## Introduction

Faraday waves refer to nonlinear standing waves, which appear on liquids enclosed by a container excited vertically with a frequency, $Ω$, close to twice the natural frequency $ωn$, of the free surface, i.e., $Ω=2ωn$. This condition is referred to as parametric resonance, since the motion of the liquid free surface is due to an excitation perpendicular to the plane of the undisturbed free surface, and thus the generated waves are also known as parametric sloshing or Faraday waves. Faraday waves are named after Faraday [1,2] who observed the fluid inside a glass container oscillates at one-half of the vertical excitation frequency. Another similar series of experiments, conducted by Mathiessen [3,4], showed that the fluid oscillations are synchronous, i.e., $Ω=ωn$. The contradiction of the two observations led Rayleigh [5,6] to make a further series of experiments with improved equipment and his observations supported Faraday's results. During that time, Mathieu [7] formulated his equations, which helped Rayleigh to explain this phenomenon mathematically. The problem was investigated again by others [8–10] who explained mathematically the discrepancy between Faraday's and Rayleigh's observations and Matthiessen's findings.

Depending on the fluid and excitation parameters, the solution of Mathieu equation can be stable or unstable. The boundaries of stability are usually given in a chart known as Ince–Strutt diagram. Analytical expressions for these boundaries are well documented [11,12]. These boundaries are emanated at excitation frequencies corrersponding to subharmonic (twice the sloshing natural frequecy) or harmonic (same as the sloshing modal frequency) and they enclose regions of instability. Outside these regions, the fluid free surface is stable. Benjamin and Ursell [10] showed that if the plane free surface is unstable, the resulting motion 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] and Matthiessen [3,4] could be correct. However, the experimental results of Benjamin and Ursell [10] only showed that damping results in a threshold excitation amplitude below which the fluid free surface is stable. Woodward [13] suggested that in most real fluids there is sufficient damping such that the unstable regions, except the first several unstable ones, will be located completely above the threshold excitation amplitude level and considerations need only be given to those modes in the lower frequency range. The fundamentals of parametric sloshing are well documented in Ref. [14]. The purpose of this paper is to present the recent advances of Faraday waves and related problems.

Generally, engineers have treated the hydrodynamic of parametric sloshing of fluids with filling depths greater or smaller than the critical fluid depth. The critical fluid depth is the depth above which the free-surface oscillations behave like a soft nonlinear spring. In the neighborhood of internal resonance (or autoparametric resonance) among sloshing modes, Faraday waves may experience complex behavior in the form of energy exchange and modes competition. In particular, physicists dealt with the physics of small layers of fluid and studied the competition of modes and the mechanisms and generation of selected wave patterns. The purpose of this review article is to bring the recent developments of Faraday waves as treated by Engineers, Physicists, and Mathematicians. Some controversies reported in the literature among different researchers regarding the formulations of nonlinear sloshing modal interactions will be assessed.

## Between Faraday Waves and Cross Waves

Faraday waves oscillate subharmonically with the external excitation frequency. They are excited as the forcing amplitude is raised above a critical value. The surface waves could also be harmonic (or synchronous) with the external excitation in the presence of large dissipation. This usually occurs in thin layers of viscous fluids at low excitation frequencies. This leads to the possibility of a bicritical point at the instability threshold point, where both subharmonic and harmonic (synchronous) surface waves can be excited for the same value of excitation amplitude acceleration. The physical interpretation of the resonance at half the excitation frequency was given in Ref. [15] and is demonstrated in Fig. 1. When a fluid layer is excited parametrically at half the excitation frequency, then when the vessel goes down, the fluid inertia tends to create a surface deformation, as in the Rayleigh–Taylor instability1. This deformation disappears when the vessel comes backup, in a time equal to a quarter-period of the corresponding wave. The decay of this deformation creates a flow which induces, for the following excitation period, the exchange of the maxima and the minima. Thus, one obtains the period of one fluid cycle to be twice the period of excitation.

Fig. 1
Fig. 1
Close modal
As mentioned in the Introduction, the fluid-free surface motion under parametric excitation may be described by a system of Mathieu equations in the form
$d2amndτ2+(Pmn-2qmncos2τ)amn=0$
(1)
where $amn$ is the nondimensional wave height of mode mn, $Pmn=4ωmn2/Ω2$, $qmn=2(ξmn/R)Z0tanh(ξmnh/R)$, $τ=ωmnt$, $ωmn$ is the mn sloshing natural frequency, $t$ is the time, $Z0$ and $Ω$ are the amplitude and frequency of parametric excitation, respectively, $ξmn$ are the roots of the first derivative of the Bessel function of the first kind, i.e., $d/dr[Jm(ξmn)]=0$, $R$ is the tank radius (for circular tank), and $h$ is the fluid depth. The domains of instability could be reduced somewhat if linear damping due to fluid viscosity be added in Mathieu equation as [16,17]:
$d2adt2+2cdadt+(ωmn2-Ω2Z0kcosΩt)a=0$
(2)
where $c=2νk2$ is the damping coefficient as assumed in Refs. [17,18] $k$ is the wave number, and $ν$ is the kinematic viscosity. The regions of instability according to Sorokin [17] are determined by the boundaries
$1-(2Z0k)2-4(2cΩ)2<(2ωmnΩ)2<1+(2Z0k)2-4(2cΩ)2$
(3a)
Comparing with the undamped case [19]
$1-2Z0k<(2ωmnΩ)2<1+2Z0k$
(3b)
The stability boundaries described by Eq. (3a) determine the critical excitation amplitude, $Zc$, below which the fluid free surface would remain stable. This amplitude is determined by equating the expression under the radical sign to zero at $Ω/(2ωmn)=1$ to give
$Zc=2ckΩ$
(4)

For $Z0>Zc$, the free-surface amplitude increases at an exponential rate. Brand and Nyborg [20] carried out an experimental investigation to measure $Zc$, i.e., the minimum values of excitation amplitude required for exciting half-frequency surface waves. The measured values of $Zc$ were found to be much greater than those predicted analytically. They attributed this difference to the lack of development in the previous theories of the free-surface damping coefficient. This will be discussed in Sec. 4.

Regarding the interface instability during vertical excitation of the free surface, Wright et al. [21] indicated that during one cycle of fluid oscillations the interface experiences stabilizing and destabilizing acceleration directed from the light to the heavy fluid and vice versa. The interplay between the Rayleigh–Taylor instability prevails during the first half-cycle of the vibration. The stabilizing influence of the fluid acceleration directed from the heavy to the light fluid, prevailing in the second half-cycle of the vibration, is responsible for temporal symmetry of the wave profiles during a complete cycle. This asymmetry should be contrasted with the symmetry of the unforced gravity or capillary waves. If the frequency of oscillations is small, the interface is subjected to destabilizing acceleration for a sufficiently long period. This allows the Rayleigh–Taylor instability to proceed into its nonlinear stages, causing the development of spikes and bubbles.

Faraday waves are distinct from other waves with crests normal to a moving boundary (wave-maker) as cross waves [22–26]. The mechanism of creating the transverse waves observed in wave tanks was discussed in Ref. [27]. It was shown that regular trains of progressive waves created by a wave-maker are unstable to disturbances along their crests, and that the disturbance has a frequency exactly half that of the progressive waves themselves. The possible relevance of this nonlinear subharmonic resonant interaction to the problem of edge wave generation was also discussed. Miles and Henderson [28] provided an account of the history of Faraday waves. They designate those waves associated with an oscillation of the effective gravitational acceleration as Faraday waves and those waves with crests normal to a wave-maker as cross waves or edge waves originally discovered by Stokes [29], which may be subharmonically excited by either an incoming wave or a disturbance moving parallel to the shore. The nonlinear dynamics of nonlinear modulated cross waves of resonant frequency $ω1$ and carrier frequency $Ω≈ω1$ was investigated by Friedel et al. [30]. In a long horizontal container with finite depth, which is subjected to a vertical oscillation of frequency $Ω=2ω1$, the wave can appear in solitary form. It is known that the solitary wave is only stable in a certain parameter regime, depending on damping and driving amplitudes. It was shown how instabilities are saturated following generic routes to chaos in time with spatially coherent structures.

Figure 2 shows different waveforms, in which cross waves are strictly 2:1 subharmonic waves produced by a wave-maker consisting in a partially submerged horizontally vibrating plate. Interaction between both end-walls was found to produce sloshing modes that exhibit new dynamics promoted by wave reflection at the end-walls. Parametric excitation is triggered by the harmonic wave field produced by the vibrating end-walls, which exhibits two distinguished components. On the one hand, two counterpropagating harmonic wave-trains (see Fig. 2(d)) aligned along the vibrating end-walls are present, which propagate (and decay by viscous dissipation) inward from the end-walls. Perez-Gracia et al. [31] examined capillary–gravity, modulated waves of nearly inviscid fluid parametrically excited by monochromatic horizontal vibrations in liquid containers whose width and depth are both large compared with the wavelength of the excited waves. A general linear amplitude equation is derived with appropriate boundary conditions that provides the threshold acceleration and associated spatiotemporal patterns, which compare very well with the experimental measurements and visualizations. The resulting (quasi-periodic) waves are generally oblique, not perpendicular to the vibrating end-walls. This article will not address this type of waves.

Fig. 2
Fig. 2
Close modal

It should be mentioned that the mathematical formulations of stability boundaries and nonlinear response under parametric excitation depend on the fluid boundary value problem. The mathematical formulation of the liquid boundary problem in closed container depends on some assumptions pertaining to the fluid properties and the container geometry. The general boundary value statement of liquid sloshing is described in Sec. 3 for incompressible viscous fluids with the inclusion of its surface tension.

## 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 equations, which are written in a moving Cartesian frame attached to the container, with the plane of x- and y-axes coincide with the unperturbed free surface and the z-axis pointing vertically. The governing equations in the moving frame attached to the container are:

• (1)
The continuity equation for incompressible flow (divergence of velocity field is zero everywhere)
$∇·v=0$
(5)
• (2)
Conservation of linear momentum (Navier–Stokes equation)
$∂v∂t-v·∇v=-1ρ∇p+ν∇2v+G$
(6)

where $v=ui+vj+wk$ is the flow velocity, $ρ$ is the fluid density, $p$ is the fluid pressure, $ν$ is the fluid kinematic viscosity, and $G=-g(t)k$ represents body forces per unit volume acting on the fluid and in the present case is due to the vertical acceleration (including gravitational acceleration). $∇=i(∂/∂x)+j(∂/∂y)+k(∂/∂z)$, $i$, $j$, and $k$ are unit vectors along x, y, and z-axes, respectively. Note that the convective term, $-v·∇v$, on the left-hand side, is the only quadratic nonlinear term and any small amplitude excitation will result in mean fields that evolve on a slower time scale. Such mean fields can either be a simple biproduct of the nonlinear vibration or couple to the primary oscillating field, affecting the dynamics of the system. At low viscosity, the primary oscillating flow satisfies the linearized momentum equation, $∂v/∂t=-∇p/ρ$, implying that the flow is potential and satisfies the condition $∇×v=0$. Note that this condition applies to the bulk and not to the oscillatory boundary layers attached to the solid walls 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 mean flow does not vanish at the outer edge of the boundary layers, but provides a nonzero velocity and a nonzero stress at the edges of the boundary layers near the solid walls and the free surface. And it is these finite boundary values that induce the viscous mean flow in the bulk. These new forcing terms are independent of the viscosity and quadratic in the amplitude of the primary waves, indicating that the effects of the mean flow cannot be neglected, even in the limit of vanishing viscosity, a striking result considering that these terms are entirely due to viscous effects.

• (3)
At the free surface, $z=η(x,y,t)$, the vertical velocity of a fluid particle located on the free surface should equal the vertical velocity of the free surface itself. This is expressed by the kinematic boundary condition
$∂η∂t=w-u∂η∂x-v∂η∂y|z=η$
(7)
• (4)
At the free surface, $z=η(x,y,t)$, the pressure of the free surface should equal the capillary pressure due to surface tension, i.e., $Δp=γ((1/r1)+(1/r2))$, where $γ$ is the surface tension,2 and $r1$ and $r2$ are the principal radii of curvature. In this case, the dynamic free-surface boundary condition takes the form
$pρ-12|v|2 + g(t)η+γ∇·[∇η1+|∇η|2]=ν∇2v$
(8)
• (5)
No slip condition at the walls
$v=0 at the boundary of the free surface and z=-h$
(9)

According to Ruvinsky et al. [33], one can assume, for low viscosity fluids, that the oscillating part of the flow in a surface wave consists of a potential component given by $∇Φ$ and a small vortex part $v1$ excited by the potential component. In the linear analysis, $|v1|/|∇Φ|≈k/δ=(2π/λ)2ν/ω$, where $k=2π/λ$ and $δ=2ν/ω$. $ω$ and $λ$ are characteristic time and space scale of gravity–capillary waves, respectively. $Φ$ is the velocity potential function. When the thickness of the viscous boundary layer is small compared to the typical wavelength of the free surface, $δ<λ$, then weak effects due to viscosity can be taken into account by introducing effective boundary conditions for the otherwise potential bulk flow. This is the basic idea of the quasi-potential approximation developed in Refs. [33] and [34]. This quasi-potential formulation was extended by Zhang and Viñals [35] to three-dimensional (3D) flow for quasi-potential fluid.

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

Continuity equation:
$∇2Φ(x,y,z;t)=0$
(10)
The boundary conditions at the free surface $z=η(x,y;t)$
$∂η∂t+∇Φ·∇η=∂Φ∂z+w(x,y;t)$
(11)
$∂Φ∂t+12(∇Φ)2 - g(t)η+2ν∂2Φ∂z2=γκ$
(12)
$∂w(x,y;t)∂t=2ν(∂∂x2+∂∂y2)∂Φ∂z$
(13)
$∂Φ∂z|z→∞=0$
(14)
where $w(x,y;t)$ is the z-component of the rotational part of the velocity field at the free surface, $g(t)=g0+gz(t)$, and $g0$ is the gravitational acceleration, and $κ$ is the mean curvature of the free surface given by the expression
$κ=∇·(-∂η∂x,-∂η∂y,1)1+(∇η)2$
(15)
Combining Eqs. (13) and (11) gives
$∂w(x,y;t)∂t=2ν(∂2∂x2+∂2∂y2)[∂η∂t+∇Φ·∇η-w]$
(16)
Note that $2ν∇Φ·∇η$ is a nonlinear viscous term and can be ignored. Furthermore, $w≈O(ν)$ and $νw≈O(νx2)$, which can also be ignored. With these approximations, Eq. (16) takes the form
$∂∂t(w-2ν∇2η)=0$
(17)
Equation (17) is solved to give
$w(x,y;t)=2ν∇2η(x,y;t)+w0-2ν∇2η0$
(18)
where $η0$ and $w0$ are the initial conditions of the wave height and the rotational velocity at the free surface. If the motion starts from rest, then $w0=0$ and also in view of the small value of $2ν∇2η0$ it can be ignored for nonlinear finite-amplitude states. In this case, the free-surface boundary conditions (11) and (12) take the form
$∂η∂t=∂Φ∂z+2ν∇2η-∇Φ·∇η$
(19)
$∂Φ∂t=g(t)η-2ν∂2Φ∂z2-12(∇Φ)2 + γκ$
(20)

The boundary values problem of liquid in closed containers comprised the continuity equation represented by Laplace's equation (for incompressible) together with dynamic and kinematic conditions of the free surface and conditions at the wetted walls and bottom where the velocity component normal to the boundaries must vanish. Large liquid-free surface oscillations in closed containers have been treated in the literature using a number of approximations. The following are selected number of theories:

1. (a)

Moiseev [36] constructed normal mode functions and characteristic numbers by integral equations in terms of Green's function of the second kind (Neumann function). Chu [37] generalized Moiseev's method by employing a perturbation technique using the characteristic functions to determine the subharmonic response to an axial excitation.

2. (b)

Penney and Price [38] carried out a successive approximation approach where the potential function is expressed as a Fourier series in space with coefficients that are functions of time. These coefficients are approximated by Fourier time series using the method of perturbation. The resulting solution is expressed in terms of a double Fourier series in space and time. This method was applied by others [19,38,39].

3. (c)

Hutton [40] expanded the dynamic and kinematic free-surface equations in Taylor series about a stationary surface position. This method was modified and used in many sloshing problems by Woodward [13,41].

4. (d)

Time-average Lagrangian developed by Miles [42]: Miles [42–44] solved the kinematical boundary value problem for nonlinear gravity waves in a cylindrical basin using a variational formulation together with the truncation and inversion of an infinite matrix. The results were applied to weakly coupled oscillations, using the time-averaged Lagrangian, and to resonantly coupled oscillations, using Poincaré's action-angle formulation. Miles [44] demonstrated that a Lagrangian formulation, in which the generalized coordinates are the coefficients in a normal-mode expansion of the free-surface displacement and are slowly modulated sinusoids, leads to a set of evolution equations for 2N slowly varying amplitudes, where N is the number of modes retained in the truncated modal expansion. The Lagrangian formulation becomes a very powerful tool in developing the fluid field equations in Lagrangian coordinates and boundary conditions [45–50]. This approach becomes very attractive in developing numerical algorithms such as finite-difference and finite element, in particular, for cases of large amplitude motions at resonance sloshing and Faraday waves.

5. (e)

Center-manifold and normal form formulation developed by Meron and Procaccia [51,52]: A detailed formulation to solve the boundary value problem using the center-manifold theory and normal form averaging was presented in Refs. [51,52]. Their approach was found in difference with the Lagrangian formulation of Miles [43,44]. The difference was in the lack of symmetry of the nonlinear terms of modal amplitude equations. However, the two approaches confirmed the essential features of experimental results.

6. (f)

A spectral technique developed by Horsley and Forbes [53] to solve the nonlinear boundary value problem of Faraday waves and reducing the system to a set of nonlinear ordinary differential equations for a set of Fourier coefficients: Time-periodic solutions of the main subharmonic resonance were obtained 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 in the dynamic behavior of Faraday waves. The influence of damping on the stability–instability boundaries together with the dynamic response of the free surface has received extensive research activities. It is important to realize that the damping factor involves uncertainties, since the measured value is not always fixed and depends on temperature and oscillations of the free surface. This is in addition to the condition of the container wetted walls. This problem is considered in Secs. 4 and 5.

## Damping Effect

Considerable efforts were given to determine the damping coefficient of liquid-free surface motion and its influence on the fluid natural frequencies and its response. The kinematic viscosity, $ν$, is a fundamental parameter for measuring the rate of vorticity and momentum by molecular transport. In fact, it is known that the vorticity is present at a distance, $δ$, outside the boundary of the body, whereas the time required for diffusing the vorticity or momentum through this distance is of order $(δ2/ν)$, which is referred to as the diffusion time. Boussinesq [54] introduced the influence of viscous damping to study progressive and standing waves in closed containers. Boussinesq's work was extended by Keulegan [55] to calculate the attenuation of solitary waves. Case and Parkinson [56] determined the damping of surface waves of small amplitude in partially filled cylindrical tank. They considered viscous dissipation in an assumed laminar boundary layer as the primary cause of damping. Van Dorn [57] conducted experimental investigations and attributed the differences between the predicted and measured results to a surface film produced by spontaneous contamination. It was stated: “while the observed attenuation agreed with that computed for solid boundaries when the water was fresh, the former tended to increase with time to some higher limiting value, usually within an hour.”

Miles [58] and Mei and Liu [59] indicated that the damping of surface waves in closed basins appears to be due to several sources. These include viscous dissipation at the boundary of the surrounding basin, viscous dissipation at the surface in consequence of surface contamination, and capillary hysteresis associated with the meniscus surrounding the free surface. The theoretical results for the logarithmic decrements of gravity waves in circular and rectangular cylinders were compared with the decay rates observed by Case and Parkinson [56] and by Keulegan [55], which typically exceeded the theoretical value based on wall damping alone by a factor of 2–3. It was concluded that both surface films and capillary hysteresis can account for these observed discrepancies. Despite of the elegant formulations of sloshing damping reported in the literature [10,14,55,60], there is always significant discrepancy between predicted and measured damping factors. The measurements become more difficult for higher modes 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, width $Ly$ in the y-direction, and for antisymmetric sloshing modes, the damping due to boundary-layer flow expressed by the damping ratio based on the theory of Keulegan [55] was given by the expression
$ζn=12ν2ωnLx2[(sinh(2knh)-knh)+knLxsinh(knh)cosh(knh)]+ν2ωnLy2=πνTn2πLy[LyLx(1+(knl/2)-knhsinh(knh)cosh(knh))+1]$
(21)

where $kn=nπ/l$, $n=1,2,...$, and $Tn$ is the period of nth sloshing mode.

Although the measured natural frequencies were found in excellent agreement with the theoretical values, Ikeda et al. [62] found significant differences between the measured values (using logarithmic decrement) and the theoretical values based on Keulegan. For example, based on Keulegan theory, the values of the damping ratios of the three sloshing modes, in a square container, were $ζ1,0=ζ0,1 = 0.003142$, $ζ2,0=ζ0,2=0.0028274$ and $ζ3,0=ζ0,3=$$0.0025638$. Curve fitting was used to match the theoretical results to the experimental data, and the damping ratios were identified as $ζm,n=0.015$. These ratios were similar to both the measured value of $ζ10=0.018$ obtained by the logarithmic decrement and to the theoretical values based on Keulegan theory. These values were obtained for a square tank of cross dimension of $100 mm×100 mm$ and liquid depth of $h=60 mm$. Note that the measured values involved significant uncertainties due to temperature differences, contamination of the free surface, etc. In general, damping is an inherent parameter that involves the most significant level of uncertainty, and for this reason, structural dynamicists adopted a probabilistic description of the damping parameter, which is usually represented by a random variable with a given probability distribution [63].

The frequency and damping rate of surface capillary–gravity waves in a bounded region depend on the conditions imposed where the free surface makes contact with the boundary. An edge condition that models the dynamics associated with moving contact lines, but not contact angle hysteresis, was obtained by making the slope of the free surface at contact proportional to its velocity as proposed by Hocking [64]. This model was used to obtain the frequency and damping rate of a standing wave between two parallel vertical walls. The effect of viscosity in the boundary layers on the walls was included and it was shown that the dissipation associated with the surface forces can exceed that produced by viscosity. Henderson [65] and Henderson and Miles [66] calculated the natural frequencies and damping ratios for surface waves in a circular cylinder based on the assumptions of a fixed contact line, Stokes boundary layers, and either a clean or a fully contaminated surface. The differences between the predicted and observed frequencies were found to be less than 0.5% for all but the fundamental axisymmetric mode with a clean surface. The difference between the predicted and observed damping ratio for the dominant mode with a clean surface was 20%.

Experiments on single-mode Faraday waves in small rectangular and circular cylinders in which both capillary and viscous effects are significant were reported by Henderson and Miles [67]. Theoretical predictions of the resonant frequency of a single mode and of the threshold amplitude for its excitation based on the hypothesis of linear boundary-layer damping were found in agreement with the measured data. In their theoretical analysis, they used the measured damping rate to predict these quantities for waves in the rectangular cylinder. Later, Henderson et al. [68] measured the damping rates and natural frequencies of the fundamental axisymmetric mode in circular cylinders when the contact angle between the water and the side walls was acute, obtuse, and about $π/2$. It was found that damping rates decrease with increasing contact angle, while the natural frequencies increase with increasing contact angle. Pritchett and Kim [69] described an apparatus 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 flat surface solution co-exist. The lower hysteresis boundary in forcing frequency space was defined by the lower boundary above which nontrivial stationary points exist. Single-mode limit cycles were found as solutions of a nonlinear evolution equation for parametrically excited standing surface waves in a rectangular container. Later, Decent and Craik [72] examined their structure analytically and numerically, and described local and global bifurcations. To calculate the linear and nonlinear damping coefficients, Decent [73] indicated that it is necessary to determine the dissipation in the main body of the liquid, the dissipation in the boundary layers at the sidewalls and at the surface, and the dissipation due to capillary hysteresis. Decent [74] extended Miles' calculations [58] to obtain the cubic damping coefficient. Decent and Craik [71] estimated experimentally the linear and cubic damping coefficients in a rectangular tank. Both the theoretical method of Decent [74] and the experimental method of Decent and Craik [71] revealed that the cubic damping coefficient for deep water is positive for water depths greater than approximately 1.2 cm. Decent [70] showed that the free-surface amplitude equation exhibits a Hopf bifurcation when the cubic damping coefficient is positive, which is absent for negative cubic damping coefficient. This Hopf bifurcation was found to give rise to a stable limit cycle solution, which corresponds to a time-modulated standing wave, where the maximum amplitude of the standing wave varies with the slow time scale.

Martel et al. [75] estimated the natural frequencies and damping rates of surface waves in a circular cylinder with pinned-end boundary conditions in terms of the gravitational Reynolds and Bond numbers, $Rg=gR3/ν$ and $B=ρgR2/γ$, respectively, together with the fluid depth ratio, $h/R$. Higher-order approximations that include the effect of viscous dissipation in the Stokes boundary layers and the bulk were considered. A comparison with clean surface experimental results [66] showed a satisfactory agreement except for the first axisymmetric mode, which exhibits a 26% discrepancy. The boundary-layer calculation was supplemented by a calculation of interior damping based on Lamb's dissipation integral for an irrotational flow [76]. The analysis yielded results of comparable accuracy within the parametric domain of experiments. The corresponding calculations for a fully contaminated surface were found to reduce the discrepancy between calculation and experiment but, in contrast to the results for a clean surface, leave a significant residual discrepancy.

The vertical oscillation of a plate partially immersed in a nonwetting fluid was found to cause a radiated wave-train when the contact line between the plate and the free surface of the fluid cannot move freely along the plate [64]. Realistic conditions to apply at the contact line when capillarity is not negligible were found to include the dynamic variation of the contact angle and contact angle hysteresis. The amplitude of the radiated waves and the energy dissipation at the contact line were calculated. Jiang et al. [77] found a complicated nonlinear relationship between wave frequency and amplitude near contact lines. The relative damping rate was found to be dependent on the wave amplitude, increasing significantly at smaller wave amplitude. These results were discussed in relation to different formulations of contact line conditions for oscillatory motions and free-surface flows.

Faraday waves can be modeled by a damped Mathieu equation [81]
$d2ηdt2+4νk2dηdt+[gk+γρk3-akcos(Ωt)]η=0$
(22)

where $η$ is the surface wave amplitude, $γ$ is the surface tension, $a$ is the excitation amplitude acceleration, and $k$ is the instability wave number. Benjamin and Ursell [10] noted that standing waves can be created even if $a/g<1$. When the water becomes shallow and viscous effects remain small, i.e., $kh≈1,kδ ≪1$, where $δ≈ν/Ω$ is the boundary layer thickness at the free surface, an equation which is nonlocal in time is needed to describe the behavior of $η$ [78,79]. The nonlocality arises because of viscous dissipation at the bottom boundary. The surface wave height obeys a nonlocal equation when the fluid is deep and viscous effects are strong $(kh>>1,kδ≈1)$. However, Cerda and Tirapegui [80,81] indicated that a local equation can quantitatively describe $η$ when the fluid becomes shallow. This equation is again a Mathieu equation, but the dependence of the damping on the wave number is much different than when viscous effects are weak. In this regime, $(kh≈1,kδ≈1)$, diffusion of momentum occurs so rapidly relative to the surface oscillations that the damping no longer depends on the history of the motion. When viscous effects are strong, the critical acceleration amplitude, $ac$, required to excite the standing waves is typically larger than the acceleration of gravity. In the deep water limit (wavelength $≪$ liquid depth), Kumar [82] indicated that the preferred wave number in the Faraday instability is primarily determined through Rayleigh–Taylor instability. In the case of shallow water (wavelength $≈$ liquid depth), the agreement between the Rayleigh–Taylor and Faraday wave numbers does not appear to be as good, probably due to the interaction between the oscillatory motion of the standing waves and the bottom boundary.

Lapuerta et al. [83] considered a horizontal heavy fluid layer supported by a large aspect ratio container, which is subjected to parametric excitation. When the density and viscosity of the fluid are small compared to their counterparts in the heavier fluid, a nonlinear analysis was found to yield a generalized Cahn–Hilliard equation for the evolution of the fluid interface. This equation revealed that the stabilizing effect of vibration is like that of surface tension and is used to analyze the linear stability of the flat state, the local bifurcation at the instability threshold, and some global existence and stability properties concerning the steady states without dry spots. A set of equations was derived for the coupled evolution of the left- and right-traveling surface waves and the associated mean flow. The viscous mean flow was found to drastically affect the dynamics of the system and the resulting surface wave patterns. Funada et al. [84] extended the work of Benjamin and Ursell [10] to purely irrotational waves on a viscous fluid. Two irrotational theories were presented. The first theory was based on viscous potential flow in which the effects of viscosity enter only through the viscous normal stress term evaluated on the potential. In the second irrotational theory, a viscous contribution was added to the Bernoulli pressure. The second theory gives rise to the same damped Mathieu equation as the dissipation method. The damping term in the second theory was found to be twice the damping rate of the first theory. The growth rates of unstable disturbances computed by viscous potential flow were found to be uniformly larger than those computed by the second theory. Comparisons with the exact solution and the Rayleigh–Taylor instability revealed that thresholds and growth rates for viscously damped waves are better described by the viscous potential flow.

Numerical algorithms were proposed in the literature for calculating the damping of liquid sloshing with small amplitudes. These algorithms are based on finite element method [85,86] and the finite volume schemed based on the volume-of-fluid (VOF) method [87–89]. The simulated results showed that the equivalent damping coefficients are directly proportional to the liquid viscosity. Surfactants, such as oil on water, can cause an increase in the damping of surface waves [58]. Miles [58] concluded that both surface contamination and capillary hysteresis might have contribution to the damping surface waves in closed basins. For detailed account of damping associated with liquid sloshing, the reader may refer to Chap. 3 in Ref. [14].

## Influence of Surfactants and Stratified Fluids

Surfactants are compounds, which may lower the interfacial tension between two liquids or between a liquid and a solid. Surface contamination has been modeled by phenomenological formulae [90–92] based on Marangoni elasticity with insoluble surfactant. 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, natural frequencies, and amplitudes of the fundamental axisymmetric Faraday waves in circular cylinders were experimentally and theoretically studied by Henderson [92]. In particular, Henderson [92] determined the effects of elastic, insoluble films on Faraday wave dynamics and examined the effects of films on the damping rates of large amplitude waves with low frequency. The water surface was contaminated with varying concentrations of surfactants such as oleyl alcohol, lecithin, diolein, arachidyl alcohol, and cholesterol. It was found that when the surface is saturated with surfactant, the damping rates are greatly in excess of both the clean surface model and inextensible surface model predictions, indicating that, although the quiescent surface had zero elasticity during wave passage, the surface had a finite elasticity. Measured damping rates were found unaffected by arachidyl alcohol and cholesterol surfactants when the wave amplitudes are large. An analysis was presented by Nicolás and Vega [93] who anticipated that surface contamination would enhance the generation of the streaming flow produced by the surface wave.

The nonlinear subharmonic resonant wave heights in rectangular tank under parametric excitation were measured by Virnig et al. [94] for liquids without and with additives. They compared the measured results with those predicted analytically by Gu et al. [95] and Fig. 3(a) shows the dependence the response amplitude on the excitation frequency for the sloshing mod (1,1) for two excitation amplitudes. The results were taken for a rectangular container of width of 17.78 cm, breadth of 22.86 cm, and liquid depth of 11.43 cm. Only the stable branches of the analytical solution are shown by solid and dashed–dotted curves. It is interesting to observe that both experimental and predicted results have good agreement for higher values of wave amplitudes. At lower wave amplitudes, the experimental results reveal a “tailing” which is repeated for both cases of increasing and decreasing the excitation frequency at constant amplitude. The observed tailing effect is attributed to the influence of surface tension. Figure 3(b) shows similar set of results after applying a surfactant (Kodak Photo Flo 200 solution). It is seen that as the surface tension is reduced the tailing effect is eliminated. The inclusion of surface tension in the analysis 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 that there is a critical fluid depth, which separates two nonlinear regimes of the fluid free surface referred to as soft and hard spring characteristics. This depth was originally predicted by Tadjbakhsh and Keller [96] and verified experimentally [97,98] for waves produced by excitations in a direction parallel to the free surface.

Fig. 3
Fig. 3
Close modal

The dependence of the liquid wave amplitude on the excitation amplitude for constant excitation frequency was obtained by Virnig et al. [94] and the results are shown in Fig. 4(a) for four different excitation frequencies after applying a surfactant. The analytical results are seen to be higher than those measured experimentally and convergence of both results appears when the excitation frequency is close to twice the natural frequency of the first mode ($2f1=4.692 Hz$). For excitation frequencies less than twice the natural frequency, the analytical results reveal the occurrence of a saddle node bifurcation at critical excitation amplitude. The unstable branch is shown by a dashed curve. The saddle node signifies the occurrence of a jump in the response amplitude and this point is governed by the system damping ratio and excitation frequency. The amplitude–frequency response is shown in Fig. 4(b) for different values of liquid depth. It is known that below the critical depth, the liquid free surface behaves like a “hard” oscillator, i.e., the amplitude increases with the excitation frequency. Above the critical depth, the “softening” characteristic takes place. For this case, the predicted critical depth is 7.7 cm, while the experimental measurements suggest that it is between 6.6 cm and 7.47 cm.

Fig. 4
Fig. 4
Close modal

The role of insoluble surfactants on the stability of parametrically driven surface waves was studied by Kumar and Matar [99]. It was found that in order to obtain time–periodic solutions, which involve Marangoni3 forces, it is necessary to consider the high-Péclet4 number limit of the surfactant transport equation. The results showed that the presence of surfactants raises or lowers the critical amplitude and wave number depending on the spatial phase shift between the surfactant-concentration variations and surface deflections. If the concentration variations are in phase with the surface deflections (maximum concentration at wave crests), they will drive a Marangoni flow that pulls fluid away from the wave crests, and this will produce larger critical amplitude. Similarly, if the concentration variations are out of phase with the surface deflections (minimum concentration at wave crests), they will drive a Marangoni flow that pulls fluid toward the wave crests, and this will produce smaller critical amplitude. For nonzero diffusivities, it was found that disturbances in the surface concentration of the surfactant simply decay exponentially on a time scale, which is inversely proportional to the surface diffusivity of the surfactant. Kumar and Matar [100] considered the thickness of the liquid layer to be much smaller than the wavelength of the interfacial disturbance. It was found that the liquid layer is unstable to long-wavelength disturbances if it is covered by surfactants, while it is stable to such disturbances if the surfactants are absent. These results were found valid for nonzero surfactant diffusivities and represent standing wave solutions in which Marangoni flows are present. Later, Kumar and Mater [101] found that surfactants can potentially lower the value of the critical amplitude relative to its value for an uncontaminated free surface. The critical wave number was found to be an increasing function of the Marangoni number.

The effect of surface contamination, modeled by Marangoni elasticity with insoluble surfactant and surface viscosity, on drift instabilities in spatially uniform standing Faraday waves was studied by Martín and Vega [102,103]. The order of magnitude of the Marangoni elasticity was taken as that already obtained by Nicolás and Vega [93] to fit the experimentally measured damping rate for contaminated water. Surface viscosity is expected to be small in contaminated water, but it can also be large in other systems. It was found that contamination enhances drift instabilities that lead to various steadily propagating and oscillatory patterns. The elastic effects of an insoluble surfactant on the formation and evolution of 2D Faraday waves were studied numerically by Ubal et al. [104–106]. The numerical results revealed that the interface is always subharmonically excited at the onset and that the presence of the surfactant requires a higher external force to induce standing waves. The magnitude of the external amplitude was found to be related to the temporal phase shift that exists between the evolution of the surfactant concentration and the free-surface shape.

Giavedoni and Ubal [107] studied the formation of Faraday waves on the free surface of a liquid layer covered by an insoluble surfactant. The linear analysis included the effects of both surface elasticity and surface viscosity. The critical force needed to form the waves, as well as the critical wave number were determined within a large range of values of the dimensionless parameters representing the physicochemical properties of the surfactant. When Marangoni number is small, the largest concentration gradient 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 interfacial velocity, slowing the motion of the liquid from the trough to the crest of the wave; therefore, the tangential velocity turns to zero before the free surface attains its maximum deformation. For typical values of Péclet number (Pe), the capillary number5 (Ca), Bond number6 (Bo), and the fluid depth ratio, it was shown that the presence of a surfactant always increases the force required to develop a wavy interface. However, the dependence of the force on Marangoni number was found nonmonotonic when the surface viscosity is negligible and Pe > 1.

Faraday waves in a thin sheet of a viscous fluid subjected to a uniform and slow rotation about the vertical axis were studied by Mondal and Kumar [108]. The rotation created a Coriolis force, which breaks the mirror symmetry of the flow due to parametrically excited surface waves. The Coriolis force was found to delay the Faraday waves on the free surface. However, the Faraday (subharmonic) waves can always be excited at the onset of surface instability if the excitation frequency is much greater than four times the angular frequency of the rotating body. The surface waves were found to be synchronous with the vertical vibration in a thin sheet of viscous fluid. Instability phenomena in a thin layer of a slowly rotating viscous fluid (whose viscosity is approximately ten times the viscosity of water) were observed. The synchronous surface waves developed additional local maxima, when the excitation acceleration amplitude was raised above certain value. This was manifested when any point on the free surface moves up and attains two different local maxima in one period of external forcing. The wave number of these waves is double that of the synchronous waves. Superharmonic waves at the instability onset were found possible in parametrically forced viscous fluids. Furthermore, different responses, such as subharmonic, harmonic, and superharmonic with different wave numbers, may co-exist for the same forcing amplitude. This leads to the interesting possibility of a tricritical point as the primary instability.

Figure 5 shows instability regions on the plane of excitation/Galileo ratio, $A/G$ (where $A=ah3/ν2$ and $G=$$gh3/ν2$, $a$ is the excitation acceleration amplitude) versus wave number, $k$. The figure shows a tricritical point and superharmonic waves in viscous sheet of water–glycerol mixture with Galileo number, $G=gh3/ν=2.7×103$, Capillary number, $Ca=ρν2/(γh)=4.7× 10-4$, and excitation frequency ratio $ϖ=Ωh2/ν=7.5$. The shaded zones and zones bounded by dots are for subharmonic and harmonic regions, respectively. Note that harmonic response goes to a tricritical point for $2ωrh2/ν$ = 2.9, plot (c) of Fig. 5, where surface waves with wave numbers $k1,1.5k1$ and $2k1$ with $k1$ as the wave number of the first harmonic response co-exist at the instability onset. Further increase in rotational rate would lead to superharmonic waves as primary instability. Stability of the free surface of thin sheets of a metallic liquid on a vertically vibrating hot plate, in the presence of a uniform and small rigid body rotation about the vertical axis, was studied by Mondal and Kumar [109]. It was found that the inhomogeneity in the surface tension due to a uniform thermal gradient across the liquid sheet prefers subharmonic response. On the other hand, it was indicated that the rigid body rotation prefers harmonic response at the fluid surface. The competition results in Marangoni and Coriolis forces acting as fine-tuning parameters in the selection of wave numbers corresponding to different instability tongues for subharmonic and harmonic responses of the fluid surface. Bicritical points were found to involve both the solutions oscillating subharmonically, harmonically, or one oscillating subharmonically and the other harmonically with respect to the vertical forcing frequency. The effect of small Marangoni and Coriolis forces on the onset of standing surface waves and bicritical points in a vertically vibrated sheet of mercury was studied by Mondal and Kumar [109]. When both subharmonic and harmonic (synchronous) surface waves can be excited for the same value of forcing excitation amplitude, the corresponding point is referred to as bicritical. The metallic fluid was subjected to a small thermal gradient and rigid body forces (Coriolis), which resulted in a variety of bicritical points and the possibility of two different kinds of tricritical points. In the absence of the Marangoni force, a tricritical point involving two harmonic solutions with different wave numbers and a subharmonic solution was reported. On the other hand, in the presence of both Marangoni and Coriolis forces, it was found that a tricritical point involves two subharmonic solutions with different wave numbers and a harmonic solution is possible. It was concluded that one may use a Marangoni number and a Coriolis force as tuning parameters to influence the nature of the multicritical point at the onset of parametrically forced surface waves. Galileo number, $G=gh3/ν$, was found to have an influence on the threshold acceleration and critical wave number in molten sodium.

Fig. 5
Fig. 5
Close modal

Faraday waves were observed on the free surface of helium-4 layer when excited vertically at low temperature $(T=700 mK$, where $T(K)=T(°C)+273.15$) [110]. Standing wave patterns were observed to appear on the surface. Threshold excitation amplitudes for the instability were clearly manifested. The difference in the threshold amplitude between the superfluid and the normal fluid is that the threshold amplitude is larger for the normal fluid than for the superfluid, and the difference was attributed to the wall damping in the cell geometry. Hysteretic behavior specific to the nonlinear waves was also observed in the superfluid phase. Higuera et al. [32] presented an overview on the small viscosity limit and the effects of symmetries and nonlinearity on the response of fluid systems. The role of viscous mean flow was found to couple the primary surface waves. Under low frequency experiments of surface rheology, the importance of fluid dynamics provided an explanation of the irreversible character of surface pressure versus surface area. An experimental study of the Faraday instability of viscoelastic fluid was presented by Cabeza and Rosen [111] who used a shear thinning polymer solution in which the elastic effects are predominant. Depending on the fluid layer depth and the driving frequency, harmonic or subharmonic regimes were developed. In addition, the onset acceleration was used to estimate the rheological properties of the fluid. Ezersky et al. [112] showed experimentally that parametric excitation of capillary waves in a liquid polymer may give rise to spatially periodic distribution of microparticles. It was found that photopolymerization makes it possible to fix position of microparticles and produce materials with controllable spatially periodic inhomogeneities.

The parametric excitation of internal 2D waves of a viscous continuously stratified fluid, completely filling a rectangular vessel under vertical oscillations, was considered in the literature [113–117]. Approximate formulae were obtained for the threshold amplitude of the vessel excitation and the boundaries of the resonance zones. The dynamics of internal gravity waves excited by parametric instability in a stable stratified medium, either at the interface between water and a kerosene layer, or in brine with a uniform gradient of salinity was studied by Benielli and Sommeria [117]. Each internal wave mode was amplified for an excitation frequency close to twice its natural frequency, when the excitation amplitude is sufficient to overcome viscous damping. In the interfacial case, each mode was made well separated from the others in frequency and was found to behave like a simple pendulum. The case of a continuous stratification is more complex as different modes have overlapping instability tongues. Foster and Craik [118] considered capillary–gravity waves of an inviscid liquid, which exhibit second- or sub-harmonic resonance at precise frequencies. Equations describing this situation were derived incorporating slight detuning from two-wave and Faraday resonances.

The case of two incompressible viscous fluids with different densities meet at a planar interface subjected to oscillating acceleration directed normal to the interface was considered by Jacqmin and Duval [119]. General viscosities and densities for the two fluids were considered but a Boussinesq equal-viscosity approximation was adopted. It was shown that the linear evolution of a perturbation to the interface subjected to an arbitrary oscillating acceleration is governed by a single integrodifferential equation. Parameter regions of subharmonic, harmonic, and untuned modes were delineated. The critical 7Stokes–Reynolds number was given in terms of the surface tension and the difference in density and viscosity between the two fluids. The critical Stokes–Reynolds number and the most unstable perturbation wavelengths were found to be insensitive to the degree of density and viscosity differences between the two fluids. The parametric excitation of internal 2D waves of a viscous continuously stratified fluid completely filling a rectangular vessel was studied by Kravtsov and Sekerzhi-Zenkovich [120]. The fluid was assumed to have a small viscosity. Approximate formulae were obtained for the threshold amplitude of the oscillations of the vessel and the boundaries of the resonance zones.

The Faraday resonance of interfacial waves in a two-layer weakly viscous system in a rectangular domain was considered in Refs. [121,122]. The scaling of the viscosity results in boundary layer corrections at the solid walls and at the interface. The damping in the meniscus region, where the interface contacts the side walls, was included. As a result of the presence of both destabilizing effects, due to vertical oscillation, and stabilizing effects, due to viscosity, a threshold condition for instability was derived. Faraday waves formed on the interface between two immiscible liquids in a cylindrical cell were studied in Ref. [123]. The effects of the volume filling ratio on the bifurcation set associated with the onset of the fundamental axisymmetric mode were examined. In particular, the study considered the subharmonic regime of the control parameter space where the response was significant. Both super- and subcritical bifurcations were uncovered, with hysteresis in the latter case. The extent of the hysteresis was observed to be strongly dependent on volume filling ratio, suggesting that nonlinear damping effects are influenced by this parameter. At large excitation amplitudes, a precessional periodic motion was found to develop via a Hopf bifurcation. This mode was observed to disappear catastrophically at an excitation frequency equal to 1.853 ± 0.006 times the natural frequency of the resonant mode. The problem of capillary–gravitational Faraday waves on the interface between fluids was studied in Ref. [124].

Sections 2 through 5 provided an overview of the pertinent features of parametric instability boundaries of liquid free surface and the boundary value problem of incompressible viscous fluids with surface tension. These sections were very useful to assess the damping associated with Faraday waves and the influence of contaminations and surfactants. However, they did not address the nonlinear response characteristics of the free surface under parametric excitation, which is only predicted by including the nonlinear effects of the free-surface boundary conditions as will be demonstrated in Sec. 6.

## Nonlinear Parametric Sloshing

In the neighborhood of parametric resonance, the amplitude of the free surface grows without limit. As the amplitude increases, geometric nonlinearities (nonlinear inertia) emerge and the motion achieves bounded amplitude. For the case of parametric excitation of liquid free surface, the response exhibits other nonlinear phenomena such as nonplanar motion, rotational motion, chaotic motion, and free-surface disintegration. Kalinichenko and Sekerzhi-Zenkovich [125] subdivided Faraday waves into three categories, namely, regular, irregular, and breaking waves. The regular Faraday waves include those whose profiles possess either time-periodic or symmetrical about vertical planes passing through the wave antinodes. The limiting angle at the crest of such waves was estimated to be 80 deg. The waves, which have disturbed temporal and spatial symmetries but retain the connectivity of the oscillating fluid volume, were considered to be irregular. Finally, waves with separate droplets shed from the free surface of the fluid or with jet launches were assigned to the class of breaking Faraday waves. It was noted that in the case of irregular and breaking waves their height can be estimated only in the statistical sense, that is, using a quantitative characteristic, such as the wave steepness, is permissible only for individual wave profiles. Among the irregular waves are those with a small depression in the wave crest and periodic triplets. In the case of breaking waves, the mechanism of jet launch formation on the wave crest was considered by Kalinichenko [126] who used the experimental setup reported by Kalinichenko et al. [127,128]. It was experimentally demonstrated that the breaking of standing waves in a rectangular reservoir starts with cavity collapse on the wave crest in process of formation. It was shown that jet launch from the wave crest is preceded by the initiation, development, and collapse of a cavity. Based on a comparison of the experimental data with an analytical model, it was suggested that cavity initiation is due to the nonlinearity of the waves themselves, namely, the presence of two small disturbances of the free-surface traveling counter to one another and forming a cavity.

The nonlinear behavior of liquid free surface under parametric excitation for filling depths greater than the free-surface wave length may be classified into weakly nonlinear and strongly nonlinear characterized by breaking waves. These two classes will be addressed in Secs. 6.1 and 6.2.

### Weakly Nonlinear Parametric Sloshing.

The weakly nonlinear dynamic behavior of the nonlinear theory can predict the steady-state response and uncover complex free-surface dynamic behavior such as quasi-periodic and chaotic motions. Some controversies were reported in the literature regarding the analytical description of the sloshing modal equations. For example, Dodge et al. [19] developed a finite-amplitude analysis for a circular cylindrical container, but their equations of motion for the modal amplitudes were found by Miles [43] to violate reciprocity conditions. Miles [43] rectified this problem by performing some algebraic manipulation that resulted in nonlinear inertia terms in the first antisymmetric mode. The amplitude modal equations of Dodge et al. [19] were developed using Laplace's equation for the velocity potential function together with the kinematic boundary conditions on the tank walls and the nonlinear kinematic and dynamic boundary conditions at the free surface. They retained terms of first-, second-, and third-orders in the amplitude of the primary mode. The sloshing amplitude equations of the first antisymmetric, $a11,$ first symmetric, $a01,$ and second symmetric, $a21,$ modes were obtained in the form
$a··11+(1-ɛν2cosντ)a11(1+K11a112+K01a01-K21a21)+0.034780λ112a··11a112+k11a11a·112+0.165a11a··01-0.198686a11a··21+k01a·01a·11-k21a·21a·11=0$
(23a)
$a¨01+λ01tanhλ01h(1−εν2cosντ)a01−a11a¨11(0.12158λ01tanhλ01h−0.198686λ112)+a112[λ01tanhλ01h(0.070796λ112−0.060741)+0.263074λ112]=0$
(23b)
$a¨21+λ21tanhλ21h(1−εν2cosντ)a21+a11a¨11(0.350807λ21tanhλ21h−0.48267λ112)+a112[λ21tanhλ21h(0.175403−0.065931λ112)−0.48267λ112]=0$
(23c)

where the modal amplitudes are nondimensionalized with respect to the length $1/(λmntanhλmnh)$, $ν=Ω/ω11$, $Ω$ is the parametric excitation frequency, $ω11$ is the natural frequency of the first antisymmetric sloshing mode, $τ=ω11t$ is the nondimensional time, $h$ is the fluid depth, $ɛ=ω112Z0/g$, $λmn$ are the roots of the first derivative of the Bessel function of the first kind, i.e., $d/dr[Jm(λmnr)]|r=R=0$, $R$ is the tank radius, and $Kmn$ and $kmn$ are constants, which depend on the the fluid depth and $λmn$ and are documented in Refs. [19,129].

Miles [43] noted that these equations differ from those derived using Lagrangian formulation. The main difference is in the presence of the expression $K11a112+K01a01-K21a21$ in Eq. (23a). The two formulations can be reconciled if the entire Eq. (23a) is divided throughout by the expression $(1+K11a112+ K01a01- K21a21)$, which is approximated by one for all nonlinear terms, while the first term $a··11$ is multiplied by the inverse of that expression, which takes the form $(1-K11a112-K01a01+K21a21)$. This processes results in identical coefficients of the terms $a··11a112$ and $a11a·112$, and likewise the coefficients of the terms $a··11a01$, $a·11a·01$, $a··11a21$, and $a·11a·21$. Figure 6 shows the experimental measurements [19] of the dependence of the liquid-free surface height on excitation frequency for three different values of excitation amplitudes. The measured amplitude was taken at $r/R=0.837$ and the wave height was estimated as the one-half the difference between the maximum and minimum liquid-free surface excursions for fluid depth ratio $h/R=2$. The figure shows the points at which the response amplitude jumps and collapses depending on whether the excitation frequency increases or decreases. The characteristics of the response are belonging to soft spring nonlinearity.

Fig. 6
Fig. 6
Close modal

Another finite-amplitude analysis for a circular cylindrical tank was developed in Ref. [130]. The free-surface boundary conditions were applied at the equilibrium position of the free surface and a correction to the resonant frequency was obtained in terms of first-order in the amplitude, which was noted by Miles [43] to be of second-order. Parametric nonlinear excitation vibration of the liquid surface in a partially filled circular cylindrical tank was studied in Refs. [131] and [132]. The analysis was extended by Takahara and Kimura [133] to the case of 3D nonlinear liquid motion in a rigid rectangular tank partially filled with liquid under pitching excitation. It was noted that the pitching excitation also causes the parametric excitation when the pitching axis does not intersect the symmetrical axis of the circular cylindrical tank. The time histories of the liquid surface displacement to the harmonic vertical excitations were calculated for the case of axisymmetric mode possessing nodal radii. Dynamical behavior of parametrically excited solitary waves in Faraday's water trough was studied by Wang et al. [134]. The endless collision of two solitary waves of like polarity, and the periodical reflection and attraction of a solitary wave by the boundary (one end of the water trough) were analyzed.

The fluid free surface changes dramatically when the shape of the container is deviated from circular to ellipsoidal as pointed out by Higuera et al. [135]. In this case, the mean flow is coupled with the amplitudes of surface waves as well as the spatial phases of the resulting pattern. The ellipticity was found to break the rotational symmetry of the primary waves and would select two standing oscillations with nodes along either the major or the minor axis of the ellipse. In Faraday waves, coupled amplitude mean flow equations were derived for patterns consisting of multimode plane waves as reported by Higuera et al. [136] who predicted coupling between the mean flows and patterns when the patterns satisfy some symmetry requirements. In a fluid with a free surface being vibrated vertically, surface waves are excited when the vibration amplitude exceeds a frequency-dependent critical value. Subharmonic standing waves are known to have the lowest threshold driving amplitudes, i.e., the wave frequency being half of the driving frequency. By using suitable cell geometry, Chen [137] generated waves with curved wave fronts. It was argued that these predictions should also be true in Faraday waves. Chen [137] measured the mean flow velocities driven by curved rolls in a pattern formation system. Curved rolls in Faraday waves were generated in experimental cells consisting of channels with varying widths. The mean flow magnitudes were found to scale linearly with roll curvatures and squares of wave amplitudes, agreeing with the prediction from the analysis of phase dynamics expansion. The effects of the mean flows on reducing roll curvatures were also reported.

Experimental investigation of Faraday crispation was reported by Cabeza et al. [138] who measured the fluid surface displacement. The results revealed the period-doubling cascade with at least two bifurcations and an incoherent signal. In other studies, Cabeza et al. [139,140] obtained the numerical reconstruction of the patterns appearing on the surface and the dynamical evolution to chaos of a localized point of the surface. The classic evolution scenario of the Faraday instability system was modified through controlling the height of the fluid layer, in a regime of highly dissipative 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, Cabeza et al. [140] observed the generation of strongly localized solitary waves, which propagate over these structures that are destroyed only when they reach the border of the trough.

Parametric excitation of a narrow rectangular container (cross section of $600 mm×60 mm$ and fluid depth of 300 mm) was considered by Jiang et al. [141] who observed mild to steep standing waves of the fundamental mode over an excitation frequency range of 3.15 Hz to 3.34 Hz. These standing waves were also simulated by a 2D spectral Cauchy integral code. The experimental results showed that contact line effects increase the viscous natural frequency and alter the neutral stability curves. The addition of a wetting agent, such as Photo Flo, was found to significantly change the stability curve and the amplitude–frequency response of the free surface. Figure 7 shows the free-surface amplitude–frequency responses for three different values of excitation amplitude: 2.5 mm, 3.0 mm, and 3.5 mm. The shown arrows indicate that the response of amplitude was obtained when the excitation frequency is changed according to the arrow direction. Solid/hollow symbols represent the experiments with increasing/ decreasing frequency. For increasing forcing frequency, the response curves (dashed lines) exhibit jump from zero to bounded amplitude. The frequency ratio, $Ω/2ωn$, where the jump 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 excitation amplitude.

Fig. 7
Fig. 7
Close modal

Figure 8(a) shows the amplitude–frequency response plots for treated water, while Fig. 8(b) is for treated water with Photo Flo. Figure 8(a) includes the analytical response based on Henderson and Miles [67] theory with excitation amplitude of 2.5 mm. It is seen that the addition of Photo Flo resulted in a significant change in the jump position as shown in Fig. 8(b). Strong modulation in the wave amplitude for some forcing frequencies higher than 3.30 Hz was observed experimentally. Over the frequency ratio range of 1.02–1.05, large modulations were observed in the time history record of the free-surface wave amplitude as shown in Fig. 9 for two different excitation frequencies and amplitudes. It is seen that the magnitude of the modulation for higher excitation frequency of 3.32 Hz is more significant with a modulation frequency of 0.03 Hz. No modulation was predicted [141] by using nonlinear numerical simulations, even with higher harmonics in the sinusoidal excitation. However, when the experimental forcing signal was used as the numerical input, modulation was predicted, albeit with less magnitude. Since no contact line effect was simulated, it was concluded that the modulation may be caused by contact line effect and sideband noise in the excitation signal.

Fig. 8
Fig. 8
Close modal
Fig. 9
Fig. 9
Close modal

The Faraday waves were revisited by Craik [142,143] and Craik and Armitage [144] who were able to describe the wavelength selection and hysteresis observed experimentally. Their experimental investigation dealt with surface gravity–capillary waves in a rectangular tank undergoing small vertical oscillations. The large aspect ratio of the tank enabled them to study the behavior of several neighboring 2D wave modes; their onset, hysteresis and instability. Two water depths, close to one and two centimeters, were investigated in detail. Hysteresis below the minimum forcing for linear wave-onset was found to imply that the excitation and nonlinear damping are both significant as confirmed by Miles [145]. Jiang et al. [141] performed boundary-integral simulations of 2D motion to determine the highly nonlinear motion of an interface with finite- and large-amplitude deformations. Their experimental investigation revealed that contact line effects increase the viscous natural frequency and alter the neutral stability curves. Furthermore, strong modulations in the wave amplitude for some forcing frequencies higher than 3.30 Hz were observed. Reducing contact line effects by Photo Flo addition suppresses these modulations.

One or more surface wave solitons8 with polar-on-like behavior were reported by Wu et al. [146] in partially filled container parametrically driven at an appropriate frequency and amplitude created. An amplitude equation in the form of a perturbed nonlinear Schrödinger equation was derived for parametric excitation of surface waves in an extended system by Elphick and Meron [147]. The existence of a stable nonpropagating kink solution was predicted. In addition, a stable nonpropagating soliton solution was found for subcritical excitation. Lioubashevski et al. [148] presented an experimental study of the onset of the Faraday instability in highly dissipative fluids. It was found that the critical acceleration for the transition to parametrically excited surface waves scales as a function of two dimensionless parameters corresponding to the ratios of the critical driving amplitude height and viscous boundary layer depth to the fluid depth. This scaling, which exists over a wide range of fluid parameters, was found to identify the proper characteristic scales and indicates that a Rayleigh–Taylor type mechanism drives the instability in this regime. An exact equation, which is nonlocal in time for the linear evolution of the surface of a viscous fluid, was derived by Cerda and Tirapegui [81]. It was shown that this equation becomes local and of second-order was used to study Faraday's instability in a strongly dissipative regime.

The mechanism responsible for the presence of complex dynamics in the damped nonlocal parametrically forced nonlinear Schrödinger equation was examined by Higuera et al. [149]. The evolution equations take the form of a pair of damped parametrically driven nonlinear Schrödinger equations9 with nonlocal coupling [150,151]. As the strength of the applied excitation increases, this equation undergoes a sequence of transitions to chaotic dynamics. The origin of these transitions is linked to the presence of heteroclinic connections between the trivial state and spatially periodic standing waves. These connections are associated with cascades of gluing and symmetry-switching bifurcations. The dynamics near the minima of the resulting resonance tongues were described by a system of coupled nonlocal Schrödinger equations with damping and parametric forcing by Vega and Knobloch [152]. Near the bicritical points where two adjacent resonance tongues intersect, a pair of coupled damped complex Duffing equations captures the properties of both pure and mixed modes, and of the periodic solutions resulting from a Hopf bifurcation on the branch of mixed modes. The coupled system of equations describing fast surface oscillations and slowly evolving mean flows in three dimensions was developed by Vega et al. [153]. A phenomenological model of parametric surface waves was introduced in the limit of small viscous dissipation that accounts for the coupling between surface motion and slowly varying streaming and large-scale flows. The analysis was limited to the simplest regular pattern consisting of stripes. Mean flows were induced by perturbation of the stripes, and their coupling to the order parameter equation affected the stability of stripe solutions. The results for the secondary instabilities of the primary wave revealed that the mean flow would lead to a weak destabilization of the base state and introduced a longitudinal oscillatory instability.

The dynamics of parametrically driven counterpropagating waves in a one-dimensional extended nearly conservative annular system were examined by Martel et al. [154]. The waves were described by two coupled damped parametrically driven nonlinear Schrödinger equations with opposite transport terms due to the group velocity and small dispersion. The system was characterized by two length scales defined by a balance between forcing and dispersion (the dispersive scale), and forcing and advection at the group velocity (the transport scale). The weakly nonlinear evolution of Faraday waves of a vertically excited annular container was studied by Martín and Vega [155]. In the small viscosity limit, the evolution of the surface waves was coupled to a nonoscillatory mean flow that develops in the bulk of the container. A system of equations for the coupled slow evolution of the spatial phase of the surface wave and the streaming flow was numerically integrated to show that the simplest reflection symmetric steady-state becomes unstable for realistic values of the parameters. The spatiotemporal evolution of the surface in the asymptotic regime was obtained by Rojas et al. [156]. The amplitude of the surface deformation was saturated to a finite value, due to the nonlinear terms. At very low Reynolds number, the temporal dynamics were found to be strongly nonlinear. The bifurcation was found to be supercritical.

The standing Faraday wave-trains that appear near threshold in a nearly conservative, parametrically excited system were studied by Mancebo and Vega [157]. Sufficiently close to threshold, the relevant equation whose cubic coefficient is extremely sensitive to wave number shifts, which can only be understood in the context of a more general quintic equation that also includes two cubic terms involving the spatial derivative. A weakly nonlinear analysis of one-dimensional viscous Faraday waves in 2D large aspect ratio containers was presented by Mancebo and Vega [158]. The surface wave was found to be coupled with a viscous long-wave mean flow.

Under parametric excitation, whose frequency is a slowly time-dependent “chirped,” i.e., $Ω(t)=Ω-μt$, where $μ$ is the constant chirp rate, Assaf and Meerson [159] showed that, when passing through resonance, resonance always occurs when the chirp rate is sufficiently small. The critical chirp rate, above which breakdown of autoresonance occurs, was found for different initial conditions. The theory of parametric autoresonance predicts that a downward chirp of the vibration frequency should cause persistent wave growth, which is only expected to terminate at large amplitudes, when an underlying constant frequency system ceases to exhibit a nontrivial stable fixed point [159]. Ben-David et al. [160] presented experimental verification of parametric autoresonance excitation of a nonlinear wave. They demonstrated that autoresonance is not hindered by moderate dissipation and the predicted (negative) frequency chirp indeed drives persistent wave growth, via the parametric autoresonance mechanism, to amplitudes that surpass the theory's region of validity.

### Breaking Surface Waves.

A surface wave whose amplitude reaches a critical level at which some process can suddenly start to cause large amount of wave energy to be transformed into turbulent kinetic energy is known as a breaking wave. During breaking, a deformation in the form of a bulge forms at the wave crest. High-frequency detail was found to be present in a breaking wave and to cause crest deformation and destabilization. After the tip of the wave overturns and the jet collapses, it creates a very coherent and defined horizontal vertex. The main vortex along the front of the wave diffuses rapidly into the interior of the wave after breaking, as eddies on the surface become more viscous. Advection (transport mechanism of a substance or property by a fluid due to the fluid's bulk motion) and molecular diffusion play a part in stretching the vortex and redistributing the vorticity, as well as the formation turbulence cascades.

Under relatively high-frequency parametric excitation, large amplitude surface waves were observed in Refs. [161–164]. This motion usually occurs in the form of typical spray-formed waves. Sometimes the surface motion becomes very violent at larger excitation levels with small vapor bubbles entrained in the liquid. The bubbles can become negatively buoyant and sink to the tank bottom. Analytical and experimental studies to examine the occurrence of surface disintegration with liquid particles spray and the wave characteristics of spray-excited low-frequency waves were reported in Refs. [129,165–167]. The surface disintegration and bubble formation in a vertically excited liquid column were theoretically and experimentally investigated by Buchanon et al. [168–180]. These studies showed that the smaller fluid height the larger is the minimum excitation amplitude required for the surface disintegration. It was also shown that the frequency of a spray-excited low-frequency wave is independent of the liquid height-to-diameter ratio.

Steep standing waves were found to undergo a surprising transformation at even large parametric excitation amplitude [181]. Small plunging breakers first appear to each side of the dimpled crest. A further increase in excitation amplitude would lead eventually to period tripling with breaking every two out of three waves. The existence regime for period tripling is shown in Fig. 10. It is seen that the measured neutral-stability curve is shifted downward due to contact line effects [141]. All experiments including those with period-tripled breaking were found to be limited to within the neutral-stability curve. Jiang et al. [181,182] showed experimentally that increasing the excitation amplitude further leads to breaking waves in three recurrent modes (period tripling): sharp crest with breaking, dimpled, or flat crest with breaking, and round crest without breaking. Interesting steep waveforms and period-tripled breaking were found to be related directly to the nonlinear interaction between the fundamental mode and the second temporal harmonic. The period-tripled breaking consists of three distinct modes: A, B, and C, as illustrated in Fig. 11. The maximum wave profile in mode A is seen to be characterized by its high elevation, sharp crest angle, violent breaking, and drop formation. Mode B follows with a dimpled or flat crest and double plungers to the sides of the crest. Mode C has a round (nonbreaking) crest similar to Penney and Price's [38] solution. The sharp-crested mode A reappears after mode C, forming a recurrent cycle with a three-wave period. The mode A wave was never realized experimentally on steep nonbreaking waves.

Fig. 10
Fig. 10
Close modal
Fig. 11
Fig. 11
Close modal

Although the first sharp crest forms an upward jet in mode A, it can also develop into a large plunger with its crest listing to one side. The appearance of an asymmetric plunger is random and does not affect the triple periodicity. A leftward plunging breaker shown in Fig. 12(a) was captured experimentally with both seed particles and dye in the water so that both the free surface and the underlying water are illuminated. Jiang et al. [181] demonstrated mode-B breaking in Fig. 12(b). Double plungers were formed at each side of the flat crest and create local bores (0.02 and 0.04 s). These postbreaking plungers slide down the wave crest at 0.06 and 0.08 s, creating irregular motions near the surface. The bulbous centre at 0.04 s to 0.08 s was found to be caused by the rebounding 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 and usually no irregular surface motions were observed as shown in Fig. 12(c).

Fig. 12
Fig. 12
Close modal

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

Steep forced waves generated by moving a tank containing water were studied experimentally and numerically by Bredmose et al. [185]. The most unusual feature was type-A table-top breaker in the form of a flat-topped wave crest with almost vertical sides was observed. A sequence of video snapshots shown in Fig. 13 reveal the generation of sharp-crested waves followed by flat-topped waves in a water tank of water depth of 400 mm. The process of steepening of successive wave crests can be observed by comparing the first two frames of Fig. 13. Table-top crests were predicted numerically by Topliss [186] using a Cauchy boundary-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 set of systematically differing initial conditions. These energetic waves are characterized by sustaining a flat top through the rise and fall of the crest. For some of the table-top breakers, a thinning of the profile during the downward motion was observed.

Fig. 13
Fig. 13
Close modal
Fig. 14
Fig. 14
Close modal

Under parametrically excited surface waves, the breaking state ejects droplets from wave peaks, when the applied forcing exceeds an acceleration threshold. The rate of breaking events approaches zero gradually with decreasing acceleration. Two properties of these ejections were studied around the ejection threshold [189]. Analysis of the ejection rate dependence on acceleration allowed the determination of the ejection threshold and an inference about the wave height distribution. A Poisson distribution was found for the times between ejections. Cabeza et al. [190] studied parametrically excited surface waves when finger structures are generated on the free surface. The intermediate states between regular patterns and droplet ejection in Faraday instability were also analyzed. It was found that the surface breaks and ejects droplets from wave peaks when the applied force exceeds an acceleration threshold. Both chaotic and turbulent bifurcation behaviors of Faraday surface waves were studied and many different transitions were reported. The spatial and temporal finger distribution was described in terms of external acceleration. Kahan et al. [191] used proper orthogonal decomposition to characterize the evolution of fingers regime in Faraday instability. The structural transition was analyzed for varying acceleration amplitude.

On a vertically vibrating fluid interface, a droplet can be indefinitely bouncing. When approaching the Faraday instability onset, the droplet was found to couple with the wave and starts to propagate horizontally. The resulting wave–particle association (referred as a walker) was shown to have remarkable dynamical properties, reminiscent of quantum behaviors [192,193]. The nature of a walker's wave field was investigated experimentally, numerically, and theoretically by Eddi et al. [194] who showed that the walker field results from the superposition of waves emitted by the droplet collisions with the interface. It was shown that each shock emits a radial traveling wave, leaving behind a localized mode of slowly decaying Faraday standing waves. As it moves, the walker keeps generating waves and the global structure of the wave field results from the linear superposition of the waves generated along the trajectory.

Surface singularities produced by the collapsing depressions of standing waves were examined in Ref. [195]. This phenomenon was first observed by Longuet-Higgins [196]. Under parametric excitation of a cylindrical tank partially filled with a fluid, it was observed that below a critical standing wave height $ηc$ the fluid surface topology remains smooth and simply connected. Above $ηc$, the collapsing wave entrains an air bubble beneath the surface and changes its topology from simply to multiple-connected. Thus, the critical height represents the threshold of the surface topology change. The resulting inertial collapse creates a singularity on the fluid surface at which the velocity and surface curvature diverge. This singularity was found to localize the kinetic energy of 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 square root of the kinematic surface tension. Their model was reconsidered by Das and Hopfinger [197] who presented some results pertaining to parametrically forced gravity waves in a circular cylinder in the limit of large fluid-depth approximation. The instability boundary in terms of the excitation amplitude ratio $Z0/R$ and excitation frequency ratio $Ω/(2ω01)$, where $Z0$ is the excitation amplitude, $R$ is the tank radius, $Ω$ is the excitation frequency, and $ω01$ is the first damping axisymmetric sloshing mode frequency is shown in Fig. 16 for Fluorocarbon (FC-72) fluid. It is seen that the axisymmetric mode exists in the range $0.94<Ω/2ω01<1.02$. The solid curve corresponds to the theoretical prediction with damping ratio $ζ=0.0022$. This curve is given by the expression

Fig. 15
Fig. 15
Close modal
Fig. 16
Fig. 16
Close modal
$Z0R=13.832tanh(3.832h/R)[ζ2+[(Ω/2)2-ω012]2(2ω012)2]$
(24)

where $h$ is fluid depth in the tank, and the detuning parameter is $β=[(Ω/2)2-ω012]/(2ɛω012)$, and forcing parameter $ɛ¯=Z0k01tanh(k01h)$. At $Ω/2ω01=1.02$, $β4=0.875$, the wave motion was found to bifurcate to the wave mode (3,1) of dimensionless wave number $k31R=4.201$. In the range $0.94<Ω/2ω01<0.974$, $β1≥β>β2$, where $β1=-0.962$ and $β2=-1.015$; the wave motion is unstable when the parametric instability threshold is crossed with exponential growth in wave amplitude up to breaking (inset image 1 in Fig. 16) with possible jet formation. Over the interval $0.974≤Ω/2ω01≤0.987$ and $β2≤β≤β3$, where $β3=-0.753$, the wave motion is unstable above the instability threshold but the amplitude remains finite (inset image 2 in Fig. 16). In the range $0.987<Ω/2ω01≤1.02$, the instability is subcritical for $Ω/2ω01<1$ and supercritical at $Ω/2ω01=1$ and above.

The period tripling event at excitation amplitude ratio $Z0/R=0.008$ is shown in Fig. 11(b). It is seen that the wave crest steepens, is flat-topped in the next period, and then takes intermediate amplitude before the cycle begins again. For three different liquids, Fig. 17 shows typical shapes of the last stable wave crests as well as the cavities, which form half a period later at the wave trough. Clearly, surface tension and viscosity affect the shape of the wave crests and consequently the cavity size by inhibiting parasitic capillary waves and by preventing Taylor instability from developing. Figure 17(a) reveals that the large viscosity of glycerin–water solution inhibits parasitic capillary waves and Taylor instability. The cavity, which is formed (lower part of image) when the wave column is accelerated downward, and impinges on the wave trough, is deeper and of smaller size than in water and FC-72. Its aspect ratio (cavity depth, $ℓ$, to cavity radius, $rca$) is $ℓ/rca≥1$. The cavity size shown in the images corresponds to the instant when the cavity just begins to contract. When the kinematic surface tension is small as in FC-72, shown in Figs. 17(d) and 17(e), a cylindrical precursor fluid column is projected upward (Taylor instability) that is then partially or completely taken over by the following wave crest; drop pinch-off may occur in some cases, as shown in Fig. 17(d), or the wave crest may be flat-topped, as shown in Fig. 17(e). Figures 17(b) and 17(c) belong to water in which upward projection of a cylindrical fluid column also occurred, but is less pronounced because of higher surface tension. Parasitic capillary waves are clearly present but because of the high surface tension, the wave crest remains axisymmetric. The wave shape shown in Fig. 17(f) is for FC-72 and container radius R = 2.5 cm resulting in a lower Bond number and, hence, a smoother wave crest.

Fig. 17
Fig. 17
Close modal

The Faraday waves' instability was studied in domains with flexible boundaries implemented by triggering this instability in floating fluid drops in Ref. [198]. An interaction of Faraday waves with the shape of the drop was observed, the radiation pressure of the waves exerted a force on the surface tension held boundaries. Two regimes were observed. These are co-adaptation of the wave pattern with the shape of the domain so that a steady configuration is reached, and the radiation pressure dominates and no steady regime is reached. The drop stretches and ultimately breaks into smaller domains that have a complex dynamics including spontaneous propagation. For the second regime, Fig. 18 shows the displacement of the boundary due to the waves did not lead to equilibrium. The formation of parallel standing waves results into a constantly increasing elongation of the drop into a snakelike structure which breaks into fragments having a large variety of dynamical behaviors. Figure 18 was obtained with ethanol of volume of $1.0±0.02 ml$, density $ρ2=789 kg/m3,$ dynamic viscosity $μ2=0.9×10-3 Pa·s,$ and surface tension $γ2=$$23 mN/m$ floating on silicon oil of corresponding properties $ρ1=965 kg/m3, μ1=100×10-3 Pa·s,$ and $γ1= 20 mN/m$. The measured interfacial tension was $γ1,2=0.7 mN/m$. The main characteristic is that ethanol is wetted by silicon oil. At rest, an oil film was observed to cover the upper surface of the drop as shown in Fig. 18(a). The instability in ethanol appears in a subcritical way; i.e., large amplitude waves form at threshold. These waves stretch the drop as shown in Fig. 18(c) but there is no convergence toward a final stable shape. The elongation is followed by buckling as shown Fig. 18(d) then breaking into several fragments as demonstrated in Fig. 18(e). Kiyashko et al. [199] experimentally investigated the dynamics of defects in the domain walls that arise at the interface of two domains under a parametric excitation of capillary waves on the fluid free surface. It was shown that in a symmetric domain wall defects move strictly along it. It was found that a single defect located near the domain wall can attract defects of domain wall and embedded in it.

Fig. 18
Fig. 18
Close modal

This section has addressed both weakly nonlinear and breaking Faraday waves. The occurrence of each type depends on the fluid and excitation parameters. However, when the natural frequencies of different modes are related to each other through nonlinear coupling of the modal equations of motion, the problem becomes complicated as the sloshing modes are sharing energy and the free surface becomes unsteady. This problem will be considered in Sec. 7.

## Parametric Excitation Under Internal Resonance

Weak resonant coupling among gravity waves on the surface of deep water can cause significant energy transfer among wave-trains. The nonlinear coupling may give rise to the occurrence of internal resonance conditions among the interacting modes. Internal resonance implies the presence of a linear algebraic relationship among the natural frequencies of the sloshing modes in the form $∑j=1nsjωj=0$, where $sj$ are integers, and $ωj$ are the natural frequencies of the coupled modes, the number $S=∑j=1n| sj |$ is known as the order of internal resonance. The problem of internal resonances in nonlinearly coupled oscillators is of interest in connection with redistribution of energy among the various natural modes. The coupling among these modes plays a crucial role in such interactions. In a straightforward perturbation theory, internal resonances lead to the problem of small divisors.

Experimental investigations conducted by McGoldrick [27] and Longuet-Higgins and Smith [200] demonstrated the presence of resonant interaction among surface waves. McGoldrick [27] showed that both second-harmonic and triadic resonances are possible for deep-water gravity–capillary waves. Hammack and Henderson [201] presented an overview of resonant interaction theories and experiments for waves on the surface of a deep layer of water. Pattern formation in parametric surface waves was studied in the limit of weak viscous dissipation in Refs. [202,203]. A multiscale expansion of the quasi-potential equations revealed the importance of triad resonant interactions, and the saturating effect of the driving force leading to a gradient amplitude equation. Minimization of the associated Lyapunov function was found to yield standing wave patterns of square symmetry for capillary waves, hexagonal patterns, and a sequence of quasi-patterns (QPs) for mixed capillary–gravity waves.

Nonlinear modal interaction was also studied for the case of spatiotemporal resonant triads in a horizontally unbounded fluid domain. The interaction takes place such that each critical wave number from linear analysis actually corresponds to a circle of critical wave-vectors. It has been argued that resonant triads may play a central role in the Faraday's wave pattern selection problem as indicated in Refs. [204–208]. Resonant triads comprised three critical wave-vectors that sum to zero, i.e., $k3=k1±k2$, where $|k1|=|k2|$ is the wave number of one critical mode and $|k3|$ is the wave number of the other critical mode. The bifurcation problem was formulated using a stroboscopic map and estimation of Floquet multipliers [209–214]. It was shown that there is a fundamental difference in the pattern selection problems for subharmonic and harmonic instabilities near the 10codimension-two point. Many experimental [204,215] and theoretical [208,216] studies attributed the formation of exotic patterns near the codimension-two (or “bicritical”) point to resonant triad interactions involving the critical or near-critical modes with different spatial wave numbers. 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 the process of pattern formation in weakly damped Faraday waves. For the case of two-frequency forcing, it was found that the strength of observed three-wave interactions depends on the frequency 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 occurring near the bicritical point where two pattern-forming modes with distinct wave numbers onset simultaneously. This triad was observed in the form of rhomboids11 and was involved in the formation of QPs and superlattices.

The period-doubling bifurcation of parametrically excited standing waves was analyzed for square and circular containers by Crawford [210,211] using reduced maps for the critical mode amplitudes. Experimental and theoretical investigations for surface waves in square containers suggested that the effective symmetry may be larger than the geometric symmetry of the container cross section. The difference between the geometric symmetry and effective symmetry is demonstrated in Fig. 19 by considering the cross section $Π$ of a square of nondimensional length $π$. For square containers, the geometric symmetry is generated by reflection about $x=π/2$ and reflection across the diagonal $(x,y)→(π-x,y)$ and $(x,y)→(y,x)$. A standing wave pattern can be smoothly extended by reflection in x and y to give a solution to the corresponding free-boundary problem posed on the larger domain $Π˜$ with periodic boundary conditions. The extra symmetries complicate the construction of normal forms and appear to stabilize effects in the experiments. These symmetries may be directly observed, if the sidewalls are deformed to a nonsquare cross section that retains square symmetry. Crawford et al. [212] indicated that there are hidden translational and rotational symmetries that further constrain the linear and nonlinear behavior of the fluid free surface. As a result, unexpected degeneracy among the linear wave frequencies and unexpected branches of nonlinear solutions in the bifurcation equations for the surface waves can occur. These additional symmetries are not obvious, since they are not symmetries of the square container and consequently do not preserve the boundary conditions of the problem.

Fig. 19
Fig. 19
Close modal

The interaction between surface wave modes leads to a variety of interesting nonlinear phenomena, including chaotic dynamics. When two or more spatial modes are simultaneously excited, the amplitude equations contain the independent dynamics of each mode and also their coupling. Symmetry considerations greatly simplify the analysis by reducing the number of allowed coupling terms to any given order in the perturbation expansion. Internal resonance arises from the coupling of two sloshing modes whose frequencies are either equal or one modal frequency is n times the frequency of another mode. The case of equal frequencies occurs naturally in a circular cylinder for which the nonaxisymmetric modes occur in degenerate pairs. These modes differ only by an azimuth rotation of $π/2$ and have the same natural frequency $ω1$. They are uncoupled (orthogonal) in the linear approximation, but are nonlinearly coupled both through direct, third-order interactions and through secondary modes that are excited at second-order and affects indirect, third-order interactions. However, the coupled motion of such a pair necessarily comprises angular momentum, which cannot be generated by parametric excitation. It is possible to have approximately coincidental modal frequencies in the doubly infinite, discrete spectrum for any liquid container [220].

A more common internal resonance may occur between two orthogonal modes with natural frequencies in the approximate ratio of 2:1. As indicated by Miles [221], the direct coupling is quadratic while the indirect coupling, through secondary modes, is of higher order and is therefore negligible. Internal resonance with $ω2=2ω1$ for two gravity-wave modes with wave numbers $k2$ and $k1$ in cylindrical tank of depth $h$ requires
$k2tanhk2h=4k1tanhk1h$
(25)

An example with a much larger coupling coefficient is resonance between the dominant antisymmetric and axisymmetric modes, for which $k1R=1.8412, k2R=3.8317$, and $h/R=0.1523$, where $R$ is the tank radius [221].

For a rectangular tank, the internal resonance of liquid modes in a rectangular tank under parametric excitation was studied in Refs. [95,222-225]. The liquid free surface under parametric excitation may be represented by the expression
$ηmn(x,y;t)=[Amn(t)cos(Ωt/2)+Bmn(t)sin(Ωt/2)] ×[cos(mπx/Lx)cos(mπy/Ly)]$
(26)
where $Amn$ and $Bmn$ are mode amplitudes which vary with time, in the weakly nonlinear regime, on a timescale much larger than $1/Ω$, m and n are integers giving the number of half-wavelengths in the x- and y-directions, and $Lx$ and $Ly$ are the width and breadth dimensions of the tank. The complex amplitude, $Cmn(t)=Amn(t)-iBmn(t)$, was governed by the first-order equation:
$C·mn(t)=αCmn(t)+βC¯mn(t)+γ|Cmn(t)|2Cmn(t)=S(Cmn,C¯mn)$
(27)
where $α$, $β$, and $γ$ are constant coefficients. In rectangular and square tanks, the evolution equations of modes (3,2) and (2,3) were obtained in the form [225]
$C·32(t)=S32(C32,C¯32)+μ32C32|C23|2+ν32C¯32C232$
(28a)
$C·23(t)=S23(C23,C¯23)+μ23C23|C32|2+ν23C¯23C322$
(28b)
where the function $S$ is defined in Eq. (27) for a single mode. The coefficients $μmn$ and $νmn$ are constants. For a square tank, these coefficients are identical and the two equations are interchangeable. This satisfies the symmetry requirements and internal resonance condition. For a square tank of 6-cm side and fluid depth of 2.5-cm, Simonelli and Gollub [225,226] measured the slowly varying amplitudes $Amn$ and $Bmn$ that contribute to the fluid free surface wave height, $η(x,y;t)$, using two problems under parametric excitation. Figure 20(a),
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 over one period of the forcing period [225].

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 over one period of the forcing period [225].

Close modal
shows the image of the free surface of mode (3,2), while Fig. 20(b) shows the superposition of the two modes (3,2) and (2,3). Figure 21(a),
Fig. 21

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

Fig. 21

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

Close modal
shows three regions of stability boundaries of three modes and their degenerate modes. Figure 21(b) shows the detailed structure of stability boundaries near the (3,2) and (2,3) resonance including additional stability region B in which the stable states are superposition of the two modes, with equal amplitudes $|C32|2=|C23|2$. In view of the symmetry properties of these modes, four equivalent mixed states are expected and were observed experimentally. Inside region D only pure states were found. Due to the symmetry, either pure state can be found depending on the initial conditions. Regions A and C were found to be hysteretic: in A the flat surface state co-exists with mixed states, while in C mixed states co-exist with pure states. The general bifurcation problem of Faraday resonance in a square container was also studied by Silber and Knobloch [227] using a 2D map. They suggested the necessity of fifth-order nonlinearity in their model in order to reproduce Simonelli and Gollub [225] diagram shown in Fig. 21.

The case of parametric excitation when the modal frequencies are in the ratio of 1:2 was studied in Refs. [95,223,228]. Holmes [228] qualitatively showed the existence of chaotic motions for certain parameter ranges close to 2:1 subharmonic resonances. Gu and Sethna [223] studied periodic, almost periodic, and chaotic wave motions in a rectangular tank subjected to vertical sinusoidal excitation. The internal resonance condition 1:2 requires the fluid height to be relatively small, which causes excessive energy dissipation. Such energy dissipation suppresses nonlinear phenomena making it impossible to verify the analytical results experimentally. When the frequencies are nearly equal, the free oscillation in a nearly square container was discussed by Bridges [229] for the case of standing waves. For the case of a nearly square container, all nonsymmetric modes have nearly equal natural frequencies independent of the fluid depth. The case of 1:1 internal resonance was considered in Refs. [227,230-232] and it was shown that the system is capable of exhibiting periodic and quasi-periodic standing and traveling waves. They were able to identify parameter values at which chaotic behavior can occur. Faraday waves in a circular cylinder, which are internally resonant with either the subharmonic mode (with frequency one-fourth that of the forcing) or the superharmonic mode (with frequency equal to that of the forcing) were investigated experimentally by Henderson and Miles [233]. For subharmonic resonance, both modes achieved comparable amplitudes that were steady or were modulated with one or two periods, or exhibited quasi-periodic or chaotic motions. For low modes, an energy exchange occurred during the initial period of growth, and precession instability was developed. For high modes for which both frequencies and wave numbers are in a 2:1 ratio, superharmonic resonance occurred irreproducibly and it was found to be prevailed by 1:1 interactions among the possible Faraday wave modes.

Nayfeh [234] presented weakly nonlinear analysis and derived the general fluid field equations. The free-surface conditions were expanded into Taylor series in terms of linear mode shapes. The governing equation of the liquid-free surface elevation was in full agreement with Miles Lagrangian formulation. In the presence of week damping, the free oscillation component of any mode that is not directly excited by the parametric resonance or indirectly excited by internal resonance will decay with time. If the condition of perfect internal resonance of two modes, $ω2=2ω1$, was relaxed, then Hopf bifurcations are possible. Nayfeh and Nayfeh [235] considered the case of 2:1 internal resonance condition when the higher mode, rather than the lower mode, is excited by a principal parametric resonance. The solutions of the modulation equations were found to be fixed-point, limit-cycle, or chaotic solutions. Multiple limit cycles with different amplitudes and periods were detected and shown to co-exist over some ranges of the external parametric resonance detuning parameter of the second mode taken as the bifurcation parameter. Some limit cycles were found to experience pitchfork bifurcation while others undergo cyclic fold bifurcation. The pitchfork bifurcation produces symmetry-breaking bifurcation and the cyclic fold bifurcation results in cyclic jumps. In the case of principal parametric resonance of the lower mode, the route to chaos was found to be a period-doubling sequence of bifurcations.

The role of weakly damped modes in the selection of Faraday wave patterns forced with rationally related frequency components $mω$ and $nω$ was examined by Topaz and Silber [236]. Symmetry considerations were used for the special importance of the weakly damped modes oscillating with twice the frequency of the critical mode, and those oscillating primarily with the “difference frequency” $| n-m | ω$ and the “sum frequency” $| n+m | ω$. The resonance effects predicted by symmetry were emerged in the perturbation results for one spatial dimension and agree with the numerical results for two dimensions. The difference frequency resonance was found to play a key role in stabilizing superlattice patterns observed by Kudrolli et al. [237]. Later, Topaz et al. [238] showed how pattern formation in Faraday waves may be manipulated by varying the harmonic content of the periodic forcing function. Their approach was based on the crucial influence of resonant triad interactions coupling pairs of critical standing wave modes with damped, spatiotemporally resonant modes. For forcing functions with arbitrarily many frequency components, it was found that there are at most five frequencies that affect each of the important triad interactions at leading order. The relative phases of those forcing components were found to make the difference between an enhancing and suppressing effect. Their approach was applied to one-dimensional periodic patterns obtained with impulsive forcing and to 2D superlattice patterns and QPs obtained with multifrequency forcing.

Moehlis et al. [239] extended the work of Feng and Sethna [232] to study periodic orbits associated with heteroclinic bifurcations in a model of the Faraday system for containers with square cross section. These periodic orbits were found to correspond to quasi-periodic surface waves in the physical system. The periodic orbits were associated with heteroclinic bifurcations. Chaotic attractors were also found in the model equations, which correspond to chaotic surface waves. The case when the sloshing modal frequencies are in the ratio of 3:1 was considered in Refs. [240,241]. For the standing gravity waves, 3:1 internal resonance between (1,1) mode and (5,2) mode was found to be the strongest for the Faraday resonance in the 2D standing gravity waves of finite depth. Wada and Okamura [241] derived the nonlinear evolution equations of the dominant free-surface modes up to fifth-order in wave amplitude. It was found that the third-order evolution equations are integrable while the fifth-order evolution equations are chaotic. This means that the fifth-order nonlinearity breaks the integrability of the third-order integrable system. The 3:1 resonance was found possible between two normal modes for 2D standing gravity waves with the dimensionless depth $hk≈0.6232$, where $k$ is the wave number of the standing gravity wave. Related to the problem of internal resonance is the problem of sloshing modal competition and associated free-surface patterns. These issues will be discussed in Secs. 8 and 9.

## Mode Competition of Faraday Waves

It is known that under parametric excitation of frequency $Ω$ the first excited mode is the one near to the parametric resonance at $Ω/2$. However, in some cases, the detuning may be of the same order for two or more adjacent modes, so that these modes can enter in competition even close to threshold. The competition between nearly degenerate modes was shown to lead to chaotic behavior in a circular container [220] and to complex selection rules in a square cell [242]. Depending on the value of the forcing frequency, the system may either display a subcritical transition from one mode to the other or a competition between the two modes, leading to the onset of a mixed mode.

At a certain critical depth-radius ratio of liquid in a circular container, the first-mode oscillation may be coupled with a higher mode motion at an integral multiple of the basic frequency. Chaotic behavior may arise due to competition between two different spatial modes or patterns [220,243,244]. An axisymmetric mode and two completely degenerate antisymmetric modes of gravity waves in a circular cylindrical container were, respectively, studied by Mack [245] and Miles [221]. A region of mode competition emerges in which the fluid surface can be described as a superposition of two modes with amplitudes having slowly varying envelopes. These slow variations can be either periodic or chaotic. Ciliberto and Gollub [243] developed a phenomenological model in the form of the nonlinear Mathieu equation
$a··mn+ζmna·mn+(ωmn2-ψmnZ0cosΩt)amn=ςmnamn3$
(29)
where $ςmn$ is the coefficient of the cubic term that limits the growth of the mode, and $ψmn$ is the gain coefficient of the parametric excitation term. The subscripts $mn$ indicate the number of angular maxima and the number of nodal circles, respectively. The cubic term on the right-hand of Eq. (29) was added to limit the growth of the response amplitude and was not based on the actual nonlinear modeling as in the case of the boundary value problem of liquid sloshing dynamics. This cubic term should be regarded as equivalent to all nonlinear inertia terms of mode mn. Figure 22
Fig. 22

Surface mode contours: (a) (m,n) = (4,3) and (b) (m,n) = (7,2) [246]

Fig. 22

Surface mode contours: (a) (m,n) = (4,3) and (b) (m,n) = (7,2) [246]

Close modal
shows the surface mode profile of the two modes (4,3) and (7,2), which were the subject of many studies in the literature for mode competition [246].

Figure 23 shows the digitized optical intensity fields formed by stable patterns of modes (4,3) and (7,2), whose natural frequencies are $f4,3=7.892 Hz$ and $f7,2=8.1042 Hz$, respectively. The crosses are experimentally determined points on the stability boundaries [243,244]. At the intersection of the two stability boundaries, both modes oscillate simultaneously. Above the stability boundaries, the liquid surface oscillates at half the driving frequency in a single stable mode. The shaded areas are regions of mode competition, in which the surface can be described as a superposition of the (4,3) and (7,2) modes with amplitudes having a slowly varying envelope in addition to the fast oscillation at half the excitation frequency, $f0/2$. These slow variations can be either periodic or chaotic. At driving amplitudes higher than those shown in Fig. 23, the surface can become chaotic even if the driving frequency is resonant, so that a single mode is dominant.

Fig. 23
Fig. 23
Close modal

As one crosses from the region of slow periodic oscillations into the chaotic region, one finds a period-doubling bifurcation followed by a transition to chaos. A typical example is shown in Fig. 24, where time history records are shown for three different driving amplitudes but fixed driving frequency of 16.05 Hz. The figure also shows the corresponding power spectra of the time series. It is seen that subharmonic bifurcation occurs at relatively higher excitation amplitude ($Z0=149 μm$ and $Z0=190 μm$). This is accompanied by a slight broadening of the peaks. Above $Z0=180 μm$, the time history record is chaotic. Ciliberto and Gollub [243,244] indicated that at amplitudes higher than $Z0=200 μm$, mode competition disappears and only the fourfold symmetric mode dominates. An experimental study was performed by Karatsu [247] to study nearly degenerate (4,1) and (1,2) surface wave modes in a circular cylinder and observed periodic and chaotic mode competition.

Fig. 24
Fig. 24
Close modal
The work of Ciliberto and Gollub [243,244] attracted the interest of some researchers in an effort to understand the appearance of few-dimensional chaos systems with an infinite number of degrees-of-freedom. For example, Meron and Procaccia [51,52] employed the center-manifold and normal-form theories to derive evolution dynamical equations in cylindrical container in terms of two time scales, $t$ and the slow time $τ=ɛt$, where $ɛ=2Ω(ω4,3-ω7,2)$ is a small parameter, in the form
$da4,3dτ=(−ζ4,3+iσ4,3)a4,3+i[Γ1a¯4,3+Γ2|a4,3|2a4,3+Γ3|a7,2|2a4,3+Γ4a¯4,3a7,22]$
(30a)
$da7,2dτ=(−ζ7,2+iσ7,2)a7,2+i[Δ1a¯7,2+Δ2|a7,2|2a7,2+Δ3|a4,3|2a4,3+Δ4a¯7,2a4,32]$
(30b)
where $σmn=(2ωmn-Ω)/(4Ω)$ is a detuning parameter, $ζmn$ are phenomenological damping parameters, and the coefficients $Γi$ and $Δi$ depend on the excitation amplitude and system parameters. Meron and Procaccia [51,52] were able to rationalize essentially all the major experimental observations such as the appearance of regular and chaotic mode competitions, the existence of asymmetry between (4,3) and (7,2) modes, and the qualitative structure of the stability diagram. However, Miles [248] and Miles and Henderson [28] disputed the analytical results of Meron and Procaccia [51,52] on the basis that Eqs. (45) and (46) violate symmetry 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 coefficients $Γ3=Δ3$ and $Γ4=Δ4$. On the other hand, Meron and Procaccia [51] neglected nonlinear inertia terms containing the time derivative on the ground that these terms are not expected to contribute significantly to the long-time behavior of the amplitude equations. In their reply, Meron and Procaccia [249] confirmed that these coefficients should be different, in principle. They also claimed that the Hamiltonian formulation of Miles [43] yields disagreement with the linearized hydrodynamics treatment with $Γ1=Δ1$. The differences between $Γ3$ and $Δ3$, also between $Γ4$ and $Δ4$, are of the same order as that between $Γ1$ and $Δ1$. It was affirmed by Meron and Procaccia that the difference between these coefficients is small for almost degenerate models of the order $(ω4,3-ω7,2)/Ω$ and can be considered as a high-order correction. Recall
$ωmn2=λmntanh(hλmn)(γλmn2ρ+g)$
(31)

where $γ$ is the surface tension and $g$ is the gravitational acceleration. Note that when the term $gλmn$ dominates the mode is a gravity wave, while when $γλmn3/ρ$ 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 were reduced into a two degrees-of-freedom system because each nonaxisymmetric mode has two completely degenerate components and the associated angular momentum of each mode tends to vanish owing to the damping and the circular symmetry. Period-doubling bifurcation and chaotic solutions with one positive Lyapunov characteristic exponent were obtained numerically. It was shown that some of the period-doubling bifurcations are related to the symmetry. Using the results of Umeki and Kambe [251] and Umeki [252], the controversy between the results of Meron and Procaccia [249] and Miles [248] concerning the existence of canonical formulations was resolved. A two-parameter analysis of the interaction between two period-doubling modes with azimuthal wave numbers $k$ and $l(l>k≥1)$ was developed by Miles et al. [253]. It was found that when one mode bifurcates subcritically and the other supercritically the pure mode branches lose stability to a branch of reflection-symmetric mixed modes. The system was found to undergo a Hopf bifurcation then to a quasi-periodic reflection-symmetric pattern. The results explain a number of observations reported by Ciliberto and Gollub [243,244] on parametrically excited surface waves in circular container.

Parametric excitation of surface waves in a container under vertical forcing was investigated by Kambe and Umeki [209]. A system of evolution equations of third-order nonlinearity was derived for the case of excitation frequency, which is close to twice the frequencies of two nearly degenerate free modes. It was found that this dynamical system yields not only excitation of a single-mode state but also interaction between two modes in which each mode oscillates either periodically or chaotically. These results were found in good agreement with the experimental observations, except for the case of strong nonlinearity. Homoclinic chaos in the Hamiltonian system of two degrees-of-freedom without damping was studied numerically. It was suggested that the chaotic mode competition observed experimentally is different from the homoclinic chaos. The analytical results for a pair of (4,3) and (7,2) modes in a circular cell were found in agreement with the experiment by Ciliberto and Gollub [220,244], and the results for the (2,3) and (3,2) mode pair in a rectangular cell are also in good agreement with the experiment by Simonelli and Gollub [225], except for the presence/absence of mixed-mode excitation. Umeki [252] showed that the periodic mode competition occurs from a Hopf bifurcation, which is supercritical for the predicted nonlinear coefficients of gravity waves but may become subcritical for slightly modified values of the coefficients, of a mixed rotating wave state for the slightly rectangular case. If the damping coefficient is sufficiently small, the periodic orbits in the phase 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 of frequency $ω$ and wave number $k$ in a square container subjected to the vertical displacement $Z0cos2ωt$ were studied based on the assumptions that $0 < ζ < kZ0,$ where $ζ$ is the linear damping ratio. The fixed points of the evolution equations comprised four solutions. These are: (1) the null solution; (2) an orthogonal pair of rolls described by either $coskx$ or $cosky$; (3) an orthogonal pair of squares described by either $coskx+cosky$ or $coskx-cosky$; and (4) coupled-mode solutions for which both modes are active and neither in phase nor in antiphase. The fixed points for rolls and squares lie on separate loci in an energy-frequency plane that intersect the null solution at a pair of pitchfork bifurcations, one of which is definitely supercritical and the other of which may be either subcritical or supercritical. Crawford et al. [254] formulated the modal interaction between two period-doubling in the presence of symmetry as a map in four dimensions, and classify the primary, secondary, and tertiary bifurcations. The observed amplitude oscillations were found to arise as a tertiary Hopf bifurcation from a mixed mode pattern provided exactly one of the primary instabilities is subcritical. Riecke [255] analyzed the stable wave number kinks in parametrically excited standing waves in large aspect ratio. It was found that for small group velocity of the underlying traveling waves the stable band of wave numbers can split up into two sub-bands, which are separated by a region of unstable wave numbers. This gives rise to solutions with stable wave number kinks which bridge the unstable regime between the sub-bands.

Modal competition between three neighboring 2D modes in a narrow rectangular tank was experimentally and theoretically studied [256]. In particular, the stability of a standing wave to perturbations from the two neighboring sideband modes was studied. Cubic damping, cubic forcing, and fifth-order conservative terms were retained in the analytical model for deep water. Retaining these higher order nonlinearities gave rise to fairly good agreement with experimental results. Faraday instability in an elongated rectangular cell was studied by Residori et al. [257] for a range of the forcing frequency for which two nearly degenerate modes enter in competition. The stability properties of the two modes were found to either co-exist in a bistable way or give rise to a mixed mode. It was indicated that different bifurcation scenarios exist depending on the relative detuning between the two modes. Slow oscillations accompanying the mixed mode were explained by the frequency beating of the two competing modes.

In the presence of small viscosity, i.e., $Cg≡ν[gh3+ (γh/ρ)]-1/2≪1$, where $Cg$ is the capillary–gravity number, it was shown that the waves couple to a streaming flow driven in oscillatory viscous boundary layers at rigid walls and the free surface [150]. This flow in turn affects the waves responsible for the oscillatory boundary layers. It was indicated by Martín et al. [258] that this coupling is responsible for different types of drift instabilities of the waves, instabilities that were observed in experiments in annular containers [259] but are absent from the theory when the coupling to the streaming flow is neglected. These instabilities were found to arise not only in annular containers but in cylindrical containers as well, and are driven by a coupling between the streaming flow and the spatial phase of the waves. Richer dynamic behavior was reported when the container is deformed into an elliptical one as shown in Higuera et al. [135]. In this case, the streaming flow was found to couple with the amplitudes of the standing waves, resulting in a much stronger coupling between the waves and the streaming flow. Higuera et al. [260] explored the consequences of this coupling in model equations derived originally by Higuera et al. [135] under the assumption that the Reynolds number of the streaming flow is small. The system was described by a five-dimensional system of ordinary differential equations. The eccentricity of the container, although small, was found of crucial importance since it results in a coupling of the streaming flow to the complex amplitudes of the two nearly degenerate modes.

The Floquet theory was used by Mancebo and Vega [261] to determine the threshold acceleration for the appearance of Faraday waves in large aspect ratio containers. Different distinguished limits were considered in the analysis. In particular, the case of nearly inviscid limits, i.e., when the capillary–gravity number $Cg≪1$ then the most dangerous mode at threshold is potential, except in two thin boundary layers near the bottom wall and the free surface, and an approximation of the critical excitation acceleration amplitude, $ac$ can be found in closed form. If that limit of the capillary–gravity number does not hold, then the most dangerous mode at threshold exhibits nonlocalized vorticity due to viscous effects. The other case is the basic limit and highly viscous sublimit, which is captured as $Cg>1$. In this case, viscous effects dominate gravity and surface tension, which can both be ignored in the analysis. The resulting simplified problems either admit closed-form solutions or are solved numerically by the well-known method introduced by Kumar and Tuckerman [262]. The critical excitation acceleration amplitude $ac$ was found to depend on excitation frequency, the capillary–gravity number, $Cg$, and the gravity–capillary balance parameter, $S=γ/(γ+ρgh2)$, which measures the ratio of surface tension to its combined effect with gravity.

Knobloch et al. [263,264] considered nearly inviscid parametrically excited surface gravity–capillary waves in 2D periodic domains of finite depth with small and large aspect ratios. Coupled equations describing the evolution of the amplitudes of resonant left- and right-traveling waves and their interaction with a mean flow in the bulk were derived. The mean flow consisted of an inviscid part together with a viscous streaming flow driven by a tangential stress due to an oscillating viscous boundary layer near the free surface and a tangential velocity due to a bottom boundary layer. Through nonlinear rectification, Reynolds stresses were generated, which drive a streaming flow in the nominally inviscid bulk. This flow in turn was found to advect the waves responsible for the boundary layers. The resulting system was described by amplitude equations coupled to a Navier–Stokes-like equation for the bulk streaming flow, with boundary conditions obtained by matching to the boundary layers. The coupling to the streaming flow was found to be responsible for various types of drift instabilities of standing waves and in appropriate regimes can lead to the presence of remarkable relaxations oscillations.

A sequence of symmetry-breaking instabilities leading to a chaotic state was observed in the surface deformations of a fluid layer subjected to a vertical oscillation by Gollub and Meyer [265]. For driving amplitudes above a critical value, a primary instability was observed in the form of circularly symmetric standing waves at half the driving frequency. A second instability at a higher threshold 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] used liquid helium and water in thin annular troughs and observed both period-doubling and quasi-periodic motions apparently involving three modes. Nonlinear evolution equations for the amplitudes of resonant capillary–gravity waves with different directions of wave number vector were derived by Yoshimatsu and Funakoshi [268]. These equations include cubic nonlinearity and the effects of viscous damping and parametric forcing obtained from the energy equation. The center-manifold was used to obtain quintic amplitude equations for unstable modes in which cubic damping and forcing were included. It was concluded that the squares are stable for sufficiently short waves, the hexagons and the eightfold QPs are stable when the wavelength is within two intermediate regions, and the stripes are always unstable. The disordered structure of the free-surface flow under relatively large harmonic excitation amplitude was also experimentally observed by Gollub and Meyer [265]. Their measurements showed a sequence of symmetry-breaking instabilities leading to chaotic state.

In the weakly inviscid regime, parametrically driven surface gravity–capillary waves generate oscillatory viscous boundary layers along the container walls and the free surface. Through nonlinear rectification these generate Reynolds stresses, which drive a streaming flow in the nominally inviscid bulk. This flow in turn advects the waves responsible for the boundary layers. The resulting system was described by amplitude equations coupled to a Navier–Stokes-like equation for the bulk streaming flow. Higuera et al. [260,269,270] examined two model systems, which include an elliptically distorted cylinder while in the second it is an almost square rectangle. The forced symmetry breaking was found to result in a nonlinear competition between two nearly degenerate oscillatory modes. This interaction was found to destabilize standing waves at small amplitudes and amplifies the role played by the streaming flow. In both systems, the coupling to the streaming flow triggered by these instabilities was found to lead to slow drifts along slow manifolds of fixed points or periodic orbits of the fast system, and to generate behavior that resembles bursting in excitable systems. The new dynamical behavior included relaxation oscillations involving abrupt transitions between standing and quasi-periodic oscillations, and exhibiting “canards.” Here, relaxation oscillations are nonlinear oscillations, which are nonsinusoidal repetitive such as those shown in Fig. 25. These limit cycles are evidently relaxation oscillations, but of an unusual type, involving slow drifts along branches of both equilibria and of periodic orbits, with fast jumps between them. Moreover, these oscillations may be symmetric or asymmetric, with the symmetry alternately present and broken in successive periodic windows. The presence of relaxation oscillations in this system can be attributed to the disparity between the decay times of free surface gravity–capillary waves and the streaming flow that prevails in the nearly inviscid regime.

Fig. 25
Fig. 25
Close modal

In the case of a slightly elliptical container, two types of standing waves, oriented along the major and minor axes of the ellipse, were found to come in close succession as the amplitude of the parametric excitation increases, and these may interact at small amplitude, producing mixed modes which are much more efficient at driving a streaming flow. In this case, the fluid flow couples to the two amplitudes, as well as to the spatial phase of the resulting pattern. Higuera et al. [260,269,270] demonstrated that this interaction can lead to relaxation oscillations characterized by switching from single frequency standing waves to two-frequency waves and back. Under appropriate conditions these relaxation oscillations can exhibit the so-called canard phenomenon in which the system follows nominally unstable solutions in the slow phase. Various global bifurcations are located as well, of which perhaps the most interesting is responsible for the appearance of chaotic dynamics right at threshold of the primary instability. In an elliptical container, Faraday waves were described by a third-order system of ordinary differential equations with characteristic slow–fast structure [271]. These equations describe the interaction of standing waves with a weakly damped streaming flow driven by Reynolds stresses in boundary layers at the free surface and the rigid walls.

Faraday waves provide a convenient experimental system for studying pattern formation due to fast time scales and large aspect ratio. The theoretical starting point of pattern formation is usually a set of deterministic equations of motion, typically in the form of nonlinear partial differential equations. The study of pattern formation in fluids has greatly been benefited from the careful and controlled experiments as well as the development of new analytic and numerical tools. Cross and Hohenberg [272] and Müller [273] presented two different review accounts on the spatiotemporal pattern formation of Faraday wave patterns. Knobloch [274] presented an overview of the theory of pattern formation in many systems of interest in physics, which exhibit spontaneous symmetry-breaking instabilities that form structures of patterns.

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

Other experimental investigations were conducted on large systems to examine the pattern selection and chaotic behavior [278,280–287]. Ezersky et al. [288] observed a grid of waves of squares on a vibrating thin layer of silicone, while three types of grids were reported by Levin and Turbnikov [276] and presented a hexagonal grid, a grid of squares, and one-dimensional grid. Christiansen et al. [283] and Edwards and Fauve [289] observed an eightfold and 12-fold QPs analogous to a 2D quasi-crystal in parametric excitation of capillary waves experiment. The observed QP was found independent of the shape of the container side walls. Regions with perfect crystal patterns separated by domain walls-chains of dislocations were found to exist in parametrically excited capillary ripples [287]. The transition of ensembles of domains to perfect crystals was found to be due to collapse of individual domains when the domain wall is a closed contour of annihilating dislocations and merging of neighboring domains due to the dislocation climb to the walls of a cell. A stable fivefold standing wave pattern exhibiting a germinal quasi-crystalline structure was observed in a symmetry breaking Faraday instability experiment with a low aspect ratio and the liquid depth being less than the surface wavelength by Torres et al. [290].

When Faraday waves are excited well beyond the threshold for pattern formation, the ordered patterned structure is lost [291]. Such spatial disorder, known also as “defect-mediated turbulence,” was found to occur in a wide class of driven nonlinear systems [277,292,293]. It was indicated that defect-mediated turbulence is well characterized. Above a critical value of the excitation acceleration amplitude, defects in the form of dislocations of pattern lines were found to spontaneously nucleate throughout the system and their nucleation rate increases with the excitation. Furthermore, their motion was found to generate rapid exponential decay of temporal and spatial correlations. The transition 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 spatially incoherent oscillatory phase, which consists of highly damped waves that propagate through the effectively elastic lattice defined by 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 the pattern selection using analytical tools. Two mechanisms for selecting the main frequency responses that are different from the first subharmonic one were identified in the literature [295]. The first mechanism occurs when two or more frequency components are introduced in the parametric excitation. Each component will tend to excite its own corresponding first subharmonic mode. Their relative amplitudes will determine which of these responses has the lowest global excitation strength threshold, which establishes the instability that is observed at onset. The second mechanism can only arise in the high viscosity regime. If the fluid layer is shallow enough, even a single component excitation with low enough frequency can excite instability different from the first subharmonic one. As the viscous boundary layer reaches the bottom of the fluid container, the threshold of the lowest unstable modes rises allowing others with higher main frequency components to become unstable at onset. In Secs. 9.1 and 9.2, this paper will provide an account of the formation and selection of Faraday patterns under single-, two-, three-, and multifrequency excitations.

### Single-Frequency Parametric Excitation.

Under single-frequency parametric excitation of a fluid layer can display Faraday waves of different types of patterns depending on the excitation level and frequency. Weak nonlinear effects can cause interactions between surface waves with different wave modes and can be important in determining free-surface wave patterns. The liquid free surface may be regarded as a superposition of interacting waves propagating in different directions. Faraday waves were observed to be especially versatile and exhibit the common 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 rhombic patterns [298], and localized waves [299]. Christiansen [300] conducted experimental investigation to measure the damping rates and the critical amplitude of the primary instability of high aspect ratio of Faraday waves. Above the primary instability, a sequence of ordered crystalline states was observed. These states include a quasi-crystalline pattern in the form of eightfold symmetry.

It was reported that if the dissipation is large, the preferred pattern consists of parallel stripes [284,301–303]. At higher driving amplitudes, stationary patterns with a hexagonal shape in the centre of the cell were observed experimentally in Ref. [242]. Standing wave patterns with different symmetries were observed and understood as superposition of two linearly unstable modes of the fluid free surface. The pattern wave number was fixed by the excitation frequency, whereas the shape was found to depend on the nonlinear interaction. Douady et al. [259] presented a study of secondary instabilities of parametrically generated standing waves in a horizontal layer of fluid subjected to vertical vibrations. It was found that when the driving frequency is increased (respectively, decreased), the system bifurcates abruptly to another standing wave pattern by nucleation (respectively, annihilation) of one wavelength. An experimental study of surface waves parametrically excited was presented by Douady [15]. Stability boundaries, wave amplitude, and perturbation characteristic time of decay were measured and found to be in agreement with an amplitude equation derived by symmetry. The measurement of the amplitude equation coefficients provided some interpretation of the observed transition, which is supercritical, and showed the effect of the edge constraint on the dissipation and eigenvalues of the various modes. The fluid surface tension was obtained from the dispersion relation measurement.

The threshold accelerations necessary to excite surface waves in a vertically excited fluid square container was measured [304,305]. Figure 26 shows the stability boundaries according to the numerical solution (solid curve), while the dashed curve represents the threshold curve in the absence of mode quantization. Because the actual thresholds are larger than the predictions, it was concluded that another source of energy dissipation was present. The small inset images in Fig. 26 are examples of patterns obtained at different frequencies. The Faraday instability was examined experimentally and theoretically by Barrio et al. [306] and found that for certain liquids, in which viscosity is very small and with peculiar physical properties such as a high molecular weight and density, there exists an additional dissipation mechanism that introduces an important nonlinear term in the equations of motion. This mechanism was found to produce the stabilization of symmetric patterns. The model equation was solved numerically and the patterns produced were analyzed and compared with experimental images. QPs were observed in two different Faraday wave experiments, one with a low-viscosity deep layer of fluid with single-frequency parametric excitation [204,205,283], and the other with a high-viscosity shallow layer of fluid and forcing with two commensurate temporal frequencies [301]. The pattern instability for a layer of a viscous fluid in a large aspect ratio container subject to vertically arbitrarily periodic excitation was studied by Chen and Wei [307]. The instabilities for Faraday water wave system under excitations of the triangle and square waves were analyzed.

Fig. 26
Fig. 26
Close modal

Under single-frequency excitation and depending on the fluid viscosity of glycerol/water mixture, the pattern of squares was observed for kinematic viscosity below $ν≈0.70 cm2/s$. For larger viscosity $ν≈1.00 cm2/s$, the pattern changes to parallel lines as shown in Fig. 27. The lines preferred to be perpendicular to the sidewalls, especially at high excitation amplitude. For the circular geometry of the container, the lines of the pattern are bowed. It is seen that the wavelength at the center is smaller than near the circumference. As the excitation amplitude increases, a spontaneous defect pair generation was observed similar to the behavior in convective rolls at low Prandtl number [308,309]. It was observed that these defects migrate outward until they vanish near the boundary and then another defect pair is produced. If the excitation amplitude is abruptly increased, patterns of circles and spirals were observed by Edwards and Fauve [301]. The spiral pattern 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 stable bound state in parametrically excited capillary ripples [310]. Note that a perfect structure consists of two mutually orthogonal pairs of standing waves. Individual dislocations in the form of such bound states may interact with each other. It was shown that the dislocations may either annihilate, if they have opposite topological charges, or are arranged into quasi-stable states in the form of a linear chain, in the case of like topological charges. Kalinichenko et al. [128,311] derived the conditions for harmonic instability of the free surface of a liquid of low viscosity in a rectangular vessel of finite horizontal dimensions.

Fig. 27
Fig. 27
Close modal

Spirals and targets were studied by Kiyashko et al. [312] using silicon oil with viscosity $ν=1.0 cm2/s$, density $ρ=0.97 g/cm3$, surface tension coefficient $γ=20.5 dyn/cm$, and fluid depth of 0.5 cm. It is known that plane standing waves are completely stationary with nonmoving nodes and antinodes. On the other hand, standing waves forming targets and spirals drift slowly toward the core [312]. The speed of the drift was found to depend on the magnitude of vertical excitation acceleration, as shown in Fig. 28. Wave drift phase velocity was found to sensitively depend on the profile of the side walls of the fluid cell. It is seen that the drift velocity is maximal for a straight vertical wall, and is reduced for a wall with a step. It was assumed that the wave drift is related to a shear flow produced near the wall by rapidly decaying surface waves at the fundamental frequency, generated by an oscillating meniscus. This shear flow was observed even in the subcritical regime when the parametric instability is absent. Near the threshold, the magnitude of velocity of the shear flow was close to 0.5 mm/s. Both contracting targets and rotating multi-armed spirals were found to persist for a long time (a few hours) in the cavity. The experimental images of the topology of the free surface were documented in Ref. [312] and it was shown that multi-armed spirals with different values of topological charge emerge as a result of dislocations interacting with the background target pattern. Dislocation pairs were observed produced by perturbing rolls at the periphery of the target. One dislocation of the pair quickly disappears at the wall, and another moves toward the center along some curved trajectory as shown in Fig. 29. The velocity of the defect was found to increase as it approaches the core. When several defects were introduced on the target, stable multi-armed spirals were observed. The evolution of a target with four dislocations (two positive and two negative) is shown in Fig. 29(a). A spiral is formed when one of the defects is already at the center while the others are still on the periphery as shown in Fig. 29(b), and later a target is restored as shown in Fig. 29(c) when all the dislocations annihilate at the center. In another experiment, a two armed spiral was formed and persisted for a long time as shown in Fig. 29(d).

Fig. 28
Fig. 28
Close modal
Fig. 29
Fig. 29
Close modal

Faraday waves in a stadium shaped container were experimentally studied by Kudrolli et al. [313]. Figure 30 shows samples of surface wave patterns generated at different parametric excitation frequencies. As pointed by Agam and Altshuler [314], the experimental findings are quite intriguing, since these patterns appear in 90% of the cases, and their magnitudes is unexpectedly large. Agam and Altshuler [314] further examined the problem and showed that, at sufficiently low frequencies, the wave patterns are “scars” selected by the instability of the corresponding periodic orbits, the dissipation at the container side walls, and interaction effects which reflect the nonlinear nature of Faraday waves. Large boundary dissipation was found to prevent the formation of random patterns as well as other modes such as the whispering gallery modes. Moreover, increasing the boundary dissipation, by lowering the fluid level, should eventually suppress all scars except the horizontal one.

Fig. 30
Fig. 30
Close modal

Two mechanisms were proposed for QP formation. The first is applied to single-frequency forced Faraday waves and was tested experimentally by Westra et al. [206]. The second was developed to explain the origin of the two length scales in superlattice patterns [219,236], which were observed in two-frequency experiments [237]. These two mechanisms were examined by Rucklidge and Silber [315] with the purpose of explaining the selection of QPs in single- and multifrequency forced Faraday wave experiments. Both mechanisms were used to generate stable QPs in a parametrically forced partial differential equation that shares some characteristics of the Faraday wave experiment. The first mechanism was found to be robust for single-frequency forcing and two different forcing strengths, for which 12-fold and 14-fold QPs were obtained. The second mechanism, which requires more delicate tuning, was used to select particular angles between wave-vectors in the QP. Nonlinear three-wave interactions between driven and weakly damped modes play a key role in determining which patterns are favored. Rucklidge and Sliber [316] performed quantitative comparisons between the predicted patterns and the solutions of the system partial differential equation based on strong damping of all modes apart from the driven pattern-forming modes. This is in conflict with the requirement for weak damping if three-wave coupling is to influence pattern selection effectively. Rucklidge and Sliber [316] distinguished two different Faraday experiments that three-wave interactions can be used to stabilize QPs, and examples of 12 -, 14 -, and 20-fold approximate QPs were presented. The first is an experiment with a low-viscosity deep layer of fluid with single-frequency excitation [204,205,283]. The second is with a high-viscosity shallow layer of fluid and forcing with two commensurate temporal frequencies [301].

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

A nonlinear analytical approach dealing with the wave pattern selection for parametric surface waves, not restricted to fluids of low viscosity was presented in Ref. [318]. A standing wave amplitude equation was derived from the Navier–Stokes for viscous fluids. The associated Lyapunov function was calculated for different regular patterns to determine the selected pattern near threshold as a function of a damping parameter $Γ=2νk02/ω0$, where $ω0=Ω/2$. For $Γ≈1$, it was shown that a single wave (or stripe) pattern is selected. For $Γ≪1$, patterns of square symmetry in the capillary regime, a sequence of sixfold (hexagonal), eightfold patterns in the mixed gravity–capillary regime, and stripe patterns in the gravity dominated regime were depicted. Figure 31 shows the symmetry of the preferred patterns predicted by Chen and Viñals [318] in the parameter space defined by the viscosity of 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 (as reported in Ref. [319]). It is seen that stripe patterns are preferred at high viscosity, whereas at low viscosity, hexagons (at low frequency) and squares (at high frequency) were observed. The shallowness of the fluid layer was accounted for the observation of a hexagonal pattern for viscosity of $1 cm2/s$ and low frequency, and not observing a quasi-periodic pattern for viscosity $0.04 cm2/s$ and excitation frequency of 27 Hz. The experiments of Binks and van de Water [204] detected this latter region in a deep fluid layer.

Fig. 31
Fig. 31
Close modal

The parametric excitation of two mutually orthogonal pairs of the quasi-monochromatic (modulated) capillary waves on the surface of a liquid layer was studied by Reutov [320]. The numerical solution predicted the formation of the modulation lattice with tetragonal cells with disorder. The disorder was manifested as local variations of the period of the vertical and horizontal lines of the lattice. The weakly nonlinear modal interaction for a finite depth of fluid subjected to a vertical oscillation was analytically developed from the full Navier–Stokes equations by Skeldon and Guidoboni [321]. The coefficients of the amplitude equations and the consequences for stability of different spatially periodic patterns in the infinite depth case were calculated. Although symmetry arguments provided a qualitative explanation for the selection of some of these patterns, quantitative analysis was found to be hindered by mathematical difficulties inherent in a time-dependent, free-boundary Navier–Stokes problem. Skeldon and Porter [322] reconsidered weakly nonlinear behavior and compared the scaling results derived from symmetry arguments in the low viscosity limit with the computed coefficients of appropriate amplitude equations using both the full Navier–Stokes equations and a reduced set of partial differential equations due to Zhang and Vináls [35,207]. An optimal viscosity range was found for locating superlattice patterns experimentally.

The phase relaxation of ideally ordered patterns of Faraday waves was examined by Kityk et al. [323]. A combined frequency–amplitude modulation of the excitation signal a periodic expansion was used and dilatation of a square wave pattern was generated. It was shown that the measured relaxation time allows a precise evaluation of the phase diffusion constant. Later, the spatiotemporal behavior of Faraday surface waves was studied by Kityk et al. [324]. The bifurcation of a doubly hexagonal superstructure out of a simple hexagonal pattern was presented. Some attempts were made to predict the surface elevation profile of Faraday surface waves [325]. Local surface deformation measurements using a focused laser beam were presented by Westra et al. [206] who obtained quantitative values of the temporal phases. Kityk et al. [326] presented the results of experimental measurements of the complete spatiotemporal Fourier spectrum of Faraday waves generated at the interface of two immiscible liquids of different density. The container consists of an aluminum ring of diameter of 18 cm with its bottom and top made of two parallel glass plates with a gap between them of 10 mm. The gap was filled by two immiscible liquids: silicone oil of viscosity of 20 m · Pa · s, density of 949 kg/m3 and an aqueous solution (sugar and nickel sulfate) of viscosity of 7.2 m · Pa · s and density of 1346 kg/m3. The liquid–liquid interfacial tension was measured to be 35 ± 2 dyn/cm yielding a capillary length of 3 mm. Kityk et al. [326] obtained the bifurcation scenario from the flat surface to the patterned state for each complex spatial and temporal Fourier component separately. For example, Fig. 32 shows the dependence of the spatial amplitude on the normalized acceleration $(ɛ=(a0-ac)/ac$, where $ac$ is the critical acceleration above which the interfacial surface in unstable) for modes 10 and 11. The amplitudes were measured at subharmonic, harmonic, and superharmonic frequencies. The figure shows the regions of pure square (see Fig. 33) and hexagonal (see Fig. 34) patterns and a narrow region of mixed patterns. It was shown that the energy is transferred from lower to higher harmonics and the nonlinear coupling generated static components in the temporal Fourier spectrum leading to a contribution of a nonoscillating permanent sinusoidal deformed surface state. A comparison of hexagonal and rectangular patterns reveals that spatial resonance can give rise to a spectrum that violates the temporal resonance conditions given by the weakly nonlinear theory.

Fig. 32
Fig. 32
Close modal
Fig. 33
Fig. 33
Close modal
Fig. 34
Fig. 34
Close modal

### Two-Frequency Parametric Excitation.

It is believed that Edwards and Fauve [289,301,327] were the first to study the two-frequency driven Faraday instability. The general form of two-frequency parametric excitation is
$Z··(t)=a[cos(χ)cos(mωt)+sin(χ)cos(nωt+φ)]$
(32a)

where the phase angle $0deg≤χ≤90deg$ represents the relative mixing between the two modes and the angle $φ$ describes their phase difference, $0<φ<2 πm/n$.

The importance of the correlation length, which determines the influence of the shape of the container on the wave pattern was discussed by Edwards and Fauve [301]. The square pattern consisting of two perpendicular standing waves was found to occur at high 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 of diameter of 12 cm and depth of 0.29 cm. They adopted the technique of Benjamin and Scott [328] using a precisely machined corner of a container filled to the prim. In this case, the fluid free surface becomes pinned at discontinuity of the slope of sidewall of the cylindrical container and the VOF is adjusted so that the surface is flat everywhere and thus has no meniscus. Thus, the brim-full state provides a homogeneous Dirichlet condition on the surface height in which the no-slip condition for viscous fluid states at a solid boundary will have zero velocity relative to the boundary. Several other containers including square, hexagonal, and octagonal were used to verify that patterns do not depend on the container geometry. The excitation was generated as a combination of two sinusoidal components of frequencies $Ω1=4ω$ and $Ω1=5ω$, with $ω/2π=14.60 Hz$, such that the general form of the excitation acceleration is
$Z··(t)=a[cos(χ)cos(4ωt)+sin(χ)cos(5ωt+φ)]$
(32b)

where the phase of $4ω$ component is zero by choice of time origin and the phase angle $φ$ is associated with $5ω$ component.

Under two-frequency excitation given by Eq. (32b) with frequency components $4ω$ and $5ω$ for $ω/2π=14.6 Hz$, $φ=75deg$, and $χ=45deg$, the free surface was found to take hexagons pattern as shown in Fig. 35(a). Hexagons-to-lines transition was found to co-exist over a finite range of bifurcation parameter $μ≡(a-ac)/ac$, where $ac$ is the threshold excitation acceleration amplitude above which the free surface is unstable. Figure 35(b) shows a 12-fold QP observed under excitation frequency $ω/2π=28.0 Hz$, $φ=68.4deg$, and $χ=72deg$.

Fig. 35
Fig. 35
Close modal

The way in which these different patterns become stable or unstable as the parameters are varied was analyzed in some detail and experimentally verified by Binks and van de Water [204]. An example of a quasi-crystalline pattern was presented by Gollub and Langer [329] and is shown in Fig. 36. Binks and van de Water [204] conducted an experimental investigation on a circular container of 44 cm diameter and fluid depth of 2 cm. They used fluid of kinematic viscosity of $ν≈0.03397 cm2/s$, density $ρ=0.8924 gm/cm3$, and surface tension $γ=1.83 J/m2=1830 dyn/cm$. Under parametric excitation, different fluid surface patterns were observed depending on the parametric excitation frequency. Figure 37 shows samples of fluid surface wave patterns in a cylindrical container of 44 cm diameter and fluid depth of 2 cm under different values of excitation frequency where fourfold square $(n=2)$, sixfold hexagonal $(n=3)$, eightfold quasi-periodic $(n=4)$, and tenfold quasi-periodic $(n=5)$ emerge close to the onset. It was found that the n = 2, 3, and 4 patterns reveal a clear long-range orientation order, with minor defects manifested by a slow bend in the case of the square pattern, and the appearance of triangularlike structures in the case of $(n=3)$. These apparent triangles are the result of a $π$-phase defect, as the alteration of hexagons and triangles can be reversed by shifting the phase at which the image is taken by $π$. For the eightfold quasi-periodic pattern, there appears to be a point defect in the top central portion of the image. The ordering of the tenfold pattern is only strong in the central region, although some tenfold orientation order can be observed throughout the image.

Fig. 36
Fig. 36
Close modal
Fig. 37
Fig. 37
Close modal

Zhang and Viňals [35,207] developed a weakly nonlinear analysis for the dynamics of small amplitude surface waves on a semi-infinite weakly inviscid fluid layer. Kudrolli et al. [237] experimentally observed that two-frequency parametrically excited waves produce an intriguing “superlattice” wave pattern near a codimension-two bifurcation point where both superharmonic and harmonic waves occurred simultaneously, but with different spatial wave numbers. The superlattice pattern is synchronous with forcing, spatially periodic on a large hexagonal lattice, and exhibits small-scale triangular structure. Silber and Proctor [330] showed that similar patterns can exist and may be stable if the nonlinear coefficients of the bifurcation problem satisfy certain inequalities. Silber et al. [214] used the spatial and temporal symmetries to indicate that weakly damped harmonic waves may be critical to understanding the stabilization of such pattern in the Faraday system.

Two-frequency-forced liquid layer systems were also studied experimentally [215,237,298,299]. These studies were conducted both in the near vicinity and far from the phase angle corresponding to the codimension-two point. They revealed a number of qualitatively superlattice type states. Silber and Skeldon [213] studied the two-frequency Faraday system in the vicinity of the codimension-two point. They considered forcing ratios $m/n$ to be either odd/even or even/odd parities, where interactions between harmonic and subharmonic waves may occur. Arbell and Fineberg [331] presented an experimental investigation of superlattice patterns generated on the surface of a fluid under parametric excitation with two commensurate frequencies. Four qualitatively different types of superlattice patterns were generated via a number of different three-wave resonant interactions. They occur either as symmetry-breaking bifurcations of hexagonal patterns composed of a single unstable mode or via nonlinear interactions between the two primary unstable modes generated by the two forcing frequencies. Besson et al. [332] showed that a transition between two patterns with different linearly unstable wavelengths can be obtained in various fluid regimes by changing the relative amplitudes of a two-frequency parametric excitation function. This transition occurs through a bicritical point, where both modes are simultaneously neutrally stable.

At low excitation frequencies, a hexagonal pattern consisting of three standing waves spaced at 120 deg was reported in Refs. [319,333,334]. The preferred surface pattern near the onset of instability is the one that minimizes a certain functional of the wave amplitudes. Regular patterns can be formed that are not spatially periodic but do have rotational symmetry, i.e., quasi-crystalline patterns [283,301]. Transitions to spatially and temporally disordered states occur were found to occur when the wave amplitudes are increasing. Some of these states, such as the hexagonal lattice, were found to disappear, while the striped phase breaks down in regions where the stripes are most strongly curved. On the other hand, if the fluid is not too viscous, so that the correlation length of the pattern is relatively long, then the symmetry imposed by the boundaries can be recovered by averaging over a large number of individually fluctuating patterns as reported by Gluckman et al. [335]. A case of strongly turbulent capillary waves was studied experimentally by Wright et al. [336]. The dynamics of a fluid surface filled with high-amplitude ripples were also studied using a diffusing light photography, which resolves the height at all locations instantaneously. When nonlinearities are strong enough to generate a cascade from long wavelength to shorter wavelength, the resulting turbulent state contains large coherent spatial structures.

Faraday instability of viscous fluids driven by an excitation of two frequencies exhibited nonlinear effects, which give rise to a hexagonal pattern, as well as an unusual “QP” with 12-fold orientational order. The development of complex states of fluid motion was illustrated experimentally on film flows by Gollub [337]. In addition, surface waves and thermal convection were considered. In one-dimensional pattern, cellular patterns bifurcate to states of spatiotemporal chaos. In 2D pattern, even ordered patterns can be surprisingly intricate when quasi-periodic patterns are included. The existence and stability of a 2D pattern with subcritical symmetrical tenfold patterns in a dissipative system described by partial differential equations were discussed by Frisch and Sonnino [338]. This state was numerically shown to be stable even when the dynamics are not derived from a free energy functional. In addition, nonsymmetric, rhomboidal patterns were also seen to be stable for some parameter values. Later, both the rhomboidal patterns and resonant QPs resulting from the above interactions were observed experimentally by Arbell and Fineberg [298]. An example rhombic lattice is shown in Fig. 38(a). The critical Fourier modes associated with the 21.8 deg superlattice pattern are indicated 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].

Fig. 38
Fig. 38
Close modal
Porter and Silber [218] considered the case of two-mode rhomboids (referred to as resonant triads) and examined the dynamical consequences of their weakly broken symmetries. The applied acceleration was composed of two commensurate frequencies $mω$ and $nω$ ($m$ and $n$ are coprime integers):
$Z··(t)=|am|cos(mωt+φm)+|an|cos(nωt+φn)$
(32c)
where an appropriate selection of $|am|$ and $|an|$ ensures that two modes (one driven primarily by the $am$ and the other by $an$) onset simultaneously. The lowest order nonlinear interaction is the one known 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-wave interaction, occurring near the bicritical point, in which two wave-vectors from one critical circle, $k1$ and $k2$, generate a third wave-vector $k3=k1+k2$ on the second critical circle as shown in Fig. 39
Fig. 39

Schematic diagrams of resonant triads at the critical point. The wave-vectors satisfy $k3 = k1+k2$: (a) $k3 = |k3|< k1 = |k1| = |k2|$ and (b) $k1 < k3 < 2k1$ [218].

Fig. 39

Schematic diagrams of resonant triads at the critical point. The wave-vectors satisfy $k3 = k1+k2$: (a) $k3 = |k3|< k1 = |k1| = |k2|$ and (b) $k1 < k3 < 2k1$ [218].

Close modal
. Patterns composed of these three modes form the two-mode rhomboids. Kumar et al. [341] studied the selection of rhombic patterns close to a bicritical point at the onset of primary surface instability in viscous fluids under two-frequency vertical vibration. Rhombic patterns were found to appear to be natural at the 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 to QPs, where the appearance of small divisors causes the standard theoretical method to fail. The symmetry-based approach developed by Tse et al. [343] was used by Rucklidge et al. [344] to analyze three observed spatial period-multiplying transitions from an initial hexagonal pattern. Three patterns are shown in Fig. 40, in which patterns (a) and (b) were both obtained using Dow–Corning silicone oil with viscosity of 47 cSt $(1cSt= mm2/s)$ and layer depth of 0.35 cm, while pattern (c) was observed for a 23 cSt oil layer of depth of 0.155 cm. All three patterns were obtained with excitation 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 and 60 Hz. Typically, the secondary bifurcations were found to occur at forcing amplitudes between 10% and 50% larger than the critical acceleration for the primary hexagonal state.

Fig. 40
Fig. 40
Close modal

Interacting surface waves, parametrically excited by two commensurate frequencies, were studied by Epstein and Fineberg [345–347] who showed that a variety of interesting superlattice type states are generated via a number of different three-wave resonant interactions. These states occur either as symmetry-breaking bifurcations of hexagonal patterns composed of a single unstable mode or via nonlinear interactions between the two different unstable modes generated by the two forcing frequencies. It was shown that this state results from the competition between two distinct nonlinear superlattice states, each with different characteristic temporal and spatial symmetries. It was demonstrated that the spatiotemporal disorder can be controlled. It was controlled to either its neighboring nonlinear states by the application of a small-amplitude excitation at a third frequency, where the spatial symmetry of the selected pattern is determined by the temporal symmetry of the third frequency. Furthermore, it was shown that all superlattice states generated by quadratic nonlinearities are grid states. Grid states are superlattices in which two correlated sets of critical wave-vectors are spanned by a sublattice whose basis states are linearly stable modes. The spatial resonances inherent in these states were found to increase their stability. The spatial and temporal structure of nonlinear states formed by parametrically excited waves on a fluid surface in a highly dissipative regime was studied by Epstein and Fineberg [348]. Short-time dynamics reveal that three-wave interactions between different spatial modes are only observed when the modes' peak values occur simultaneously. The temporal structure of each mode was described by the Hill's equation.

### Three- and Multifrequency Parametric Excitation.

The addition of a third frequency component was introduced by Müller [296] for the purpose of breaking the spatial phase symmetry in the subharmonic regime and thus controlled the transition between triangles and hexagons. This motivated Arbell and Fineberg [331] to add a third frequency in an attempt to stabilize asymmetric QPs by modifying the parametric excitation to the following form
$Z··(t)=A[a1cos(p1ω0t)+a2cos(p2ω0t+φ1)+a3cos(p3ω0t+φ2)]$
(33a)
where $A$ is the total excitation acceleration amplitude and the normalized amplitude ratios by $a1:a2:a3$ are adjusted such that $a1+a2+a3=1$. $p1:p2:p3$ are the three-frequency ratios such that $p1 and $φ1$ and $φ2$ are the phase differences with respect to the $p1$ components. By using 2:3:4 driving frequencies, Arbell and Fineberg [331] observed perfect eightfold QP and Fig. 41
Fig. 41

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

Fig. 41

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

Close modal
shows images and their spatial power spectra at two temporal phases. This state was found to be subharmonic in time and can be observed in the region where tenfold QPs and eightfold distorted QPs were observed.

Some differences were reported to exist between the two-frequency and three-frequency phase diagrams. For example, the region of stable hexagons becomes much smaller with the addition of the third frequency term. For two-frequency excitation, there is a large region of hexagons, but for three-frequency excitation, hexagons almost disappear. Another difference is that the 12-fold QP becomes the primary instability when the amplitude of the first component exceeds a certain value and most of the region corresponding to hexagons for two-frequency excitation converts to 12-fold QPs with the addition of the finite third component. Furthermore, the transition between ordered patterns and a disordered state was found to occur at lower driving amplitudes.

Nonlinear three-wave resonant interactions were recognized to play a key role in pattern selection in Faraday wave experiments and other situations where complex patterns are found [301]. For example, some studies [213,214,217–219,236,238] used symmetry considerations to understand pattern selection in Faraday wave experiments with two-frequency forcing and three-wave resonant interactions in the context of weakly broken Hamiltonian structure. This approach was capable to explain several of the experimentally observed superlattice patterns and suggested ways of designing multifrequency forcing functions that could be used to control which patterns would emerge [218,236,238]. The approach was used to determine which additional frequencies to add to the excitation function in order to make observed patterns more robust [331,347,349]. The pattern selection process under multifrequency parametric excitation was considered by Rucklidge and Silber [316] who were able to predict which patterns would be found for different parameter values. They were able to predict the amplitudes and range of stability of the patterns.

Ding and Umbanhowar [349] conducted an experimental investigation to examine the changes in pattern selection which occur with the addition of a third driving frequency at twice the difference frequency of two different even modes dominant frequency ratios 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 of radius of 7.0 cm with a polyvinyl chloride sidewall, a 0.8-cm-thick glass bottom, and a plexiglass top covered with a thin, white plastic sheet, which serves as a light diffuser. Under excitation frequency of 80 Hz, the wave number k of the pattern just above onset is 11.8 cm−1. The corresponding dissipation length, defined as $ld=2νk/f$ is approximately 0.15 cm, where $ν$ is the kinematic viscosity and f is the wave oscillation frequency. Since $kld≈117$ and $kh≈7.8$, the experiments are considered in the weakly dissipative and deep fluid layer limits as indicated by Cerda and Tirapegui [81]. They adopted a three-frequency excitation acceleration of the form
$Z··(t)=amcos(mωt)+ancos(nωt+φn)+apcos(pωt+φp)$
(33b)

In order to demonstrate the differences between surface patterns for two-frequency excitation (i.e., with $ap=0$) near the bicritical point and those for three-frequency excitation, Ding and Umbanhowar [349] generated the phase plots for the two cases. For example, Figs. 42(a) and 42(b) show the phase plots for two-frequency excitations selected such that $m:n=4:5$ or $6:7$. For the case $m:n=4:5$, a region in parameter space well above onset where 12-fold QPs were observed. Figure 42(a) reveals that for hexagons and 12-fold QPs, the $4ω$-excitation component is dominant and the surface waves oscillate at $2ω$. It is also seen that for squares and square two-mode superlattice patterns (2MS) the $5ω$-excitation frequency component is dominant. Dashed lines indicate the region where the higher acceleration resolution measurements were performed. With $p1:p2=6:7$ superlattice patterns were observed as shown in Fig. 42(b). For hexagons and superlattice patterns, the $6ω$-excitation frequency component is dominant. By adding a small third-frequency component with correct phase to the excitation acceleration 33(b), both QPs and superlattice patterns appeared near onset. The stability of these patterns was found to be strongly influenced by the third frequency component $p3$. Figures 43(a) and 43(b) show the phase diagrams for two-frequency excitation $m:n=4:5$ and three-frequency excitation $m:n:p=$$4:5:2$, respectively. For Fig. 43(b), the phase angle $φp=2=32deg$ and $ap=2=0.8g=0.59a2c$, where $a2c=1.36g$ is the critical acceleration for single frequency excitation at $2ω$. The main differences between two- and three-frequency excitations are seen in the fact that the region of stable hexagons with the addition of the third frequency component was significantly diminished. The 12-fold QP becomes the primary instability for $a5≥5g$ and most of the region corresponding to hexagons for two-frequency excitation is converted to the 12-fold QPs with the addition of the third frequency component. Another difference is observed in the transition between ordered patterns and disordered state at lower excitation amplitudes.

Fig. 42
Fig. 42
Close modal
Fig. 43
Fig. 43
Close modal

The evolution of patterns in large aspect ratio driven capillary wave experiments were described by Milner [350] who derived the amplitude equations including nonlinear damping terms. Above the excitation threshold amplitude, a reduction of the number of marginal modes yielded a simple form of the amplitude equations, which have a Lyapunov functional. This functional determines the wave number and symmetry (square) of the most stable uniform state. Frisch and Sonnino [338] studied the existence and stability of a 2D pattern with a tenfold orientational order in a dissipative system described by partial differential equations. The pattern was found to appear in a subcritical way and result from a superposition of two linearly unstable patterns with different wave numbers. Pattern selection phenomena in parametrically 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 two line patterns intersecting at an arbitrary angle were obtained in the gravity–capillary waves of arbitrary depth. Classification, stability analysis and bifurcation study of the fixed points of the dynamical equations with linear damping were performed in the cases of three symmetric linear modes and general modes. It was shown that 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]. The first is when two waves of the shorter wavelength interact with one wave of the longer, while the second type is when two waves of the longer wavelength interact with one wave of the shorter. The two types of three-wave interactions were found to provide an explanation of some of the Faraday wave phenomena in the experiments of Huepe et al. [295].

The relationship between the linear surface wave instabilities of a shallow viscous fluid layer and the shape of the periodic, parametric-forcing function that excites them was studied by Huepe et al. [295]. Huepe et al. [295] adopted a set of parametric excitation functions parametrized by the power $n$
$f(ωt)=N[2.5cos(ωt)+3ncos(3ωt)-5ncos(5ωt)]$
(33c)
where $ω$ is the fundamental frequency of oscillation and $N$ is a normalization constant which is defined such that $max(|f(ωt)|)=1$. Note that the acceleration amplitude, $a$ when multiplied by $f(ωt)$ the result is the actual parametric excitation acceleration. Figure 44,
Fig. 44

Time records of parametric excitation $f(ωt)$ and their corresponding stability boundaries on acceleration amplitude $a$ versus wave number k for (a) $n = -2$, (b) $n = -0.3$, (c) $n = 0.5$, and (d) $n = 1$. The plots are for $ω = 20π rad/s$, h = 0.3 cm, excitation acceleration amplitude $a$ is units of g, fluid density $ρ = 0.96 g/cm3$, kinematic viscosity $ν = 46 cm2/s$, and surface tension $γ = 20 dyn/cm$. The resonance tongues labeled by H and SH refer to regions with harmonic or subharmonic instabilities, respectively. Their envelope shown by dashed lines change with n [295].

Fig. 44

Time records of parametric excitation $f(ωt)$ and their corresponding stability boundaries on acceleration amplitude $a$ versus wave number k for (a) $n = -2$, (b) $n = -0.3$, (c) $n = 0.5$, and (d) $n = 1$. The plots are for $ω = 20π rad/s$, h = 0.3 cm, excitation acceleration amplitude $a$ is units of g, fluid density $ρ = 0.96 g/cm3$, kinematic viscosity $ν = 46 cm2/s$, and surface tension $γ = 20 dyn/cm$. The resonance tongues labeled by H and SH refer to regions with harmonic or subharmonic instabilities, respectively. Their envelope shown by dashed lines change with n [295].

Close modal
shows a set of time history records of $f(ωt)$ for four different values of $n$ = −2, −0.3, 0.5, and 1. The corresponding neutral stability boundaries present the usual resonance tongue structure. The harmonic and subharmonic tongues indicate regions where surface waves become unstable, oscillating with a main frequency component that is an integral multiple $(ω,2ω,3ω,...)$ or an odd half-multiple $(ω/2,3ω/2,$$5ω/2,...)$ of the fundamental excitation frequency, respectively. The tongues at higher modes ($k$ is the wave number) correspond to instabilities with shorter surface wavelengths and higher oscillation frequencies. As $n$ is increased, the excitation function changes from a simple rounded triangular shape with only two extrema per cycle to shapes with richer structure. It is seen that the envelope defined by the tongue minima, indicated by a dashed line changes from a simple convex function with a single minimum to a set of convex segments, each with its own minimum. Huepe et al. [295] indicated that the changes in the critical instabilities shown in Fig. 44 cannot be explained by a simple switch to a different dominant forcing frequency in $f(ωt)$. Indeed, as n is increased to one the lowest unstable region becomes the second harmonic tongue (with main frequency component equal to $2ω$), which does not correspond to the fundamental harmonic or subharmonic responses (with equal or half the frequency, respectively) to any of the three frequency components of $f(ωt)$: $ω,3ω,and 5ω$.

### Numerical Simulation of Faraday Waves.

Numerical simulations of faraday wave patterns were considered for two- and 3D flow in rectangular and circular containers of large aspect ratios. Numerical algorithms, such as trapezoidal, second-order Adams–Bashforth, marker-and-cell, and finite-difference projection methods, were adopted in the literature. For example, Zhang and Viñals [354] presented a numerical simulation of parametrically driven surface waves in fluids of low viscosity based on the linear damping quasi-potential equations, which remain rotationally invariant. The equations governing fluid motion in the bulk and the appropriate boundary conditions at the free surface are approximated by a nonlocal set of equations involving surface coordinates alone. The numerical analysis was performed for large aspect ratio. They used the trapezoidal scheme for linear terms and a second-order Adams–Bashforth scheme (multiple step method) for nonlinear terms. The numerical simulations predicted stable patterns of square symmetry above onset in the capillary dominated regime. A sequence of patterns of lower symmetry were predicted in the vicinity of surface tension parameter, $γk03/(ρω02)=1/3$, where $k0$ is a typical wave number, and $ω0$ is the fundamental natural frequency of the free surface, as the damping parameter is increased.

The dynamics of 2D standing periodic waves at the interface between two inviscid fluids with different densities, subject to monochromatic oscillations normal to the unperturbed interface, was numerically simulated under normal- and low-gravity conditions by Wright et al. [21,355]. The numerical simulation was based on boundary-integral method that is applicable when the density of one fluid is negligible compared to that of the other, and a vortex-sheet method that is applicable to the more general case of arbitrary densities. Viscous dissipation was simulated by means of a phenomenological damping coefficient added to the Bernoulli equation or to the evolution equation for the strength of the vortex sheet. A comparative study revealed that the boundary-integral method is generally more accurate for simulating the motion over an extended period of time, but the vortex-sheet formulation is significantly more efficient. Nonlinear effects for noninfinitesimal amplitudes were manifested at the extremes of the interfacial oscillation; growth of harmonic waves with wave numbers in the unstable regimes of the Mathieu stability diagram; and formation of complex interfacial structures including paired traveling waves.

Standing surface waves on a viscous fluid driven parametrically by a vertical harmonic oscillation were examined by Murakami and Chikano [356] based on direct numerical simulations of the 2D Navier–Stokes equation, together with appropriate boundary conditions. The condition for the onset of the waves in the experiments by Lioubashevski et al. [357] was reproduced using numerical simulation. The form of the surface elevation was analyzed and the dependence of the saturated amplitude on the forcing strength exhibited normal bifurcation. Murakami and Chikano [356] discussed the velocity fields of the 2D standing waves developed near the instability onset. Nonlinear wave dynamics in parametrically driven surface waves were studied in numerical simulations of the 2D Navier–Stokes equation by Chen [358]. Modulating behavior of primary wave modes in a particular parameter range and in time scales much longer than the underlining wave periods was reported. Valha et al. [359] performed a numerical study to examine the behavior of a gas liquid interface in a vertical cylindrical vessel subjected to a sinusoidal vertical motion. The computational method was based on the simplified marker-and-cell method and includes a continuum surface model for the incorporation of surface tension. The numerical results indicated that the surface tension has very little effect on the period and amplitude of oscillations of the interfacial waves. The stability of the interfacial waves was found to depend on the initial pressure pulse disturbance, and exponential growth of the interfacial wave was predicted in some cases. The results were found in good agreement with available experimental and analytical solutions.

A numerical analysis of 2D Faraday waves was presented by Ubal et al. [360] based on direct numerical simulation of Navier–Stokes and continuity equations with appropriate boundary conditions. Stability maps on the excitation amplitude-wave number plane for viscous liquid layers with equilibrium depths between $5×10-5 meter$ and $10-5 meter$ were presented. The results were compared with those obtained by Benjamin and Ursell [10] for an inviscid fluid, and by Kumar and Tuckerman [262] for a viscous fluid. The results confirmed the previous findings obtained by linear stability analysis. In particular, stronger excitation forces are needed to produce unstable waves as the thickness of the film is reduced. The lower boundary of the unstable regions in the stability charts appeared to move toward higher wave number values.

The Faraday waves were examined numerically using some simplifications and reductions. For example, a set of simplified equations referred to as the coupled amplitude streaming flow equations were derived for the case of weakly nonlinear small viscosity flow by Martín et al. [258] and Higuera et al. [135]. Two- and 3D Faraday waves were studied with periodic boundary conditions by O'Connor [361]. The full Navier–Stokes equations were solved including the complex dynamics of the free-surface waves to gain a better understanding of the interplay between the viscous boundary layers, the nonlinear streaming flow, and the bulk flow. The formation of a square wave pattern was predicted and found in qualitative agreement with experiments of Kudrolli and Gollub [319]. Nonlinear effects of standing waves in fixed and vertically excited tanks were numerically investigated by Frandsen and Borthwick [362]. Numerical solutions of the governing nonlinear potential flow equations were obtained using a finite-difference time-stepping scheme on adaptively mapped grids. A horizontal linear mapping was applied, so that the resulting computational domain is rectangular, and consists of unit square cells. The small-amplitude free-surface predictions in the fixed and vertically excited tanks were in agreement with second-order small perturbation theory. For steep initial amplitudes, the predictions were found to differ considerably from the small perturbation theory solution, demonstrating the importance of nonlinear effects.

Périnet et al. [363] performed numerical simulations of Faraday waves in three-dimensions for two incompressible and immiscible viscous fluids. The Navier–Stokes equations were solved using a finite-difference projection method coupled with a front-tracking method for the interface between the two fluids. The critical accelerations and wave numbers, as well as the temporal behavior at onset were compared with the results of the linear Floquet analysis by Kumar and Tuckerman [262]. The finite-amplitude results were found in agreement the experimental results of Kityk et al. [326]. The detailed spatiotemporal spectrum of both square and hexagonal patterns was reproduced for excitation acceleration of $30.0 m/s2$, which is the acceleration used in the experimental investigation of Kityk et al. [326,364] for square patterns. Figures 45(a) and 45(b) represent examples of the patterns obtained at saturation and are taken from the same simulation at the two different instants. The symmetries characterizing the squares (reflections and π/2 rotation invariance) are clear, showing a first qualitative agreement with Kityk et al. [326] where both structures were observed. The pattern oscillates subharmonically, at 2 T, where T is the forcing period. Figure 45(a) is taken when the interface attains its maximum height, while Fig. 45(b) is taken at a time 0.24 × 2 T later. At this later time, the dominance of a higher wave number was observed.

Fig. 45
Fig. 45
Close modal

A direct numerical simulation of Faraday waves was carried out by Périnet et al. [365] in a minimal hexagonal domain. They observed alternation of patterns referred to as quasi-hexagons and beaded stripes. Starting from zero velocity and an initial random perturbation of the flat interface, their numerical simulations produced a hexagonal pattern which oscillates subharmonically with the same spatiotemporal spectrum as reported by Kityk et al. [326]. It was reported that hexagons are transient and difficult to be stabilized experimentally and are competing with squares and disordered states as indicated by Wagner et al. [366] and Kityk et al. [326]. In their numerical simulations and after about ten subharmonic periods, Périnet et al. [365] predicted a drastic departure from hexagonal symmetry. A fully nonlinear numerical simulation of 2D Faraday waves between two incompressible and immiscible fluids was performed by Takagi and Matsumoto [367] who adopted the phase-field method developed originally by Jacqmin [368]. In the nonlinear regime, qualitative comparison was made with an earlier vortex-sheet simulation of two-dimensional Faraday waves by Wright et al. [21].

The majority of Faraday wave studies have been carried out on traditional viscous and inviscid fluids under deterministic excitation. Nontraditional fluids, such as magnetic fluids, ferrofluids, and liquid crystals subjected to nontraditional sources of parametric excitation, including random excitation, have received the attention of few studies as will be demonstrated in Secs. 10 and 11.

## Faraday Waves of Other Media

Nontraditional sources of parametric excitation include electrostatic forces and convective temperature gradient. The parametric response of the interface between two dielectric liquids under an alternating electrostatic force was studied in Refs. [369–373]. Their studies showed the stability of the interface required the applied voltage to be high enough to suppress surface tension effects and lower than a certain analytically determined critical value. For voltages greater than this critical value, Reynolds [369] indicated that the interface is unstable. The stability of the equilibrium was found to depend not only on the mean temperature gradient, as in Rayleigh's problem [374], but also on the amplitude and frequency modulation. The free-surface oscillations of a magnetic liquid were studied in Refs. [375–378]. The parametric excitation of weakly nonlinear surface waves in a magnetic fluid subject to a periodically oscillating magnetic field was examined by Pursi and Malik [379]. Transverse homoclinic orbits leading to a chaotic transition were obtained together with necessary conditions for the existence of chaos. Reimann et al. [380] conducted a Faraday experiment with magnetic fluid in a direct current magnetic field driven externally by accelerative modulation. A pattern of standing twin peaks was observed whose origin was believed in the simultaneous excitation of two different wave numbers in the nonmonotonic regime of the dispersion relation.

Gershuni and Zhukhovitskii [381] discovered a parametric resonance called convective instability in a fluid body subject to a periodically varying temperature gradient. Later, Gershuni and Zhukhovitskii [382] found that the modulation of the vertical temperature gradient had the same influence as the modulation of the angular velocity of rotation of the fluid as a rigid body. The stability of the equilibrium was found to depend not only on the mean temperature gradient, as in Rayleigh's problem [374], but also on the amplitude and frequency modulation.

Dissipative patterns caused by spin-wave instabilities in insulating ferromagnetic films driven by out-of-plane parallel pumping were studied by Elmer [383]. It was found that the only stable patterns are squares, hexagons, and quasi-periodic patterns based on three standing waves. Quantitative results strongly support the suggestion that these patterns should be experimentally observable by means of Faraday rotation. At the surface of a magnetic fluid, parametric waves can be excited by an alternating magnetic field, parallel to the surface. With this anisotropic system, Bacri et al. [384] observed a pattern transition from parallel rolls, perpendicular to the magnetic field, to a rectangular array of cross rolls. The free-surface waves of a magnetic fluid subjected to a normal magnetic field were examined experimentally by Browaeys et al. [385]. The waves were generated by a small modulation at frequency of the vertical field. It was concluded that a linear and inviscid analysis is sufficient to fit well the experimental data, except in the vicinity of the critical field where a surface instability occurs. A linear stability analysis of the free surface of a horizontally unbounded ferrofluid layer of arbitrary depth subjected to vertical vibrations and a horizontal magnetic field was presented by Mekhonoshin and Lange [386]. A nonmonotonic dependence of the stability threshold on the magnetic field was found at high frequencies of the vibrations. It was found that the magnetic field can be used to select the first unstable pattern of Faraday waves. In particular, a rhombic pattern as a superposition of two different oblique rolls can occur.

Another type of fluid medium known as ferrofluid has been the subject of some studies. A ferrofluid is a liquid which becomes strongly magnetized in the presence of a magnetic field. Ferrofluids are made of nanoscale ferromagnetic particles suspended in a carrier fluid (usually an organic solvent or water). Each tiny particle is thoroughly coated with a surfactant to inhibit clumping. A stability theory for the onset of parametrically driven surface waves of a ferrofluid was developed by Müller [273] who considered the effects of viscous dissipation and finite depth effects. It was shown that a careful choice of the filling level permits the normal and anomalous dispersion branches to be measured. Furthermore, it was demonstrated that the parametric driving mechanism may lead to a delay of the Rosensweig instability. When a paramagnetic fluid is subjected to a strong vertical magnetic field, the surface forms a regular pattern of peaks and valleys. This effect is known as the normal-field instability or Rosensweig instability. A bicritical situation can be achieved when Rosensweig and Faraday waves interact.

The singular surface electromagnetic waves are guided by a plane boundary of a chiroplasma half-space if chirality and plasma parameters are properly matched. They form a complete set of surface polaritons jointly with the Rayleigh surface waves and generalized surface waves [387,388] in the Voigt geometry. Fisanov and Marakasov [389] indicated that a parallel-plate waveguide supports under certain conditions singular propagating modes whose characteristics drastically depend on the type of the boundary conditions.

The nonlinear stability of magnetized standing waves on the plane interface separating two immiscible inviscid magnetic fluids in a cylindrical container in the presence of both a uniform magnetic field and a periodic acceleration normal to the free surface was studied by Elhefnawy [390]. It was found that the modulation of the amplitudes and phases of the two-mode resonant waves are governed by a system of nonlinear first-order differential equations which in turn utilized to determine the steady-state solutions and consequently investigating their stability. It was concluded that the frequency-response curves exhibit the transcritical and Hopf bifurcations. The nonlinear evolution of the three-dimensional instability of standing surface waves along the interface of a weakly viscous, incompressible magnetic liquid within a rectangular basin was studied by Sirwah [391–393]. A combination of the Rosensweig instability with Faraday instability was developed, where the system is assumed to be stressed by a normal alternating magnetic field together with an external vertical oscillating force. The nonlinear equations of the complex amplitudes corresponding to the ideal fluid case were modified by adding the linear damping. It was shown that the liquid viscosity rather than the magnetic field affects the qualitative behavior of the wave motion and the system response alternates between the regular periodic and chaotic behavior depending on the specific values of some parameters.

Liquid crystals, such as smectic and nematic, are characterized by properties between those of conventional liquid and those of solid crystal. For instance, a liquid crystal may flow like a liquid, but its molecules may be oriented in a crystal-like way. One of the phases of liquid crystals is known as the thermotropic liquid crystals, which occur in a certain temperature range. Many thermotropic liquid crystals exhibit a variety of phases as temperature is changed. For instance, a particular type of liquid crystal molecule may exhibit various smectic and nematic phases as temperature is increased. The linear stability of Faraday waves on the surface of a smectic-A liquid crystal and polymer gel–vapor systems of finite-thicknesses was studied by Ovando-Vazquez et al. [394]. For the case of highly viscoelastic gel media, it was found that there are co-existing surface modes of harmonic and subharmonic types that correspond to peaks in the plot of the critical acceleration as a function of wave frequency. Larger frequencies were found to lead to subsequent peaks of co-existing subharmonic waves. A quantitative theoretical linear analysis of the dependence of the forcing acceleration and wave number on the strength of the external excitation frequency for the Faraday instability in smectic-A (Sm-A) liquid crystals was presented by Hernández-Contreras [395]. A layer of Sm-A under a constant horizontal magnetic field applied in the direction of the wave-vector that orients the molecules in a stack of lamellar layers perpendicular to the air–liquid interface was considered. Hernández-Contreras determined the dispersion relation and instability boundary in terms of acceleration-wave number of the surface waves in finite-thickness layers. Hernández-Contreras [395] found that Faraday waves develop in thin (0.1 mm and 0.05 m) smectic-A liquid crystal layers at low frequencies of external driving acceleration and in the long wavelength limit. For Sm-A under an external magnetic field, it was found that there are alternating subharmonic–harmonic branches with almost the same vanishing critical excitation acceleration as occurs in ideal inviscid fluids. An increase of the modulating frequency was found to make the subharmonic waves to preempt the harmonic ones with a lower threshold nonzero acceleration.

Time reversal of the excitation parameters has some consequences on the stability of a periodically driven dynamic system. In particular, the dissipative pattern-forming system of electrohydrodynamic convection of nematics driven by time-periodic external electric fields was studied by Heuer et al. [396]. The dynamic model describing the linear stability at the beginning was represented by a set of two differential equations for the amplitudes of the dynamic variables. It was found that under time reversal of any periodic excitation function, the dynamic equations reproduce the same stability thresholds, pattern types, and critical pattern wavelengths. Nematic electrohydrodynamic convection under excitation with superimposed harmonic wave was studied by Pietschmann et al. [397]. A time reversal of the excitation was found to have no effect on the threshold voltages and pattern wavelengths obtained in linear stability analysis. This symmetry with respect to time reversal of the excitation was found to break down close to the transition from the conduction regime to the dielectric regime. Later, Pietschmann et al. [398] demonstrated that the threshold parameters for the stability of the ground state are insensitive to a time inversion of the excitation function. The Faraday system was found to share this property with standard electroconvection in nematic liquid crystals. In general, time inversion of the excitation was found to affect the asymptotic stability of a parametrically excited system, even when it is described by linear ordinary differential equations. The pattern selection of the Faraday waves above threshold, on the other hand, discriminates between time-mirrored excitation functions.

The parametric instability in nematic liquid crystal layers was studied by Hernández-Contreras [399] using linear stability theory. The critical acceleration and wave number of the unstable stationary waves were found discontinuous at the nematic-isotropic transition temperature and conform to similar sharp changes experienced by the viscosity and surface tension as a function of temperature. Due to Marangoni flow, the curve of the critical acceleration as a function of excitation frequency was found to exhibit a minimum. If the Marangoni flow is neglected and the dynamical viscosity is increased, a monotonously increasing dependence of the acceleration in terms of oscillation frequency was observed. A bicritical instability was found to occur for a layer thickness of a few millimeters. A well-defined subharmonic wave was achieved when the thickness of the layer was further increased.

## Random Excitation of Faraday Waves

Different sources of parametric excitations are encountered in different applications of liquid sloshing dynamics. For example, Bechhoefer and Johnson [400] considered a fluid container driven parametrically by a triangle waveform. Parametrically excited surface waves excited by a repeating sequence of N delta-function impulses, were studied by Catllá et al. [401]. With impulsive forcing, the linear stability analysis may lead to an implicit equation for neutral stability boundaries. The familiar situation of alternating subharmonic and harmonic resonance tongues was found to emerge only if an asymmetry is introduced in the spacing between the impulses. By varying the spacing between the up and down impulses making up the $2π$-periodic forcing function for N = 2 impulses per period, it was found that the magnitude of the 1:2 spatiotemporal resonance effect depends dramatically on the corresponding asymmetry parameter, which measures deviation from equal spacing of the impulses. This resonance was found to occur for impulsive forcing even when harmonic resonance tongues are absent from the neutral stability curves. An experimental study of the parametric resonance and finite-amplitude parametric oscillations arising in a liquid-filled U-tube subject to alternating vertical excitation was presented by Briskman et al. [402]. Two forms of oscillations in the liquid together with their corresponding ranges of unstable equilibrium with respect to small random perturbations were reported. The studies of random parametric sloshing have not yet reached the maturity stage as its counterpart of the deterministic studies. However, these studies have been carried out for the cases of external random parametric excitation in regular gravitational field and due to g-jitter field. The next two subsections will address both cases.

### Random Excitation Under Regular Field.

The stochastic stability of a liquid surface under random parametric excitation can be studied in terms of one of the stochastic modes of convergence. 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 differential equation of the sloshing mode mn, i.e.,
$Amn"+2ζmnAmn'+[1+ξ"(τ)]Amn=0$
(34)
where $Amn$ is a dimensionless free liquid surface amplitude of mode mn, a prime denotes differentiation with respect to the nondimensional time parameter $τ=ωmnt$, $ωmn$ is the natural frequency of the sloshing mode mn, $ζmn$ is the corresponding damping ratio, and $ξ"(τ)$ is a dimensionless vertical wide band random acceleration of spectral density $2D$. The mean square stability condition of the response of Eq. (34) was determined in Refs. [404–408], which is given by the inequality
$D/2ζmn<1$
(35a)
On the other hand, the sample stability condition is
$D/2ζmn<2$
(35b)

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

As the excitation level decreases, the response changes to predominantly harmonic. The wide-band excitation covers the first 15 symmetric modes and two excitation levels were applied. The higher level exhibited subharmonic response. An abrupt transition between harmonic and subharmonic responses was observed when the excitation acceleration level was reduced. An important feature of the results showed that low-level harmonic response to random Gaussian excitation was nearly Gaussian. However, when large amplitude subharmonic response was excited, the probability distribution changed abruptly into a double-exponential distribution. Dalzell [409] conducted a least-square fitting algorithm to develop a phenomenological probability distribution of the liquid free surface elevation. The following double exponential distribution was proposed:
$P(X)=Exp{-Exp[-(π6X+γ)]}$
(36)

where $γ$ is the Euler constant = 0.577215665, $X=(η-η¯)/σ∧η$, $η¯$ is the mean value of the fluid elevation, and $σ∧η$ is the RMS of the fluid elevation.

The nonlinear motion of the free liquid surface under random parametric excitation involves the estimation of stochastic stability and response statistics of the free surface [407,409,411–413]. The free liquid surface height of a sloshing mode $mn$ in a cylindrical container was found to be governed by the nonlinear differential equation
$Amn"+2ζmnAmn'+[1+ξ"(τ)]Amn(1-K1Amn-K2Amn2)+K3Amn'2+K4AmnAmn"+K5AmnAmn'2+K6Amn2Amn"=0$
(37)

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

1. (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 damping force prevents any motion of the liquid. The excitation level for the first antisymmetric and axisymmetric modes are given by the following ranges, respectively,
$0
(38a)
$0
(38b)
2. (b)
On–off intermittent motion of the free liquid surface. This intermittent motion takes place over the following ranges of excitation level for the two modes
$1.55
(39a)
$4.98
(39b)
A corresponding regime, known as undeveloped sloshing was predicted by Ibrahim and Soundararajan [412]. This regime was characterized differently by very small motion of the liquid free surface with an excitation level
$2.0
(40)
The spatiotemporal intermittency of liquid free surface under parametric excitation was later studied by Bosch and van de Water [285] and Bosch et al. [414].
3. (c)
Partially developed random sloshing characterized by undeveloped sloshing where significant liquid-free surface motion occurs for a certain time-period and then ceases for another period. At higher excitation levels, the time-period of liquid motion exceeds the period of zero motion. The excitation levels of this regime for the two modes are, respectively
$1.82
(41a)
$6.8
(41b)

Heinrich [415] and Ibrahim and Heinrich [410] observed the development of circular motion of a central spike for the case of first symmetric mode excitation. Occasionally, the spike is displaced from the center of the tank and precesses in such a way that preserves the azimuth symmetry in a time-average sense. This motion was first reported by Gollub and Meyer [265] who studied the amplitude dependence on the precession frequency under harmonic parametric excitation.

4. (d)

Fully developed sloshing characterized by continuous random liquid motion for all exci