 5 29 Hz, and (d) tenfold quasi-periodic f 5 27 Hz [204]



surprisingly intricate when quasi-periodic patterns are included.The existence and stability of a 2D pattern with subcritical sym-metrical tenfold patterns in a dissipative system described by par-tial differential equations were discussed by Frisch and Sonnino[338]. This state was numerically shown to be stable even whenthe dynamics are not derived from a free energy functional. Inaddition, nonsymmetric, rhomboidal patterns were also seen to bestable for some parameter values. Later, both the rhomboidal pat-terns and resonant QPs resulting from the above interactions wereobserved experimentally by Arbell and Fineberg [298]. An exam-ple rhombic lattice is shown in Fig. 38(a). The critical Fouriermodes associated with the 21.8 deg superlattice pattern are indi-cated in Fig. 38(b). Figures 38(c) and 38(d) show 12-fold and 14-fold quasi-lattices, up to 11th-order and 7th-order, respectively[339].

Porter and Silber [218] considered the case of two-mode rhom-boids (referred to as resonant triads) and examined the dynamicalconsequences of their weakly broken symmetries. The appliedacceleration was composed of two commensurate frequencies mxand nx (m and n are coprime integers):

€ZðtÞ ¼ amj j cosðmxtþ /mÞ þ anj j cosðnxtþ /nÞ (32c)

where an appropriate selection of amj j and anj j ensures that twomodes (one driven primarily by the am and the other by an) onsetsimultaneously. The lowest order nonlinear interaction is the oneknown as the resonant triads because they produce a strong

coupling between the phases of the three waves involved[236,340]. Porter and Silber [218] considered a simple three-waveinteraction, occurring near the bicritical point, in which two wave-vectors from one critical circle, k1 and k2, generate a third wave-vector k3 ¼ k1 þ k2 on the second critical circle as shown inFig. 39. Patterns composed of these three modes form the two-mode rhomboids. Kumar et al. [341] studied the selection ofrhombic patterns close to a bicritical point at the onset of primarysurface instability in viscous fluids under two-frequency vertical

Fig. 38 Different surface patterns: (a) rhombic lattice with an angle h between the pri-mary wave-vectors, (b) hexagonal superlattice with an angle of 21:8 deg between themost closely spaced wave-vectors. (c) and (d) 12-fold and 14-fold quasi-lattices, up to11th-order and 7th-order, respectively [203,316].

Fig. 39 Schematic diagrams of resonant triads at the criticalpoint. The wave-vectors satisfy k3 5 k1 þ k2: (a) k3 5 k3j j< k1 5 k1j j5 k2j j and (b) k1 < k3 < 2k1 [218].



vibration. Rhombic patterns were found to appear to be natural atthe primary instability in the form of a bicritical point.

Under two-frequency parametric excitation, Rucklidge [342]considered pattern formation in large domains with an attention toQPs, where the appearance of small divisors causes the standardtheoretical method to fail. The symmetry-based approach devel-oped by Tse et al. [343] was used by Rucklidge et al. [344] to ana-lyze three observed spatial period-multiplying transitions from aninitial hexagonal pattern. Three patterns are shown in Fig. 40, inwhich patterns (a) and (b) were both obtained using Dow–Corningsilicone oil with viscosity of 47 cSt ð1cSt ¼ mm2=sÞ and layerdepth of 0.35 cm, while pattern (c) was observed for a 23 cSt oillayer of depth of 0.155 cm. All three patterns were obtained withexcitation containing two frequencies in the ratio 2 : 3. Pattern (a)was obtained under driving frequencies of 50 and 75 Hz, pattern(b) with frequencies of 70 and 105 Hz, and pattern (c) with 40 and60 Hz. Typically, the secondary bifurcations were found to occur

at forcing amplitudes between 10% and 50% larger than the criti-cal acceleration for the primary hexagonal state.

Interacting surface waves, parametrically excited by two com-mensurate frequencies, were studied by Epstein and Fineberg[345–347] who showed that a variety of interesting superlatticetype states are generated via a number of different three-wave res-onant interactions. These states occur either as symmetry-breaking bifurcations of hexagonal patterns composed of a singleunstable mode or via nonlinear interactions between the two dif-ferent unstable modes generated by the two forcing frequencies. Itwas shown that this state results from the competition betweentwo distinct nonlinear superlattice states, each with different char-acteristic temporal and spatial symmetries. It was demonstratedthat the spatiotemporal disorder can be controlled. It was con-trolled to either its neighboring nonlinear states by the applicationof a small-amplitude excitation at a third frequency, where thespatial symmetry of the selected pattern is determined by the tem-poral symmetry of the third frequency. Furthermore, it was shownthat all superlattice states generated by quadratic nonlinearitiesare grid states. Grid states are superlattices in which two corre-lated sets of critical wave-vectors are spanned by a sublatticewhose basis states are linearly stable modes. The spatial resonan-ces inherent in these states were found to increase their stability.The spatial and temporal structure of nonlinear states formedby parametrically excited waves on a fluid surface in a highlydissipative regime was studied by Epstein and Fineberg [348].Short-time dynamics reveal that three-wave interactions betweendifferent spatial modes are only observed when the modes’ peakvalues occur simultaneously. The temporal structure of each modewas described by the Hill’s equation.

9.3 Three- and Multifrequency Parametric Excitation.The addition of a third frequency component was introduced by

M€uller [296] for the purpose of breaking the spatial phase symme-try in the subharmonic regime and thus controlled the transitionbetween triangles and hexagons. This motivated Arbell and Fine-berg [331] to add a third frequency in an attempt to stabilizeasymmetric QPs by modifying the parametric excitation to the fol-lowing form

€ZðtÞ¼A a1 cosðp1x0tÞþa2 cosðp2x0tþ/1Þþa3 cosðp3x0tþ/2Þ½ �(33a)

where A is the total excitation acceleration amplitude and the nor-malized amplitude ratios by a1 : a2 : a3 are adjusted such thata1 þ a2 þ a3 ¼ 1. p1 : p2 : p3 are the three-frequency ratios suchthat p1 < p2 < p3 and /1 and /2 are the phase differences withrespect to the p1 components. By using 2:3:4 driving frequencies,Arbell and Fineberg [331] observed perfect eightfold QP andFig. 41 shows images and their spatial power spectra at two tem-poral phases. This state was found to be subharmonic in time andcan be observed in the region where tenfold QPs and eightfolddistorted QPs were observed.

Fig. 40 Experimental patterns: Patterns (a) and (b) were bothobtained using Dow–Corning silicone oil with viscosity of 47cSt and layer depth of 0.35 cm, while pattern (c) was observedfor a 23 cSt oil layer of depth of 0.155 cm. All three patterns areobtained with excitation containing two frequencies in the ratioof 2:3. Pattern (a) was obtained under driving frequencies of 50and 75 Hz, pattern (b) with frequencies of 70 and 105 Hz, andpattern (c) with 40 and 60 Hz [344].

Fig. 41 Two temporal phases of an eightfold QP observed for three-frequency driving ((a) and (b)). This state wasobserved in a 50/75/100 Hz experiment with driving amplitude ratio a1 : a2 : a3 5 0.16:0.36:0.48 and a phase differenceof 180 deg between the 100-Hz component and the two other components [331].



Some differences were reported to exist between the two-frequency and three-frequency phase diagrams. For example, theregion of stable hexagons becomes much smaller with the additionof the third frequency term. For two-frequency excitation, there isa large region of hexagons, but for three-frequency excitation,hexagons almost disappear. Another difference is that the 12-foldQP becomes the primary instability when the amplitude of the firstcomponent exceeds a certain value and most of the region corre-sponding to hexagons for two-frequency excitation converts to12-fold QPs with the addition of the finite third component. Fur-thermore, the transition between ordered patterns and a disorderedstate was found to occur at lower driving amplitudes.

Nonlinear three-wave resonant interactions were recognized toplay a key role in pattern selection in Faraday wave experimentsand other situations where complex patterns are found [301]. Forexample, some studies [213,214,217–219,236,238] used symme-try considerations to understand pattern selection in Faraday waveexperiments with two-frequency forcing and three-wave resonantinteractions in the context of weakly broken Hamiltonian struc-ture. This approach was capable to explain several of the experi-mentally observed superlattice patterns and suggested ways ofdesigning multifrequency forcing functions that could be used tocontrol which patterns would emerge [218,236,238]. Theapproach was used to determine which additional frequencies toadd to the excitation function in order to make observed patternsmore robust [331,347,349]. The pattern selection process undermultifrequency parametric excitation was considered by Ruck-lidge and Silber [316] who were able to predict which patternswould be found for different parameter values. They were able topredict the amplitudes and range of stability of the patterns.

Ding and Umbanhowar [349] conducted an experimental inves-tigation to examine the changes in pattern selection which occurwith the addition of a third driving frequency at twice the differ-ence frequency of two different even modes dominant frequencyratios of 4:5 and 6:7. Their experiment was conducted on a 0.65-cm-deep layer of 20 cS silicone oil is held in a cylindrical cell ofradius of 7.0 cm with a polyvinyl chloride sidewall, a 0.8-cm-thickglass bottom, and a plexiglass top covered with a thin, white plas-tic sheet, which serves as a light diffuser. Under excitation fre-quency of 80 Hz, the wave number k of the pattern just aboveonset is 11.8 cm�1. The corresponding dissipation length, definedas ld ¼ 2�k=f is approximately 0.15 cm, where � is the kinematicviscosity and f is the wave oscillation frequency. Since kld � 117and kh � 7:8, the experiments are considered in the weakly dissi-pative and deep fluid layer limits as indicated by Cerda and Tira-pegui [81]. They adopted a three-frequency excitationacceleration of the form

€ZðtÞ ¼ am cosðmxtÞ þ an cosðnxtþ /nÞ þ ap cosðpxtþ /pÞ(33b)

In order to demonstrate the differences between surface patternsfor two-frequency excitation (i.e., with ap ¼ 0) near the bicriticalpoint and those for three-frequency excitation, Ding and Umban-howar [349] generated the phase plots for the two cases. Forexample, Figs. 42(a) and 42(b) show the phase plots for two-frequency excitations selected such that m : n ¼ 4 : 5 or 6 : 7. Forthe case m : n ¼ 4 : 5, a region in parameter space well aboveonset where 12-fold QPs were observed. Figure 42(a) reveals thatfor hexagons and 12-fold QPs, the 4x-excitation component isdominant and the surface waves oscillate at 2x. It is also seen thatfor squares and square two-mode superlattice patterns (2MS) the5x-excitation frequency component is dominant. Dashed linesindicate the region where the higher acceleration resolution meas-urements were performed. With p1 : p2 ¼ 6 : 7 superlattice pat-terns were observed as shown in Fig. 42(b). For hexagons andsuperlattice patterns, the 6x-excitation frequency component isdominant. By adding a small third-frequency component with cor-rect phase to the excitation acceleration 33(b), both QPs andsuperlattice patterns appeared near onset. The stability of thesepatterns was found to be strongly influenced by the thirdfrequency component p3. Figures 43(a) and 43(b) show thephase diagrams for two-frequency excitation m : n ¼ 4 : 5 and

Fig. 42 Phase diagrams for (a) excitation frequency ratio m:n 5 4:5, with x=2p 5 20 Hzand /5 5 16 deg; (b) excitation frequency ratio m:n 5 6:7, with x=2p 5 16:5 Hz and/7 5 40 deg [349]

Fig. 43 Phase diagrams for varying a4 and a5 for (a) two-frequency excitation m : n 5 4 : 5 and (b) three-frequency exci-tation m : n : p 5 4 : 5 : 2, where x=2p 5 20 Hz and /5 5 16 deg.(*) Hexagons, (w) QPs, ( 1 ) squares and 2MS (two-mode super-lattice patterns), and (•) disordered. Unmarked regions belongto a flat surface state [349].



three-frequency excitation m : n : p ¼ 4 : 5 : 2, respectively. ForFig. 43(b), the phase angle /p¼2 ¼ 32 deg and ap¼2 ¼ 0:8g¼ 0:59a2c, where a2c ¼ 1:36g is the critical acceleration for singlefrequency excitation at 2x. The main differences between two- andthree-frequency excitations are seen in the fact that the region of sta-ble hexagons with the addition of the third frequency component wassignificantly diminished. The 12-fold QP becomes the primary insta-bility for a5 5g and most of the region corresponding to hexagonsfor two-frequency excitation is converted to the 12-fold QPs with theaddition of the third frequency component. Another difference isobserved in the transition between ordered patterns and disorderedstate at lower excitation amplitudes.

The evolution of patterns in large aspect ratio driven capillarywave experiments were described by Milner [350] who derivedthe amplitude equations including nonlinear damping terms.Above the excitation threshold amplitude, a reduction of the num-ber of marginal modes yielded a simple form of the amplitudeequations, which have a Lyapunov functional. This functionaldetermines the wave number and symmetry (square) of the moststable uniform state. Frisch and Sonnino [338] studied the exis-tence and stability of a 2D pattern with a tenfold orientationalorder in a dissipative system described by partial differentialequations. The pattern was found to appear in a subcritical wayand result from a superposition of two linearly unstable patternswith different wave numbers. Pattern selection phenomena inparametrically excited surface waves were studied by Umeki[351,352] for the case of weakly nonlinear system of three modes.The third-order coefficients of nonlinear interaction between twoline patterns intersecting at an arbitrary angle were obtained in thegravity–capillary waves of arbitrary depth. Classification, stabilityanalysis and bifurcation study of the fixed points of the dynamicalequations with linear damping were performed in the cases ofthree symmetric linear modes and general modes. It was shownthat squares are the most preferred pattern in capillary waves,while lines are selected in gravity waves. Two types of three-wave interactions were considered by Rucklidge et al. [353]. Thefirst is when two waves of the shorter wavelength interact withone wave of the longer, while the second type is when two wavesof the longer wavelength interact with one wave of the shorter.The two types of three-wave interactions were found to providean explanation of some of the Faraday wave phenomena in theexperiments of Huepe et al. [295].

The relationship between the linear surface wave instabilities ofa shallow viscous fluid layer and the shape of the periodic,parametric-forcing function that excites them was studied byHuepe et al. [295]. Huepe et al. [295] adopted a set of parametricexcitation functions parametrized by the power n

f ðxtÞ ¼ N 2:5 cosðxtÞ þ 3n cosð3xtÞ � 5n cosð5xtÞ½ � (33c)

where x is the fundamental frequency of oscillation and N is anormalization constant which is defined such thatmaxðjf ðxtÞjÞ ¼ 1. Note that the acceleration amplitude, a whenmultiplied by f ðxtÞ the result is the actual parametric excitationacceleration. Figure 44 shows a set of time history records off ðxtÞ for four different values of n¼�2, �0.3, 0.5, and 1. Thecorresponding neutral stability boundaries present the usual reso-nance tongue structure. The harmonic and subharmonic tonguesindicate regions where surface waves become unstable, oscillatingwith a main frequency component that is an integral multipleðx; 2x; 3x; :::Þ or an odd half-multiple ðx=2; 3x=2; 5x=2; :::Þ ofthe fundamental excitation frequency, respectively. The tonguesat higher modes (k is the wave number) correspond to instabilitieswith shorter surface wavelengths and higher oscillation frequen-cies. As n is increased, the excitation function changes from asimple rounded triangular shape with only two extrema per cycleto shapes with richer structure. It is seen that the envelope definedby the tongue minima, indicated by a dashed line changes from asimple convex function with a single minimum to a set of convex

segments, each with its own minimum. Huepe et al. [295] indi-cated that the changes in the critical instabilities shown in Fig. 44cannot be explained by a simple switch to a different dominantforcing frequency in f ðxtÞ. Indeed, as n is increased to one thelowest unstable region becomes the second harmonic tongue (withmain frequency component equal to 2x), which does not corre-spond to the fundamental harmonic or subharmonic responses(with equal or half the frequency, respectively) to any of the threefrequency components of f ðxtÞ: x; 3x; and 5x.

9.4 Numerical Simulation of Faraday Waves. Numericalsimulations of faraday wave patterns were considered for two-and 3D flow in rectangular and circular containers of large aspectratios. Numerical algorithms, such as trapezoidal, second-orderAdams–Bashforth, marker-and-cell, and finite-difference projec-tion methods, were adopted in the literature. For example, Zhangand Vi~nals [354] presented a numerical simulation of parametri-cally driven surface waves in fluids of low viscosity based on thelinear damping quasi-potential equations, which remain rotation-ally invariant. The equations governing fluid motion in the bulkand the appropriate boundary conditions at the free surface areapproximated by a nonlocal set of equations involving surfacecoordinates alone. The numerical analysis was performed forlarge aspect ratio. They used the trapezoidal scheme for linearterms and a second-order Adams–Bashforth scheme (multiplestep method) for nonlinear terms. The numerical simulations pre-dicted stable patterns of square symmetry above onset in the capil-lary dominated regime. A sequence of patterns of lower symmetrywere predicted in the vicinity of surface tension parameter,ck3

0=ðqx20Þ ¼ 1=3, where k0 is a typical wave number, and x0 is

the fundamental natural frequency of the free surface, as thedamping parameter is increased.

The dynamics of 2D standing periodic waves at the interfacebetween two inviscid fluids with different densities, subject tomonochromatic oscillations normal to the unperturbed interface,was numerically simulated under normal- and low-gravity condi-tions by Wright et al. [21,355]. The numerical simulation wasbased on boundary-integral method that is applicable when thedensity of one fluid is negligible compared to that of the other,and a vortex-sheet method that is applicable to the more generalcase of arbitrary densities. Viscous dissipation was simulated bymeans of a phenomenological damping coefficient added to theBernoulli equation or to the evolution equation for the strength ofthe vortex sheet. A comparative study revealed that the boundary-integral method is generally more accurate for simulating themotion over an extended period of time, but the vortex-sheet for-mulation is significantly more efficient. Nonlinear effects for non-infinitesimal amplitudes were manifested at the extremes of theinterfacial oscillation; growth of harmonic waves with wave num-bers in the unstable regimes of the Mathieu stability diagram; andformation of complex interfacial structures including paired trav-eling waves.

Standing surface waves on a viscous fluid driven parametricallyby a vertical harmonic oscillation were examined by Murakamiand Chikano [356] based on direct numerical simulations of the2D Navier–Stokes equation, together with appropriate boundaryconditions. The condition for the onset of the waves in the experi-ments by Lioubashevski et al. [357] was reproduced using numeri-cal simulation. The form of the surface elevation was analyzedand the dependence of the saturated amplitude on the forcingstrength exhibited normal bifurcation. Murakami and Chikano[356] discussed the velocity fields of the 2D standing wavesdeveloped near the instability onset. Nonlinear wave dynamics inparametrically driven surface waves were studied in numericalsimulations of the 2D Navier– Stokes equation by Chen [358].Modulating behavior of primary wave modes in a particular pa-rameter range and in time scales much longer than the underliningwave periods was reported. Valha et al. [359] performed a numeri-cal study to examine the behavior of a gas liquid interface in a



vertical cylindrical vessel subjected to a sinusoidal verticalmotion. The computational method was based on the simplifiedmarker-and-cell method and includes a continuum surface modelfor the incorporation of surface tension. The numerical resultsindicated that the surface tension has very little effect on the pe-riod and amplitude of oscillations of the interfacial waves. Thestability of the interfacial waves was found to depend on the initialpressure pulse disturbance, and exponential growth of the interfa-cial wave was predicted in some cases. The results were found ingood agreement with available experimental and analyticalsolutions.

A numerical analysis of 2D Faraday waves was presented byUbal et al. [360] based on direct numerical simulation of Navier–-Stokes and continuity equations with appropriate boundary condi-tions. Stability maps on the excitation amplitude-wave numberplane for viscous liquid layers with equilibrium depths between5� 10�5 meter and 10�5 meter were presented. The results werecompared with those obtained by Benjamin and Ursell [10] for aninviscid fluid, and by Kumar and Tuckerman [262] for a viscous

fluid. The results confirmed the previous findings obtained by lin-ear stability analysis. In particular, stronger excitation forces areneeded to produce unstable waves as the thickness of the film isreduced. The lower boundary of the unstable regions in the stabil-ity charts appeared to move toward higher wave number values.

The Faraday waves were examined numerically using somesimplifications and reductions. For example, a set of simplifiedequations referred to as the coupled amplitude streaming flowequations were derived for the case of weakly nonlinear small vis-cosity flow by Mart�ın et al. [258] and Higuera et al. [135]. Two-and 3D Faraday waves were studied with periodic boundary con-ditions by O’Connor [361]. The full Navier–Stokes equationswere solved including the complex dynamics of the free-surfacewaves to gain a better understanding of the interplay between theviscous boundary layers, the nonlinear streaming flow, and thebulk flow. The formation of a square wave pattern was predictedand found in qualitative agreement with experiments of Kudrolliand Gollub [319]. Nonlinear effects of standing waves in fixedand vertically excited tanks were numerically investigated by

Fig. 44 Time records of parametric excitation f ðxtÞ and their corresponding stability boun-daries on acceleration amplitude a versus wave number k for (a) n 5 22, (b) n 5 20:3, (c)n 5 0:5, and (d) n 5 1. The plots are for x 5 20p rad=s, h 5 0.3 cm, excitation acceleration ampli-tude a is units of g, fluid density q 5 0:96 g=cm3, kinematic viscosity m 5 46 cm2=s, and surfacetension c 5 20 dyn=cm. The resonance tongues labeled by H and SH refer to regions with har-monic or subharmonic instabilities, respectively. Their envelope shown by dashed lineschange with n [295].



Frandsen and Borthwick [362]. Numerical solutions of the gov-erning nonlinear potential flow equations were obtained using afinite-difference time-stepping scheme on adaptively mappedgrids. A horizontal linear mapping was applied, so that the result-ing computational domain is rectangular, and consists of unitsquare cells. The small-amplitude free-surface predictions in thefixed and vertically excited tanks were in agreement with second-order small perturbation theory. For steep initial amplitudes, thepredictions were found to differ considerably from the small per-turbation theory solution, demonstrating the importance of nonlin-ear effects.

P�erinet et al. [363] performed numerical simulations of Faradaywaves in three-dimensions for two incompressible and immiscibleviscous fluids. The Navier–Stokes equations were solved using afinite-difference projection method coupled with a front-trackingmethod for the interface between the two fluids. The critical accel-erations and wave numbers, as well as the temporal behavior atonset were compared with the results of the linear Floquet analy-sis by Kumar and Tuckerman [262]. The finite-amplitude resultswere found in agreement the experimental results of Kityk et al.[326]. The detailed spatiotemporal spectrum of both square andhexagonal patterns was reproduced for excitation acceleration of30:0 m=s

2, which is the acceleration used in the experimental

investigation of Kityk et al. [326,364] for square patterns. Figures45(a) and 45(b) represent examples of the patterns obtained at

saturation and are taken from the same simulation at the two dif-ferent instants. The symmetries characterizing the squares (reflec-tions and p/2 rotation invariance) are clear, showing a firstqualitative agreement with Kityk et al. [326] where both structureswere observed. The pattern oscillates subharmonically, at 2 T,where T is the forcing period. Figure 45(a) is taken when the inter-face attains its maximum height, while Fig. 45(b) is taken at atime 0.24� 2 T later. At this later time, the dominance of a higherwave number was observed.

A direct numerical simulation of Faraday waves was carriedout by P�erinet et al. [365] in a minimal hexagonal domain. Theyobserved alternation of patterns referred to as quasi-hexagons andbeaded stripes. Starting from zero velocity and an initial randomperturbation of the flat interface, their numerical simulations pro-duced a hexagonal pattern which oscillates subharmonically withthe same spatiotemporal spectrum as reported by Kityk et al.[326]. It was reported that hexagons are transient and difficult tobe stabilized experimentally and are competing with squares anddisordered states as indicated by Wagner et al. [366] and Kityket al. [326]. In their numerical simulations and after about ten sub-harmonic periods, P�erinet et al. [365] predicted a drastic departurefrom hexagonal symmetry. A fully nonlinear numerical simulationof 2D Faraday waves between two incompressible and immisciblefluids was performed by Takagi and Matsumoto [367] whoadopted the phase-field method developed originally by Jacqmin[368]. In the nonlinear regime, qualitative comparison was madewith an earlier vortex-sheet simulation of two-dimensional Fara-day waves by Wright et al. [21].

The majority of Faraday wave studies have been carried out ontraditional viscous and inviscid fluids under deterministic excita-tion. Nontraditional fluids, such as magnetic fluids, ferrofluids,and liquid crystals subjected to nontraditional sources of paramet-ric excitation, including random excitation, have received theattention of few studies as will be demonstrated in Secs. 10and 11.

10 Faraday Waves of Other Media

Nontraditional sources of parametric excitation include electro-static forces and convective temperature gradient. The parametricresponse of the interface between two dielectric liquids under analternating electrostatic force was studied in Refs. [369–373].Their studies showed the stability of the interface required theapplied voltage to be high enough to suppress surface tensioneffects and lower than a certain analytically determined criticalvalue. For voltages greater than this critical value, Reynolds [369]indicated that the interface is unstable. The stability of the equilib-rium was found to depend not only on the mean temperature gra-dient, as in Rayleigh’s problem [374], but also on the amplitudeand frequency modulation. The free-surface oscillations of a mag-netic liquid were studied in Refs. [375–378]. The parametric exci-tation of weakly nonlinear surface waves in a magnetic fluidsubject to a periodically oscillating magnetic field was examinedby Pursi and Malik [379]. Transverse homoclinic orbits leading toa chaotic transition were obtained together with necessary condi-tions for the existence of chaos. Reimann et al. [380] conducted aFaraday experiment with magnetic fluid in a direct current mag-netic field driven externally by accelerative modulation. A patternof standing twin peaks was observed whose origin was believed inthe simultaneous excitation of two different wave numbers in thenonmonotonic regime of the dispersion relation.

Gershuni and Zhukhovitskii [381] discovered a parametric reso-nance called convective instability in a fluid body subject to aperiodically varying temperature gradient. Later, Gershuni andZhukhovitskii [382] found that the modulation of the vertical tem-perature gradient had the same influence as the modulation of theangular velocity of rotation of the fluid as a rigid body. The stabil-ity of the equilibrium was found to depend not only on the meantemperature gradient, as in Rayleigh’s problem [374], but also onthe amplitude and frequency modulation.

Fig. 45 Example of square pattern showing the height of inter-face over the horizontal coordinates, when the height is maxi-mal. Resolution in x, y, and z is 80 3 80 3 160. Note that thevertical scale is stretched with respect to the horizontal scale:(a) at a given time t and (b) at another instant, time 0:24 3 2Tafter time t [363].



Dissipative patterns caused by spin-wave instabilities in insulat-ing ferromagnetic films driven by out-of-plane parallel pumpingwere studied by Elmer [383]. It was found that the only stable pat-terns are squares, hexagons, and quasi-periodic patterns based onthree standing waves. Quantitative results strongly support thesuggestion that these patterns should be experimentally observ-able by means of Faraday rotation. At the surface of a magneticfluid, parametric waves can be excited by an alternating magneticfield, parallel to the surface. With this anisotropic system, Bacriet al. [384] observed a pattern transition from parallel rolls, per-pendicular to the magnetic field, to a rectangular array of crossrolls. The free-surface waves of a magnetic fluid subjected to anormal magnetic field were examined experimentally by Bro-waeys et al. [385]. The waves were generated by a small modula-tion at frequency of the vertical field. It was concluded that alinear and inviscid analysis is sufficient to fit well the experimen-tal data, except in the vicinity of the critical field where a surfaceinstability occurs. A linear stability analysis of the free surface ofa horizontally unbounded ferrofluid layer of arbitrary depth sub-jected to vertical vibrations and a horizontal magnetic field waspresented by Mekhonoshin and Lange [386]. A nonmonotonicdependence of the stability threshold on the magnetic field wasfound at high frequencies of the vibrations. It was found that themagnetic field can be used to select the first unstable pattern ofFaraday waves. In particular, a rhombic pattern as a superpositionof two different oblique rolls can occur.

Another type of fluid medium known as ferrofluid has been thesubject of some studies. A ferrofluid is a liquid which becomesstrongly magnetized in the presence of a magnetic field. Ferro-fluids are made of nanoscale ferromagnetic particles suspended ina carrier fluid (usually an organic solvent or water). Each tiny par-ticle is thoroughly coated with a surfactant to inhibit clumping. Astability theory for the onset of parametrically driven surfacewaves of a ferrofluid was developed by M€uller [273] who consid-ered the effects of viscous dissipation and finite depth effects. Itwas shown that a careful choice of the filling level permits thenormal and anomalous dispersion branches to be measured. Fur-thermore, it was demonstrated that the parametric driving mecha-nism may lead to a delay of the Rosensweig instability. When aparamagnetic fluid is subjected to a strong vertical magnetic field,the surface forms a regular pattern of peaks and valleys. Thiseffect is known as the normal-field instability or Rosensweiginstability. A bicritical situation can be achieved when Rose-nsweig and Faraday waves interact.

The singular surface electromagnetic waves are guided by aplane boundary of a chiroplasma half-space if chirality and plasmaparameters are properly matched. They form a complete set of sur-face polaritons jointly with the Rayleigh surface waves and general-ized surface waves [387,388] in the Voigt geometry. Fisanov andMarakasov [389] indicated that a parallel-plate waveguide supportsunder certain conditions singular propagating modes whose charac-teristics drastically depend on the type of the boundary conditions.

The nonlinear stability of magnetized standing waves on theplane interface separating two immiscible inviscid magnetic fluidsin a cylindrical container in the presence of both a uniform mag-netic field and a periodic acceleration normal to the free surfacewas studied by Elhefnawy [390]. It was found that the modulationof the amplitudes and phases of the two-mode resonant waves aregoverned by a system of nonlinear first-order differential equa-tions which in turn utilized to determine the steady-state solutionsand consequently investigating their stability. It was concludedthat the frequency-response curves exhibit the transcritical andHopf bifurcations. The nonlinear evolution of the three-dimensional instability of standing surface waves along the inter-face of a weakly viscous, incompressible magnetic liquid within arectangular basin was studied by Sirwah [391–393]. A combina-tion of the Rosensweig instability with Faraday instability wasdeveloped, where the system is assumed to be stressed by a nor-mal alternating magnetic field together with an external verticaloscillating force. The nonlinear equations of the complex

amplitudes corresponding to the ideal fluid case were modified byadding the linear damping. It was shown that the liquid viscosityrather than the magnetic field affects the qualitative behavior ofthe wave motion and the system response alternates between theregular periodic and chaotic behavior depending on the specificvalues of some parameters.

Liquid crystals, such as smectic and nematic, are characterizedby properties between those of conventional liquid and those ofsolid crystal. For instance, a liquid crystal may flow like a liquid,but its molecules may be oriented in a crystal-like way. One of thephases of liquid crystals is known as the thermotropic liquid crys-tals, which occur in a certain temperature range. Many thermo-tropic liquid crystals exhibit a variety of phases as temperature ischanged. For instance, a particular type of liquid crystal moleculemay exhibit various smectic and nematic phases as temperature isincreased. The linear stability of Faraday waves on the surface ofa smectic-A liquid crystal and polymer gel–vapor systems offinite-thicknesses was studied by Ovando-Vazquez et al. [394].For the case of highly viscoelastic gel media, it was found thatthere are co-existing surface modes of harmonic and subharmonictypes that correspond to peaks in the plot of the critical accelera-tion as a function of wave frequency. Larger frequencies werefound to lead to subsequent peaks of co-existing subharmonicwaves. A quantitative theoretical linear analysis of the depend-ence of the forcing acceleration and wave number on the strengthof the external excitation frequency for the Faraday instability insmectic-A (Sm-A) liquid crystals was presented by Hern�andez-Contreras [395]. A layer of Sm-A under a constant horizontalmagnetic field applied in the direction of the wave-vector that ori-ents the molecules in a stack of lamellar layers perpendicular tothe air–liquid interface was considered. Hern�andez-Contrerasdetermined the dispersion relation and instability boundary interms of acceleration-wave number of the surface waves in finite-thickness layers. Hern�andez-Contreras [395] found that Faradaywaves develop in thin (0.1 mm and 0.05 m) smectic-A liquid crys-tal layers at low frequencies of external driving acceleration andin the long wavelength limit. For Sm-A under an external mag-netic field, it was found that there are alternating subharmoni-c–harmonic branches with almost the same vanishing criticalexcitation acceleration as occurs in ideal inviscid fluids. Anincrease of the modulating frequency was found to make the sub-harmonic waves to preempt the harmonic ones with a lowerthreshold nonzero acceleration.

Time reversal of the excitation parameters has some consequen-ces on the stability of a periodically driven dynamic system. Inparticular, the dissipative pattern-forming system of electrohydro-dynamic convection of nematics driven by time-periodic externalelectric fields was studied by Heuer et al. [396]. The dynamicmodel describing the linear stability at the beginning was repre-sented by a set of two differential equations for the amplitudes ofthe dynamic variables. It was found that under time reversal ofany periodic excitation function, the dynamic equations reproducethe same stability thresholds, pattern types, and critical patternwavelengths. Nematic electrohydrodynamic convection underexcitation with superimposed harmonic wave was studied byPietschmann et al. [397]. A time reversal of the excitation wasfound to have no effect on the threshold voltages and patternwavelengths obtained in linear stability analysis. This symmetrywith respect to time reversal of the excitation was found to breakdown close to the transition from the conduction regime to thedielectric regime. Later, Pietschmann et al. [398] demonstratedthat the threshold parameters for the stability of the ground stateare insensitive to a time inversion of the excitation function. TheFaraday system was found to share this property with standardelectroconvection in nematic liquid crystals. In general, timeinversion of the excitation was found to affect the asymptotic sta-bility of a parametrically excited system, even when it isdescribed by linear ordinary differential equations. The patternselection of the Faraday waves above threshold, on the otherhand, discriminates between time-mirrored excitation functions.



The parametric instability in nematic liquid crystal layers wasstudied by Hern�andez-Contreras [399] using linear stabilitytheory. The critical acceleration and wave number of the unstablestationary waves were found discontinuous at the nematic-isotropic transition temperature and conform to similar sharpchanges experienced by the viscosity and surface tension as afunction of temperature. Due to Marangoni flow, the curve of thecritical acceleration as a function of excitation frequency wasfound to exhibit a minimum. If the Marangoni flow is neglectedand the dynamical viscosity is increased, a monotonously increas-ing dependence of the acceleration in terms of oscillation fre-quency was observed. A bicritical instability was found to occurfor a layer thickness of a few millimeters. A well-defined subhar-monic wave was achieved when the thickness of the layer was fur-ther increased.

11 Random Excitation of Faraday Waves

Different sources of parametric excitations are encountered indifferent applications of liquid sloshing dynamics. For example,Bechhoefer and Johnson [400] considered a fluid container drivenparametrically by a triangle waveform. Parametrically excited sur-face waves excited by a repeating sequence of N delta-functionimpulses, were studied by Catll�a et al. [401]. With impulsive forc-ing, the linear stability analysis may lead to an implicit equationfor neutral stability boundaries. The familiar situation of alternat-ing subharmonic and harmonic resonance tongues was found toemerge only if an asymmetry is introduced in the spacing betweenthe impulses. By varying the spacing between the up and downimpulses making up the 2p-periodic forcing function for N¼ 2impulses per period, it was found that the magnitude of the 1:2spatiotemporal resonance effect depends dramatically on the cor-responding asymmetry parameter, which measures deviation fromequal spacing of the impulses. This resonance was found to occurfor impulsive forcing even when harmonic resonance tongues areabsent from the neutral stability curves. An experimental study ofthe parametric resonance and finite-amplitude parametric oscilla-tions arising in a liquid-filled U-tube subject to alternating verticalexcitation was presented by Briskman et al. [402]. Two forms ofoscillations in the liquid together with their corresponding rangesof unstable equilibrium with respect to small random perturba-tions were reported. The studies of random parametric sloshinghave not yet reached the maturity stage as its counterpart of thedeterministic studies. However, these studies have been carriedout for the cases of external random parametric excitation in regu-lar gravitational field and due to g-jitter field. The next two sub-sections will address both cases.

11.1 Random Excitation Under Regular Field. The sto-chastic stability of a liquid surface under random parametric exci-tation can be studied in terms of one of the stochastic modes ofconvergence. These modes include convergence in probability,convergence in the mean square and almost sure convergence[403]. The linear stability analysis is based on the stochastic dif-ferential equation of the sloshing mode mn, i.e.,

A00mn þ 2fmnA0mn þ 1þ n00ðsÞ½ �Amn ¼ 0 (34)

where Amn is a dimensionless free liquid surface amplitude ofmode mn, a prime denotes differentiation with respect to the non-dimensional time parameter s ¼ xmnt, xmn is the natural fre-quency of the sloshing mode mn, fmn is the correspondingdamping ratio, and n‘‘ðsÞ is a dimensionless vertical wide bandrandom acceleration of spectral density 2D. The mean square sta-bility condition of the response of Eq. (34) was determined inRefs. [404–408], which is given by the inequality

D=2fmn < 1 (35a)

On the other hand, the sample stability condition is

D=2fmn < 2 (35b)

Dalzell [409] conducted an experimental investigation to measurethe spectral density of the liquid free surface elevation under narrowand wide-band random parametric excitations. Different tests wereconducted at different values of excitation level ranging from 0.09 gto 0.18 g, root-mean-square (RMS). It was observed that such varia-tion did not have significant influence on the response spectral den-sity, implying a saturation feature. The liquid free surface amplitudespectral density revealed also another component at the excitationfrequency. Another important feature was that the RMS accelerationlevel defining the beginning of the one-half-subharmonic responsewas not bracketed. This feature motivated Ibrahim and Heinrich[410] to conduct another experimental investigation, which revealedthe occurrence of “on–off” intermittency.

As the excitation level decreases, the response changes to pre-dominantly harmonic. The wide-band excitation covers the first15 symmetric modes and two excitation levels were applied. Thehigher level exhibited subharmonic response. An abrupt transitionbetween harmonic and subharmonic responses was observedwhen the excitation acceleration level was reduced. An importantfeature of the results showed that low-level harmonic response torandom Gaussian excitation was nearly Gaussian. However, whenlarge amplitude subharmonic response was excited, the probabil-ity distribution changed abruptly into a double-exponential distri-bution. Dalzell [409] conducted a least-square fitting algorithm todevelop a phenomenological probability distribution of the liquidfree surface elevation. The following double exponential distribu-tion was proposed:

PðXÞ ¼ Exp �Exp � pffiffiffi6p X þ c

� �� (36)

where c is the Euler constant¼ 0.577215665, X ¼ g� �gð Þ=rg, �gis the mean value of the fluid elevation, and rg is the RMS of thefluid elevation.

The nonlinear motion of the free liquid surface under randomparametric excitation involves the estimation of stochastic stabil-ity and response statistics of the free surface [407,409,411–413].The free liquid surface height of a sloshing mode mn in a cylindri-cal container was found to be governed by the nonlinear differen-tial equation

A00mn þ 2fmnA0mn þ 1þ n00ðsÞ½ �Amnð1� K1Amn � K2A2mnÞ

þ K3A02mn þ K4AmnA00mn þ K5AmnA

02mn þ K6A2

mnA00mn ¼ 0 (37)

The last four terms in Eq. (37) represent quadratic (for symmetricmodes) and cubic (for asymmetric modes) inertia nonlinearities.Equation (37) represents the nonlinear modeling of any mode mnand does not include nonlinear coupling with other sloshingmodes. Ibrahim and Heinrich [410] observed the followingregimes of liquid free surface state:

(a) Zero free liquid surface motion is characterized by a delta-Dirac function of the response probability density function.The free surface is always flat because the liquid dampingforce prevents any motion of the liquid. The excitationlevel for the first antisymmetric and axisymmetric modesare given by the following ranges, respectively,

0 < D=2f11 < 1:55 (38a)

0 < D=2f01 < 4:98 (38b)

(b) On–off intermittent motion of the free liquid surface. Thisintermittent motion takes place over the following rangesof excitation level for the two modes



1:55<D=2f11<1:82 (39a)

4:98 < D=2f01 < 46:8 (39b)

A corresponding regime, known as undeveloped sloshingwas predicted by Ibrahim and Soundararajan [412]. Thisregime was characterized differently by very small motionof the liquid free surface with an excitation level

2:0 < D=2fmn < 4:0 (40)

The spatiotemporal intermittency of liquid free surfaceunder parametric excitation was later studied by Bosch andvan de Water [285] and Bosch et al. [414].

(c) Partially developed random sloshing characterized byundeveloped sloshing where significant liquid-free surfacemotion occurs for a certain time-period and then ceases foranother period. At higher excitation levels, the time-periodof liquid motion exceeds the period of zero motion. Theexcitation levels of this regime for the two modes are,respectively

1:82 < D=2f11 < 14:05 (41a)

6:8 < D=2f01 < 9:44 (41b)

Heinrich [415] and Ibrahim and Heinrich [410] observed thedevelopment of circular motion of a central spike for the caseof first symmetric mode excitation. Occasionally, the spike isdisplaced from the center of the tank and precesses in such away that preserves the azimuth symmetry in a time-averagesense. This motion was first reported by Gollub and Meyer[265] who studied the amplitude dependence on the preces-sion frequency under harmonic parametric excitation.

(d) Fully developed sloshing characterized by continuous ran-dom liquid motion for all excitation levels exceeding theprevious regime. When the first symmetric sloshing modeis excited, higher sloshing modes are excited as well. Thisis why the bandwidth of the symmetric mode excitationwas narrowed in the test. The same observation of the inter-action with other modes was reported by Dalzell [409].With regard to the first antisymmetric mode, another non-linear phenomenon was observed that is the rotationalmotion of the nodal diameter. It is known that such phe-nomenon is created due to the coupling of the modes abouttwo orthogonal axes of the circular container.

Figures 46(a) and 46(b) show the dependence of mean andmean square responses of the first antisymmetric sloshing modeon the excitation spectral density level D=2f11. The squares andcrosses refer to the values as the excitation level increases thendecreases, respectively. It is seen that the values of the mean andmean square responses remain zero up to a critical excitation levelabove which the free surface begins to lose stability. The intermit-tent sloshing region is included in the zero-response motion.Figure 47 shows the mean square response of the first axisymmet-ric sloshing mode. This figure demonstrates the intermittentregion and another region where higher modes co-exist with thatmode. Figures 48(a) and 48(b) show the measured probabilitydensity function of the first antisymmetric and axisymmetricsloshing modes for excitation levels D=2f11 ¼ 7:39 andD=2f01 ¼ 9:43, respectively. The dashed curve represents theGaussian probability density function. The diamond points referto the measured probability density, while the solid curve isaccording to the double exponential proposed by Dalzell [409]. Itis seen that the double exponential curve gives a moderately goodfit for most of the experimental results, except the region that isclose to the mean value of the liquid elevation.

Note that the excitation spectral level was limited to a lowerlevel within a narrow-band in order to avoid parametric excitation

with other modes. Mixed mode interaction under random excita-tion has not been treated in the literature. The measured probabil-ity density function of the liquid response was found to benon-Gaussian for regions of large subharmonic motion with non-zero mean. In view of the inertia nonlinearity in the equation ofmotion (37), there is limited number of technical approaches topredict the response statistics. The stochastic averaging carriedout to second order of the deterministic part, is very powerful buttedious. Another alternative is the Monte Carlo simulation, whichis also very useful in studying modal interaction.

11.2 Random Excitation Due to g-Jitter. Under microgravi-tational field, g-jitter acts as a parametric random excitation [416].During space missions, microgravity experiments revealed signifi-cant levels of residual accelerations referred to as g-jitter[417–420]. The residual acceleration field can be decomposed

Fig. 46 Dependence of (a) mean and (b) mean squareresponse of the first antisymmetric sloshing mode on excita-tion spectral density level [410]

Fig. 47 Dependence of the mean square of the first symmetricsloshing mode on excitation spectral density parameter [410]



into a quasi-steady (or systematic component) and a fluctuatingcomponent known as g-jitter. Typical values of the quasi-steadycomponent gsk k are around 10�6gE (gE is the gravitational accel-eration on the earth’s surface). The fluctuating contribution, gðtÞ,is random in nature and has a characteristic frequency of 1 Hz orhigher. Alexander [421] provided experimental results pertainingto the sensitivity of liquid behavior to residual acceleration at lowgravitational field.

The influence of g-jitter on fluid flow ranges from order of mag-nitude estimates to detailed numerical calculations [421,422].These include modeling the acceleration field by some simpleanalytic functions in which the acceleration is typically decom-posed into steady and time-dependent components. The time-dependent component was considered periodic by Farooq andHomsy [423]. Statistical description of the residual accelerationfield on board spacecraft was considered by Zhang et al. [424],and Thomson et al. [419,425]. The effects of g-jitter on thecapillary surface motion were studied by Chao et al. [426]. Thestate-of-knowledge of g-jitter problems in microgravity was docu-mented by Nelson [418]. Each spatial Fourier mode of the free-surface displacement in a slightly viscous fluid satisfies theequation of a damped parametric harmonic oscillator in the linearregime. Zhang et al. [424] described the free-surface motion by alinear differential equation with a random coefficient and exam-ined the stochastic stability boundary of the equilibrium state. Theequation of motion of fluid wave height was given in the form

d2gdt2þ 4�k2 dg

dtþ k½g0 � gzðtÞ�gþ

rq

k3g ¼ 0 (42a)

where g ¼ gðk; tÞ is the 2D Fourier transform of the interface dis-placement gðx; y; tÞ, g0 is a constant background gravitationalfield, and gzðtÞ is the fluctuating component of the residual accel-eration in the z-direction. Equation (42a) can be written in theform

d2gdt2þ c

dgdtþ ½x2

0 � nðtÞ�g ¼ 0 (42b)

where c ¼ 4�k2, x20 ¼ g0k þ rk3=q, and nðtÞ ¼ �kgzðtÞ. Zhang

et al. [424] modeled the residual acceleration field, nðtÞ, as a zero-mean Gaussian narrow-band random noise defined by theexpression

nðtÞ ¼ G1ðtÞ cos Xtþ G2ðtÞ sin Xt (43)

where G1ðtÞ and G2ðtÞ are two independent stationary Gaussianprocesses with zero mean and correlation functions

RGðt� t0Þ ¼< n2 > dije�jt�t0 j=s; i; j ¼ 1; 2 (44)

where s is a correlation time. The correlation function of nðtÞ is

Rnðt� t0Þ ¼< n2 > e�jt�t0 j=s cos½Xðt� t0Þ� (45)

The corresponding power spectral density function is

SgðxÞ ¼1

2p< g2 >

1

1þ s2ðXþ xÞ2þ 1

1þ s2ðX� xÞ2

" #(46)

Analytical results for the stability of the response second momentwere presented in the limits of low-frequency oscillations andnear the region of subharmonic parametric resonance. Casade-munt et al. [427] extended the work of Zhang et al. [424] andincluded an additive term due lateral acceleration component. Fora given spectrum of g-jitter, they found a band of unstable modesat low frequencies. They explained the behavior of free surface ofa liquid in a rectangular container undergoing vertical randomforcing characterized by a narrow-band spectrum close to subhar-monic resonant conditions. The analysis was carried out theoreti-cally by means of a weakly nonlinear analysis, assuming the ratiobetween the acceleration of the tank and the gravitational acceler-ation to be small. It was shown that the range of unstable frequen-cies significantly widens. The maximum equilibrium amplitude ofthe free-surface waves for the random excitation case was foundto be smaller than that of the single-frequency excitation and itdecreases as the spectrum width is increased.

Repetto and Galletta [428] estimated the time evolution of theensemble average, Xj jh i, of the surface amplitude for differentvalues of the excitation spectrum width s and for a wave to excita-tion frequency ratio close to 0.48. Their results, shown in Fig. 49,reveal that a constant value of Xj jh i is achieved at high values ofthe slow time scale s ¼ at=g, where a is the amplitude of vertical

Fig. 48 Comparison of measured and predicted of liquid freesurface response probability density function [410]. (a) Firstantisymmetric mode and (b) first symmetric mode.

Fig. 49 Time evolution of the ensemble average Xj jh i for differ-ent values of the spectrum width s: ______: sinusoidal accelera-tion, _ _ _ _: s 5 0.1; - - - -: s 5 0.4; ......: s 5 0.8 [428]



acceleration of the tank. Furthermore, it is seen that the asymp-totic values of Xj jh i are smaller than the equilibrium amplitudereached in the case of sinusoidal excitation. Figure 50 shows thebifurcation diagram of the average equilibrium amplitude Xj jh iplotted as a function of the wave frequency ratio. It is seen thatthe random forcing produces a significant widening of the unsta-ble range of frequencies with respect to the sinusoidal excitationcase. This effect becomes more marked as far as the spectrumwidth is increased. Figure 50 also shows that the maximum aver-age equilibrium amplitude of the Faraday waves is invariablysmaller in the case of the random oscillations than in the case of asinusoidal excitation. The work of Repetto and Galletta [428] wasextended by Blass and Romero [429] and Blass et al. [430] whoexamined the stochastic stability of capillary–gravity waves to atime-varying gravitational field arising from a random verticalexcitation. The evolution equations for weakly nonlinear Faradaywaves in a cylinder subjected to a narrow-band, random accelera-tion were developed by Miles [431] who showed these equationsto be similar (isomorphic) those obtained by to Repetto andGalletta’s [428] for the 2D problem.

The aperiodic Mathieu equation was studied by Poulin andFlierl [432] to demonstrate that aperiodicity can be both stabiliz-ing and destabilizing. As the basic state detunes from a particularfrequency, the maximum growth rates of instability tend todecrease, but the parameter regime over which the instabilitiesoccur widens. It was found that in the case of moderate stochastic-ity, a stochastic mode with large growth rates occurred. This sto-chastic mode was defined as a mode of instability that occursuniquely for a stochastic perturbation, and can push the nearly lin-ear solution into the fully nonlinear regime where the dynamicsbecome inherently more complicated, and quite often chaotic. Inaddition to the classical parametric modes, there is also an unsta-ble stochastic mode that arises only for moderate stochasticity.The power spectrum of the solutions revealed that an increase instochasticity tends to narrow the width of the subharmonic peakand increase the decay away from this peak. Stastna and Poulin[433] considered the stability of a shallow fluid layer parametri-cally excited by a random function of time. They used theoreticallinear stability analysis and high-resolution numerical simulations,including both individual realizations and ensemble calculations,of the nonlinear system of equations. It was found that two differ-ent stochastic modes of instability exist. Both modes find theirexpression in finite-amplitude oscillations of the free surface thatexhibit sharp crests and broad troughs that resemble the classicalStokes wave. The subdominant instability was found to resembleclassical parametric resonance that can exist in a harmonicallyoscillated layer of fluid, and occurs even when the flow is alwayssnapshot stable (or the gravitational acceleration is non-negative).

12 Closing Remarks and Conclusions

This article provided an overview of experimental, analyticaland numerical results pertaining to the problem of parametricsloshing, which was originally observed by Faraday in 1831. Thisproblem has attracted the interest of Engineers, Physicists andMathematicians as reflected by the exponential flow of publishedresults. For example, aerospace Engineers have been involved inthe study of Faraday waves, among other problems of liquidsloshing dynamics, since the time of space travel using liquid pro-pellant rockets. Generally, Engineers have treated the hydrody-namic of parametric sloshing of fluids with filling depths greateror smaller than the critical fluid depth. The critical fluid depth isthe depth above which the free-surface oscillations behave like asoft nonlinear system. In the neighborhood of internal resonance(or autoparametric resonance) among sloshing modes, Faradaywaves may experience complex behavior in the form of energyexchange and mode competition. In particular, Physicists dealtwith the physics of small layers of fluid and studied the competi-tion of modes and the mechanisms and selection of wave patternsgeneration.

The damping associated with Faraday waves has receivedextensive analytical and experimental studies. Note that the meas-ured values involved significant uncertainties due to temperaturedifferences, contamination of the free surface, etc. The differencebetween the predicted and observed damping ratio for the domi-nant mode with a clean surface was 20%. Henderson et al. [68]measured the damping rates and natural frequencies of the funda-mental axisymmetric mode in circular cylinders when the contactangle between the water and the side walls was acute, obtuse, andnear p=2. It was found that damping rates decrease with increas-ing contact angle, while the natural frequencies increased withincreasing contact angle. In general, damping is an inherentparameter that involves the most significant level of uncertainty,and for this reason, structural dynamicists adopted a probabilisticdescription of the damping parameter, which is usually repre-sented by a random variable with a given probability distribution[63]. Future research should involve developing statistical phe-nomenological modeling of the damping of liquid sloshing basedon extensive experimental tests conducted at nearly the sameenvironmental temperature, same container surface finish for agiven particular fluid. It should be noted that an additional nonlin-ear effect due to damping was reported in the literature. For exam-ple, it was indicated that it is necessary to determine thedissipation in the main body of the liquid, the dissipation in theboundary layers at the sidewalls and at the surface, and to deter-mine the dissipation due to capillary hysteresis. Decent [74]extended Miles’ calculations [58] to obtain the cubic dampingcoefficient. Both theoretical and the experimental results revealedthat the cubic damping coefficient for deep water is positive forwater depths greater than approximately 1.2 cm. Furthermore, itwas shown that the free-surface amplitude equation exhibits aHopf bifurcation when the cubic damping coefficient is positive,which is absent for negative cubic damping coefficient. This Hopfbifurcation was found to give rise to a stable limit cycle solution,which corresponds to a time-modulated standing wave, where themaximum amplitude of the standing wave varies with the slowtime scale.

It was shown that the presence of a surfactant results in anincrease of the force required for developing wavy interface.However, the dependence of the force on Marangoni number wasfound nonmonotonic when the surface viscosity is negligible.When Marangoni number is small, the largest concentration gradi-ent was formed when the convective transport is largest. Then, atthis instant the Marangoni traction produces its maximum effecton the interfacial velocity, slowing the motion of the liquid fromthe trough to the crest of the wave; therefore, the tangential veloc-ity turns to zero before the free surface attains its maximum defor-mation. In thin layers of viscous fluids at low excitationfrequencies, there is a possibility of a bicritical point at the

Fig. 50 Bifurcation diagram based on the ensemble averagevalues of the equilibrium Xj jh i for different values of the spec-trum width: s: ______: sinusoidal acceleration, _ _ _ _: s 5 0.1; - - - -:s 5 0.4; ......: s 5 0.8 [428]



instability threshold point, where both subharmonic and harmonicsurface waves can be excited for the same value of excitation am-plitude acceleration. When the thin layer of a viscous fluid is sub-jected to a uniform and steady rotation, and in the absence of theMarangoni force, a tricritical point involving two harmonic solu-tions with different wave numbers and a subharmonic solutionwas reported. On the other hand, in the presence of both Maran-goni and Coriolis forces, it was found that a tricritical pointinvolves two subharmonic solutions with different wave numbersand a harmonic solution can exist.

Nonlinear Faraday waves may exhibit nonlinear phenomenasuch as nonplanar motion, rotational motion, chaotic motion, andfree-surface disintegration. As the strength of the applied excita-tion increases, the response undergoes a sequence of transitions tochaotic dynamics. The origin of these transitions is linked to thepresence of heteroclinic connections between the trivial state andspatially periodic standing waves. These connections are associ-ated with cascades of gluing and symmetry-switching bifurca-tions. When regular waves are disturbed in their temporal andspatial symmetries but retain the connectivity of the oscillatingfluid volume they become irregular. Breaking Faraday wavesoccur when separate droplets shed from the free surface of thefluid. It was shown that the smaller fluid height the larger is theminimum excitation amplitude required for the surface disintegra-tion. Furthermore, the frequency of a spray-excited low-frequencywave was found to be independent of the liquid height-to-diameterratio. Experimental investigations of breaking waves revealed fas-cinating motions of different modes such as high elevation sharpcrest angle, flat crest with double plunges to the sides of the crest,and round crest. The most unusual feature is table-top breaker inthe form of a flat-topped wave crest with almost vertical sides.

The development of the theory of Faraday waves has witnesseda number of controversies regarding the analytical treatment ofsloshing modal equations and modes competition. For example,Dodge et al. [19] developed a finite-amplitude analysis for a circu-lar cylindrical container, but their equations of motion for themodal amplitudes were found by Miles [43] to violate reciprocityconditions. Miles [43] rectified this problem by performing somealgebraic manipulation that resulted in nonlinear inertia terms inthe first antisymmetric mode. On the other hand, Miles [248] andMiles and Henderson [28] disputed the analytical results of Meronand Procaccia [51,52] on the basis of the lack of symmetry condi-tions in the modal equations and thus do not lead to the canonicalformulations. Meron and Procaccia [249] claimed that the Hamil-tonian formulation of Miles [43] yields disagreement with the lin-earized hydrodynamics treatment. It was affirmed by Meron andProcaccia that the difference between these coefficients is smallfor almost degenerate models of the order ðx4;3 � x7;2Þ=X andcan be considered as a high-order correction.

The problem of nonlinear coupling of sloshing modes resultedin nonlinear resonance referred to as internal or autoparametricresonance. For example, internal resonance condition 1:2 requiresthe fluid height to be relatively small, which causes excessiveenergy dissipation. Such energy dissipation suppresses nonlinearphenomena making it impossible to verify the analytical resultsexperimentally. When the frequencies are nearly equal, the freeoscillation in a nearly square container all nonsymmetric modeshave nearly equal natural frequencies independent of the fluiddepth. The case of 1:1 internal resonance was found to result inperiodic and quasi-periodic standing and traveling waves. Chaoticbehavior may arise due to competition between two different spa-tial modes or patterns in a circular container. An axisymmetricmode and two completely degenerate antisymmetric modes ofgravity waves in a circular cylindrical container exhibited a regionof mode competition emerges in which the fluid surface can bedescribed as a superposition of two modes with amplitudes havingslowly varying envelopes. These slow variations were found to beeither periodic or chaotic. Symmetry breaking in circular andsquare containers was found to result in a nonlinear competitionbetween two nearly degenerate oscillatory modes. This interaction

destabilizes standing waves at small amplitudes and amplifies therole played by the streaming flow.

Faraday waves provide a convenient experimental system forstudying pattern formation due to fast time scales and large aspectratio. When Faraday waves are excited well beyond the thresholdfor pattern formation, the ordered patterned structure is lost. Thetransition from an ordered pattern to disorder corresponding todefect-mediated turbulence was found to be mediated by a spa-tially incoherent oscillatory phase, which consists of highlydamped waves that propagate through the effectively elastic lat-tice defined by the pattern. Two mechanisms for selecting themain frequency responses that are different from the first subhar-monic one were identified. The first mechanism occurs when twoor more frequency components are introduced in the parametricexcitation. Each component will tend to excite its own corre-sponding first subharmonic mode. Their relative amplitudes willdetermine which of these responses has the lowest global excita-tion strength threshold, which establishes the instability that isobserved at onset. The second mechanism can only arise in thehigh viscosity regime. As the viscous boundary layer reaches thebottom of the fluid container, the threshold of the lowest unstablemodes rises allowing others with higher main frequency compo-nents to become unstable at onset.

Faraday waves were observed to be especially versatile andexhibit the common patterns familiar in convection such asstripes, squares, hexagons, and spirals. These patterns include tri-angles, QPs, superlattice patterns, time-dependent rhombic pat-terns and localized waves. Under two-frequency parametricexcitation, transition between two patterns with different linearlyunstable wavelengths can be obtained in various fluid regimes bychanging the relative amplitudes of the excitation function. Thistransition was found to occur through a bicritical point, whereboth modes are simultaneously neutrally stable. Faraday instabil-ity of viscous fluids driven by an excitation of two frequenciesexhibited nonlinear effects, which give rise to a hexagonal pattern,as well as an unusual QP with 12-fold orientational order. Somedifferences were reported to exist between the two-frequency andthree-frequency phase diagrams. For example, the region of stablehexagons becomes much smaller with the addition of the third fre-quency term. For two-frequency excitation, there is a large regionof hexagons, but for three-frequency excitation, hexagons almostdisappear. Another difference is that the 12-fold QP becomes theprimary instability when the amplitude of the first componentexceeds a certain value and most of the region corresponding tohexagons for two-frequency excitation converts to 12-fold QPswith the addition of the finite third component.

The pattern formation and selection due to fast time scales hasbeen extensively studied by Physicists experimentally, analyti-cally and numerically. This problem together with Faraday wavesof other media such as dielectric liquids, magnetic liquids, ferro-fluids, smectic and nematic liquid crystal layers constitutes themajor theme of the published results considered in this article.Significant effort was made in order to understand and predict thepattern selection using analytical and numerical tools. Mecha-nisms for selecting the main frequency responses that are differentfrom the first subharmonic one were identified in the literature.One of the significant contributions is that the energy is trans-ferred from lower to higher harmonics and the nonlinear couplinggenerated static components in the temporal Fourier spectrumleading to a contribution of a nonoscillating permanent sinusoidaldeformed surface state.

The problem of random parametric excitation under gravita-tional field and g-jitter of liquid free surface has been discussed interms of stochastic stability conditions for one mode excitation.The random excitation was considered band limited in terms ofstochastic modes of convergence. Experimentally, the excitationbandwidth is limited around the mode in question. As the excitationspectral density is allowed to increase, the free surface exhibits com-plex motion. The fluid surface amplitude under different values ofthe excitation spectrum width may reveal different scenarios



depending on the excitation statistical parameters. Mixed mode exci-tation under wide band forcing function of different values of powerspectral density is still an open area for new studies. The fluid depthwill play an important factor including small fluid layers. The litera-ture has reported few studies on random parametric excitation of liq-uid layers. This issue deserves further experimental and analyticalstudies together with Monte Carlo simulation.

One of the main objectives of Engineers is to stabilize the freeliquid sloshing under parametric resonance. The phenomenon ofnoise-enhanced stability has not yet been explored in an attemptof delaying the exit time of the free-surface response to unstabledomain by imposing additive noise to the fluid container. If anadditive random excitation is imposed to the system one would beinterested in estimating the mean value of first passage time forthe response to reach a target value. Under an additive randomnoise, the mean exit time of a Brownian particle moving in apotential field was found to decrease with noise intensity[434,435] or some universal scaling function of the same parame-ters [436,437]. The dependence of the mean exit time on the noiseintensity for metastable and unstable systems was revealed tohave resonance character [438]. Noise can modify the stability ofthe system in a counterintuitive way such that the system remainsin the meta-stable state for a longer time than in the deterministiccase [439–441]. The escape time has a maximum at some noiseintensity. Note that the phenomenon of noise-enhanced stabilityonly results in an increase of the escape time rather than to causeabsolute stabilization of the originally unstable system. A compre-hensive assessment of the problem of stabilizing the parametricexcited systems through multiplicative and additive randomnoises was presented by Ibrahim [442].

Nomenclature

Amn ¼ a dimensionless free liquidsurface amplitude of modemn

amn ¼ nondimensional waveheight of mode mn

B ¼ qgR2=c ¼ Bond numbersc ¼ 2�k2 ¼ damping coefficient

cp ¼ specific heat at constantpressure (J/kg K)

Ca ¼ q�2=ðchÞ ¼ capillary numberCg � � gh3 þ ðch=qÞ½ ��1=2� 1

where Cg is the capillary–gravity number

dc ¼ characteristic dimensionD ¼ mass diffusion coefficient

(m2/s)G ¼ gh3=� ¼ Galileo number

G ¼ �gðtÞk ¼ body force per unit volumeacting on the fluid

h ¼ fluid depthi, j, and k ¼ unit vectors along x, y, and

z-axes, respectivelyk ¼ wave number�k ¼ thermal conductivity

(W/m K)L ¼ characteristic length

Lx and Ly ¼ rectangular tank length inthe x-direction and widthin the y-direction,respectively

Ma ¼ �ðdc=dTÞðLDT=laÞ ¼ Marangoni numberp ¼ fluid pressure

Pe ¼ LU=a ¼ P�eclet numberPr ¼ �=a ¼ cpl=�k ¼ Prandtl number

r1 and r2 ¼ principal radii of curvatureR ¼ tank radius (for circular

tank)

Rg ¼ffiffiffiffiffiffiffiffigR3

p=� ¼ gravitational Reynolds

S ¼ c= cþ qgh2ð Þ ¼ gravity–capillary balanceparameter

Stk ¼ ~tU0=dc ¼ Stokes numbert ¼ the time in s~t ¼ relaxation time

T ¼ temperature in KTn ¼ period of nth sloshing

modeU ¼ velocity (m/s)

v ¼uiþ vjþ wk ¼ the flow velocity vectorZc ¼ critical excitation

amplitudeZ0 ¼ the amplitude and

frequency of parametricexcitation

a ¼ �k=ðqcpÞ ¼ the thermal diffusivityc ¼ surface tension

d ¼ffiffiffiffiffiffiffiffiffiffiffi2�=x

p¼ boundary-layer thickness

r ¼ ið@=@xÞ þ jð@=@yÞ þ kð@=@zÞ ¼ gradient operatore ¼ x2

11Z0=gfn ¼ damping ratiog ¼ fluid free surface wave

heightj ¼ mean curvature of the free

surfacek ¼ wave length

kmn ¼ the roots of the first deriva-tive of the Bessel functionof the first kind, i.e.,d=dr JmðkmnrÞ½ �jr¼R ¼ 0

l ¼ dynamic viscosity� ¼ kinematic viscosity

nmn ¼ the roots of the first deriva-tive of the Bessel functionof the first kindd=dr JmðnmnÞ½ � ¼ 0

n‘‘ðsÞ ¼ dimensionless verticalwide band random acceler-ation of spectral density2D

q ¼ fluid densitys ¼ xmnt ¼ nondimensional time

parameterU ¼ the velocity potential

functionxn ¼ the natural frequency of

the free surface of mode nxmn ¼ the mn sloshing natural

frequencyX ¼ parametric excitation

frequency

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