5.1. Discretized frequency bands

In order to detail the nature of the frequency bands in a discretized system excited by two frequencies, consider figure 5, where the frequency dependence of the thresholds is given for several values of ${\it\chi}$ when $l/m=3/4$ , as an example. Here, the single-frequency systems are depicted for ${\it\chi}=0^{\circ }$ in figure 5(a) and ${\it\chi}=90^{\circ }$ in figure 5( f), which are identical upon proper scaling of the basic frequency ${\it\omega}$ . A sequence of dips in the threshold in each panel is present, corresponding to different cell modes. The cusps in the threshold therefore represent codimension-two points, where two different cell modes are simultaneously unstable.

Figure 5.

Stability of the $l/m=3/4$ system of figure 1 with stress-free cylindrical ( $R=2.55~\text{cm}$ ) discretization at various angles ${\it\chi}$ : (a) 0°; (b) 20°; (c) 30°; (d) 40°; (e) 50°; ( f) 90°. The filled circles track the threshold for fixed $f=2.4~\text{Hz}$ , and correspond to the marked thresholds in figure 6(a).

To investigate the effect of two-frequency excitation, we focus on the frequency range of 2.3–2.8 Hz, where the $(0,1)$ and $(3,1)$ modes are excited in the ${\it\chi}=0^{\circ }$ system, and the $(2,2)$ , $(0,2)$ and $(3,2)$ modes are excited in the ${\it\chi}=90^{\circ }$ system (see the insets of panels (a) and (f) in figure 5). Increase in ${\it\chi}$ from 20° to 50°, as shown in panels (b)–(e), depicts a dynamic process by which the ${\it\chi}=0^{\circ }$ frequency bands (solid line) give way to the ${\it\chi}=90^{\circ }$ bands (dashed line). At ${\it\chi}=20^{\circ }$ in figure 5(b), the same modes are excited as in the inset of panel (a), then frequency bands at which $(3,2)$ and $(1,3)$ modes due to the $4{\it\omega}$ excitation appear at ${\it\chi}=30^{\circ }$ in (c), followed by the addition of $(2,2)$ and $(0,2)$ modes at ${\it\chi}=40^{\circ }$ in (d). For ${\it\chi}=50^{\circ }$ in figure 5(e), only a small $(3,1)$ mode band due to the $3{\it\omega}$ excitation remains, and further increase results in a system where only modes excited primarily by the $4{\it\omega}$ component remain.

5.2. Discretized ${\it\chi}$ bands

The influence of a second frequency component on the primary $l{\it\omega}$ component threshold is studied by fixing the basic frequency ${\it\omega}$ and varying the mixing angle  ${\it\chi}$ . Typical results are presented in figure 6(a) for both $l/m=3/4$ and $3/2$ excitation with $f=2.4~\text{Hz}$ . It is seen that the threshold for each case consists of two ${\it\chi}$ bands that excite different modes, joined by a codimension-two point at ${\it\chi}={\it\chi}_{c}$ . For values of ${\it\chi}<{\it\chi}_{c}$ , the $(0,1)$ mode is primarily excited by the $3{\it\omega}$ component for both excitations of figure 6(a). Above ${\it\chi}_{c}$ , the $(2,2)$ and $(1,1)$ modes are primarily excited by the $4{\it\omega}$ and $2{\it\omega}$ , respectively. Neither of the curves in figure 6(a) shows significant dependence upon the phase angle ${\it\phi}$ .

Figure 6.

Fixed basic frequency (  $f=2.4~\text{Hz}$ ) stability for the cylindrical system of figure 5: (a) overall threshold amplitudes $A$ with $l/m=3/2$ and $3/4$ , ${\it\phi}=0^{\circ }$ ; and (b) primary $l{\it\omega}$ component amplitudes with various $l/m$ ( ${\it\phi}=0^{\circ }$ except for $l/m=1/2$ ) for the system of figure 1 with cylindrical ( $R=2.55~\text{cm}$ ) stress-free discretization. The filled circles in (a) correspond to the thresholds as marked in figure 5(a–f).

Figure 7.

Primary $l{\it\omega}$ component thresholds for various frequency ratios $l/m$ at fixed ${\it\chi}=20^{\circ }$ and fixed primary-component frequencies of (a) 7.2 Hz and (b) 7.65 Hz. Only the $l/m=1/2$ thresholds depend upon ${\it\phi}$ , which vary monotonically between the labelled ${\it\phi}=0$ and 180° points.

As was first discussed for a laterally infinite system in figure 2, the behaviour of the overall amplitude $A$ at criticality does not reveal the effect of a second frequency on the primary-component threshold. To isolate this behaviour we replot the amplitudes $A\text{cos}{\it\chi}$ for the $(0,1)$ mode ${\it\chi}$ bands of figure 6(a) in figure 6(b). Here, the primary-component threshold increases with ${\it\chi}$ for $3/4$ excitation and decreases for $3/2$ excitation. Similar adjustment is seen for $1/3$ and $3/1$ excitation of the $(0,1)$ mode. In each of these excitations, addition of a higher/lower-frequency component increases/decreases the $l{\it\omega}$ component threshold. An exception to this classification is the $1/2$ excitation, which can either increase or decrease the $l{\it\omega}$ threshold, and shows dependence upon the phase lag  ${\it\phi}$ .

To further differentiate between the addition of a higher ( $l<m$ ) or lower ( $l>m$ ) frequency, we present the $l{\it\omega}$ component threshold at which the $(0,1)$ mode is excited for several frequency ratios at fixed ${\it\chi}=20^{\circ }$ in figure 7. Many frequency ratios $l/m$ have been included here only to clarify trends in the threshold adjustment, and will not be considered elsewhere in the study. The frequencies are $l{\it\omega}/2{\rm\pi}=7.2~\text{Hz}$ in figure 7(a) and 7.5 Hz in figure 7(b), which correspond to conditions above (tuned) and below (detuned) the $(0,1)$ mode natural frequency. It is clearly seen that the sign of the adjustment to the $l{\it\omega}$ component threshold depends upon whether $l/m$ is greater than or less than 1, and also on whether the mode is tuned or detuned. This behaviour occurs because addition of a second frequency produces a slight shift in the natural frequency of a mode, to lower frequencies when $l<m$ and to higher frequencies when $l>m$ . The adjustment to the threshold amplitude at the new natural frequency is very slight. While this adjustment becomes more pronounced as the mode tuning or detuning is increased, it remains quite small relative to the single-frequency threshold, viz. figures 6(b) and 7(a,b). A key two-frequency experimental result that will be presented is that the nonlinear response of the system is significantly altered even when the linear threshold is not. Next we discuss a case where both the amplitude and phase of the second component have a significant effect on the primary-component threshold.

5.3. The case of $l/m=1/2$ excitation showing strong ${\it\phi}$ dependence

Strong dependence of the threshold upon the phase lag ${\it\phi}$ is seen for $1/2$ excitation of harmonic modes. That is, ${\it\omega}$ and $2{\it\omega}$ components are used to simultaneously excite the same mode harmonically and subharmonically, respectively. In figure 8(a,b) we present the overall and component threshold amplitudes for $1/2$ excitation for three values of ${\it\phi}$ . The basic frequency has been set to $f=3.6~\text{Hz}$ , such that the $(0,1)$ mode is harmonically excited by the ${\it\omega}$ component, and subharmonically excited by the $2{\it\omega}$ component. As a result, a codimension-two point between two different cell modes does not appear in this system. With ${\it\phi}=0^{\circ }$ , it can be seen that the $(0,1)$ mode threshold diverges for small ${\it\chi}$ , and there exists a ${\it\chi}$ band at significantly higher amplitudes where the system instead chooses a $(3,1)$ mode. The other sets of thresholds presented for ${\it\phi}=90$ and 135° do not exhibit this divergence in the threshold, although both still depend noticeably on  ${\it\phi}$ .

Figure 8.

Fixed-frequency (  $f=3.6~\text{Hz}$ ) stability of the cylindrical system for $l/m=1/2$ excitation: (a) overall threshold amplitudes $A$ with ${\it\phi}=0$ , 90 and 135°; and (b) corresponding $l{\it\omega}$ component thresholds. The $(0,1)$ mode is excited for each $l/m$ over the entire range of ${\it\chi}$ except for ${\it\chi}=[4,7^{\circ }]$ for ${\it\phi}=0^{\circ }$ where the $(3,1)$ mode is excited. The disparity in the axis scales of (b) emphasizes the significance of the small- ${\it\chi}$ behaviour.

The explanation for this divergence in the ${\it\phi}=0$ threshold in figure 6(b) is aided by figure 9(a–d), where the stability diagrams are presented for values of ${\it\chi}=0$ , 4, 8 and 12°. It is evident that addition of the $2{\it\omega}$ component produces a profound effect on the second, third and fourth tongues of figure 9(a). Circles are used to denote the wavenumber ( $150.2~\text{m}^{-1}$ ) and thresholds at which the $(0,1)$ mode is excited harmonically. Then, in figure 9(b), with ${\it\chi}=4^{\circ }$ , the harmonic tongue is damped considerably as slight amounts of the $2{\it\omega}$ component are added, accounting for the sharp jump observed in figure 6(b). The sharp decrease then occurs due to the creation of ‘islands’ of instability as seen for ${\it\chi}=8$ and 12° in figure 9(c,d). The damping of the second tongue and subsequent appearance of islands does not occur for the cases of ${\it\phi}=90$ or 135°. Islands can also be observed in figure 1 and in the stability diagrams of Besson et al. (Reference Besson, Edwards and Tuckerman1996). In each of these cases, the islands form by ‘pinching’ of the new tongues created by the $m{\it\omega}$ component, as seen in figure 1(c,d). However, the $1/2$ case presented here is different because the $2{\it\omega}$ component does not generate new tongues. Instead, the original single-frequency tongues of figure 9(a) are directly modified by the $2{\it\omega}$ component, which leads to island formation.

Figure 9.

Stability of the horizontally infinite system with $f=3.6~\text{Hz}$ , $l/m=1/2$ , ${\it\phi}$ of 0° and varying ${\it\chi}$ : (a) 0°; (b) 4°; (c) 8°; (d) 12°.

The effect of ${\it\phi}$ in this example is significant and well within the level of detection of our experiment. We now set out to provide experimental evidence of this effect in addition to other aspects of the two-frequency theory, in a manner similar to that conducted by Batson et al. (Reference Batson, Zoueshtiagh and Narayanan2013) for the single-frequency theory. Specifically, experimental data sets will be presented and compared to the predictions of figures 5, 6 and 8. We will also present bifurcation data in which we have measured the nonlinear response of the system. To these ends, we re-introduce the experimental system that was developed in addition to the novel methods used for characterization of the vibrational input signal.

6.1. Electromechanical shaker

Aside from the fluid choices and cell design, the critical aspect of the experimental design was control of the parametric excitation produced by the electromechanical shaker. The set-up is the same as that used by Batson et al. (Reference Batson, Zoueshtiagh and Narayanan2013), and its mechanical details are provided by Yoshikawa et al. (Reference Yoshikawa, Zoueshtiagh, Caps, Kurowski and Petitjeans2010). In summary, the set-up, pictured in figure 10, is designed to drive the test platform with sinusoidal speeds of up to $500~\text{cm}~\text{s}^{-1}$ and accelerations of $3500~\text{cm}~\text{s}^{-2}$ at frequencies between 0.1 and 10 Hz. The counterweight platform moves with phase opposite to the test platform in order to reduce transmission of vibrations to the set-up structure. The vibrational function $G(t)$ is directly programmed with the MotionPerfect 2 software interface, and the desired parameters $A$ , ${\it\chi}$ , $l/m$ and ${\it\phi}$ are independently chosen.

Figure 11.

Sample images during one wave period of a $f=7.65~\text{Hz}$ subharmonic excitation of the $(0,1)$ mode. A grid is placed behind the cell to improve the wave contrast. All wave amplitudes reported in the work are taken to be half the length of the white arrows, which are used to denote the vertical distance from the central $(0,1)$ deflection to the sidewall interfacial contact line. The parametric signal is measured and analysed by recording a fixed point on the front of the cell, such as the small black marker pointed to in panel (e).

In the single-frequency experiments of Batson et al. (Reference Batson, Zoueshtiagh and Narayanan2013), we acknowledged that, while the measured frequency control was excellent (to within 0.1 % of the set point), the vibrational amplitude supplied to the experiment was normally less than the programmed input. Thus, to precisely determine the parametric input for each experiment, we directly measured the vibrational cell motion with high-speed imaging. While the amplitude of a single-frequency signal can be calculated simply by measuring the trough-to-crest displacement, Fourier analysis is necessary for a multiple-frequency signal. We will now detail these means by which we obtain the cell motion signal and precisely determine the parametric input. The excitation of a $(0,1)$ mode with a single frequency is used as an example, but the methods apply equally to two-frequency excitation. What we will show is that, in all cases, the parametric input to the system is determined to within 2 % of its actual value, which is then used to generate the model predictions to which the experiments are compared.

6.2. Wave deflection time series

Time series of both the cell motion and the interface deflection were obtained by recording the cell at 2000 frames per second (f.p.s.) using a stationary Photron SA3 camera. A level was used on both the camera and the experimental cell to ensure both that the cell motion was vertical and that the camera was directed normal to this motion. Sample images of single-frequency excitation of the $(0,1)_{sh}$ mode are presented in figure 11. A composite image of both the central wave deflection and the cell motion time series is presented in figure 12. Here, both the wave deflection and the cell motion time series are obtained by side-by-side stacking of a vertical slice of a single pixel width from each recorded image. To illustrate this, observe the correspondence between the white arrows in figure 11 and those in figure 12, and it becomes evident that the crests and troughs of figure 12 depict the time evolution of the central wave deflection of the $(0,1)$ mode. This time series is constructed from 1024 camera images, and, considering the frame rate and the excitation frequency of 7.65 Hz, the wavy lines of figure 12 correctly exhibit nearly four oscillations. In the results section, these series are transformed from the stationary laboratory frame to the vertically oscillating frame. Notably, this transformation renders the wavy lines from the back grid straight and more intuitively represents the wave amplitude in time (see figures 17, 20, 23 and 26).

Figure 12.

(a) Time series data for the excitation depicted in figure 11. All images of time series data are created by horizontally stacking a single pixel-wide slice of data from images recorded at 2000 f.p.s. The numbered white arrows of this image directly correspond to data marked by the white arrows in the images of figure 11. The bottom half of this image is time series data obtained from the vertical motion of the black marker on the far right of the images of figure 11 for the duration of the excitation. (b) The harmonic content of the parametric input to each experiment is calculated via fast Fourier transform of this signal.

Quantitatively, we use these data to measure the maximum deflection of a saturated wave, which allows us to characterize bifurcation of the system with respect to the critical parameters $A$ , ${\it\chi}$ and ${\it\phi}$ . Qualitatively, while the time series data generally do not contain information about the spatial structure of the wave, observe that the reflection patterns in both the crests and the troughs of figure 12 are nearly identical. This is remarkable given the complexity of light refraction across the interface, and suggests that the wave is very nearly periodic from crest to crest and from trough to trough. On the other hand, in the results section, we will conclude that two-frequency waves are not periodic in this manner because we observe that the amplitudes and reflection patterns are not identical from crest to crest or from trough to trough.

6.3. Cell motion time series and digitization

The cell motion at the bottom of figure 12 was obtained by recording the motion and stacking the time series of a fixed point on the exterior of the cell. To facilitate quick and automated processing, the fixed point was created by attaching a strip of white paper with a very thin ( ${\leqslant}1~\text{mm}$ ) horizontal line to the cell. This is denoted by a black tick and is pointed out by the arrow in the fifth image of figure 11. The sharp contrast of the image allowed quick and repeatable conversion from greyscale to the depicted binary image. We used ImageJ’s ‘Skeletonize’ command to trim the signal to a single black pixel per time frame, representing the cell position at each step. Finally, the image is processed in MATLAB to obtain a vector of the cell displacement sampled at the frame rate of 2000 f.p.s. We estimate that the error associated with this digitization is no more than a pixel, which equates to less than 2 % for the smallest amplitude measurements (image resolution varied between 200 and $300~\text{pixels}~\text{cm}^{-1}$ ). Otherwise the error associated with the method decreases linearly with the signal amplitude and is less than 1 % for the vast majority of the presented data. To calculate harmonic content and the parameters $A$ , ${\it\chi}$ and ${\it\phi}$ , we use MATLAB’s discrete Fourier transform routine for a single period of the cell displacement (for example, 784 frames of a 2.55 Hz excitation). By calculating the parametric input of a single period, we assume that the cell motion is constant for the duration of the experiment. This assumption was verified by analysing the motion at different times during an excitation and observing no change, for example, well before onset and after the saturation of a wave. In the next section we discuss the harmonic characterization for two example signals.

An alternative to consider for this type of problem is to use a MEMS-based accelerometer to measure the instantaneous acceleration felt by the cell, and incorporate this measurement into a feedback loop. Of course, this method also requires its own calibration. Instead, we have incorporated a posteriori analysis to characterize the oscillatory behaviour. Nevertheless, automated codes in both ImageJ and MATLAB place the limiting rate of analysis on the camera data transfer rate, which meant that about two minutes of time was necessary to fully determine the parametric input for a given experiment. Given that the growth and saturation of the instability has a delay time of its own, the cell motion analysis was seamlessly integrated into the experimental procedure and data sets such as the onset behaviour or bifurcation data were collected efficiently and analysed.

6.4. Cell motion Fourier analysis

To detail the method by which the parametric input is characterized, we present the measured pixel displacement of the cell for four periods of a sample $1/2$ excitation ( ${\it\omega}=3.6~\text{Hz}$ ) and its Fourier spectrum in figure 13. The abscissa of the spectrum has been rescaled to present the frequencies as multiples of the basic frequency, ${\it\omega}$ . The peak amplitudes have been converted to real amplitudes, and are calculated as the square root of the product of the Fourier coefficient with its conjugate. Peaks appear precisely at the programmed frequencies of $l{\it\omega}$ and $m{\it\omega}$ , and this is consistent with all reported data. The angle ${\it\chi}$ evaluates to 7.53° and is calculated using the relation ${\it\chi}=\arctan (A_{m}/A_{l})$ , where $A_{l}$ and $A_{m}$ are the component amplitudes. Then the overall amplitude is 5.98 mm via $A=A_{l}/\!\text{cos}{\it\chi}=A_{m}/\!\sin {\it\chi}$ . The relative phase angle ${\it\phi}$ is 74.6° and is calculated using the angles of the complex Fourier coefficients.

Figure 13.

Measured (a) parametric signal and (b) Fourier component amplitudes of a sample $l/m=1/2$ signal. The peak at $l=3$ in (b) indicates the presence of undesired harmonic content in the signal (a).

The final question about the electromechanical shaker output that must be addressed is the significance of undesired signal harmonics. We found that such harmonics consistently arose for low values of $A$ and high values of ${\it\chi}$ . To illustrate an extreme case, we present another signal and its Fourier spectrum in figure 14. Here, the parameters $[A,{\it\chi},{\it\phi}]$ for this $3/4$ signal ( ${\it\omega}=2.33~\text{Hz}$ ) are calculated to be $[1.03~\text{mm},44.0^{\circ },-6.3^{\circ }]$ . The amplitudes of the undesired peaks appearing at frequencies of ${\it\omega}$ , $2{\it\omega}$ , $5{\it\omega}$ and $6{\it\omega}$ are far too small to excite modes of their own. However, in principle, these undesired components interact with the $l{\it\omega}$ and $m{\it\omega}$ components to alter the onset and the nonlinear dynamics of the instability. The only set of experiments we observed that exhibit an influence from an undesired component was the single-frequency harmonic excitation of the $(0,1)$ mode. Here we correlate fluctuations in the saturated wave response to the amplitude of the (undesired) $2{\it\omega}$ component in the input signal. We have already shown in figures 6 and 9 that this interaction can be significant, and this result further motivated us to study the $1/2$ excitation. Otherwise, within the accuracy of our measurement techniques, we do not observe significant effects due to these undesired components.

Figure 14.

Measured (a) parametric signal and (b) Fourier component amplitudes of a sample $l/m=3/4$ signal. The peaks at $l=1$ , 2, 5, 6 and 7 in (b) all indicate undesired harmonic content in the signal (a).

6.5. Measurement of the onset of instability

The experimental onset of instability is difficult to characterize owing to the non-ideality of the sidewall meniscus and the nature of the bifurcation, both of which were discussed in detail by Batson et al. (Reference Batson, Zoueshtiagh and Narayanan2013). To summarize, the contact line behaviour of the immiscible fluids creates a complex three-phase problem with the glass sidewall. When the system is at rest, this behaviour presents itself as the formation of a tiny convex-down meniscus, indicating that silicone oil preferentially wets the sidewall compared to the FC-70. This acts as a source of imperfect bifurcation, as initializing oscillation of the system causes the emission of travelling waves to the interior of the interface when excitation is either above or below the instability threshold. The amplitudes of these waves are normally quite small, and the difference between a Faraday-stable and -unstable system is obvious and can be observed by eye. However, as discussed in Batson et al. (Reference Batson, Zoueshtiagh and Narayanan2013), these waves are amplified when the $(0,1)$ mode is excited harmonically, and can lead to ‘early’ observation of the linear onset. Additionally, the response of the wave amplitude near the threshold is quite small when the excitation frequency is higher than the natural frequency, consistent with supercritical bifurcation.

Therefore, to provide a better representation of the experimental onset, we will present two types of data. The first type corresponds simply to sets of stable and unstable conditions, as determined by eye. The experimental onset that may be compared to the model predictions is then inferred as the interpolation between these conditions. Nearest the onset, it was common for three to five minutes to pass before the instability would appear and grow. Thus a point was labelled as ‘unstable’ if this appearance was observed, and ‘stable’ if only the meniscus waves persisted after a sufficient wait. A significant effort was made to narrowly pinpoint the onset by reducing the gap between stable and unstable points as much as possible. This method improves considerably upon the work of Batson et al. (Reference Batson, Zoueshtiagh and Narayanan2013), where only unstable points were presented.

The second type of data we report is bifurcation of the saturated wave deflection $A_{W}$ for the $(0,1)$ mode with respect to the parametric amplitude $A$ or angle ${\it\chi}$ . These data allow us to demonstrate the subcritical and supercritical nature of the bifurcation. We take $A_{W}$ to be half the maximum vertical difference between the sidewall contact and the crest elevation, as indicated by arrow no. 2 in figures 11 and 12. By comparing this maximum amplitude from crest to crest in a time series, we are also able to draw conclusions about the periodicity of the saturated waves. The presence of the sidewall meniscus causes this method to give a finite, but small, reading for $A_{W}$ when there is no instability.

8.1. The $3/2$ subharmonic excitation of the $(0,1)$ mode

Figures 18–20 present experimental data for $3/2$ parametric excitation. A data set similar to figure 15 is presented in figure 18, where the variable-frequency onset behaviour is probed for an average angle of ${\it\chi}=51.9^{\circ }$ . The dashed lines have been included along with the model prediction for the average (solid line) to reflect the minimum and maximum values of ${\it\chi}$ in this data set. The associated variance in the model prediction is small, and it can be concluded that the agreement with the data appears to be identical to that for the single-frequency data of figure 15. Here the $(2,1)_{sh}$ and $(0,1)_{sh}$ modes are primarily excited by the $3{\it\omega}$ component, in general agreement with the model predictions. However, the model does in fact predict the appearance of a $(1,1)_{sh}$ mode in resonance with the $2{\it\omega}$ component at $f=2.55~\text{Hz}$ , whereas it is the $(0,1)_{sh}$ mode that is first excited. This discrepancy results from larger sidewall dissipation of the $(1,1)$ mode relative to the $(0,1)$ mode. Two multi-mode excitations near the codimension-two points are observed, the first consisting of the $(2,1)$ and the $(0,1)$ mode at 2.33 Hz, and the second consisting of the $(0,1)$ mode and the $(3,1)$ mode at 2.58 Hz.

Figure 19.

Saturated wave response of the central $(0,1)$ mode deflection for $l/m=3/2$ excitation at (a) 2.45 Hz and (b) 2.55 Hz, as marked in figure 18. Parameters for the fit (7.1) are $[{\it\alpha},{\it\beta},A_{c}]=[-1.5\times 10^{-2},4.3\times 10^{-4},1.25]$ in (a) and $[1.3\times 10^{-2},0,1.17]$ in (b). Both fitting parameters $A_{c}$ are labelled on figure 18.

Figure 20.

(a) Time series data for a $(0,1)$ mode excited with $[l/m,f,A,{\it\chi}]=[3/2,2.55~\text{Hz},1.28~\text{mm},50.7^{\circ }]$ and (b) the input parametric signal. Two periods of the $f=2.55~\text{Hz}$ excitation are depicted. These parametric conditions are marked by the star in figure 18.

The data of figure 18(a) exhibit good agreement between the model and the experiment for a fixed value of ${\it\chi}$ . Figure 18(b) describes the threshold behaviour of the system for $3/2$ excitation for all values of ${\it\chi}$ , where the onset of instability is reported for a fixed frequency of 2.4 Hz. Looking first at ${\it\chi}=0^{\circ }$ , corresponding to the single-frequency 7.2 Hz data of figure 15, it is seen that the discrepancy between the model and the data remains constant for the entire range of ${\it\chi}$ for which the $(0,1)$ mode is excited. The discrepancy again results from sidewall dissipation that is unaccounted for, and thus we conclude that the model is equally satisfactory at predicting the two-frequency onset of instability as previously observed for the single-frequency case. In figure 18(b), the $(0,1)$ mode is excited in the experiment until a value of about ${\it\chi}_{c}=70^{\circ }$ , where both the $(0,1)$ and a $(1,1)$ mode are excited. The $(1,1)$ mode is then excited primarily by the $2{\it\omega}$ component for ${\it\chi}>{\it\chi}_{c}$ . These thresholds have been replotted in the inset of figure 18(b) as the individual component amplitudes $A\text{cos}{\it\chi}$ and $A\sin {\it\chi}$ . It is difficult to say whether or not the experimental thresholds show this interaction (which is predicted to be very small). The phase lag ${\it\phi}$ is calculated to have virtually no effect on the thresholds of figure 18(a) or (b), and no experimental evidence was found to claim otherwise.

While the onset is not significantly altered by two-frequency excitation, this is not true of the nonlinear response. Figure 19(a,b) presents the fixed-frequency saturated wave response for ${\it\omega}=2.45$ and 2.55 Hz, $3/2$ excitation, with average ${\it\chi}=52.4$ and 51.9°, respectively. The responses exhibit the same transition from subcritical to supercritical bifurcation as seen in single-frequency excitation. However, as presented, there is a noticeable scatter in the data, contrasting the ordered single-frequency responses of figure 16(a,b). The scatter arises from the measurement, which was made by sampling a single crest of the saturated wave. This can be visualized by a time series of the central $(0,1)$ mode deflection, presented in figure 20 (parametric conditions marked in figure 18). The response is sampled for two periods of an $f=2.55~\text{Hz}$ excitation at an amplitude near the onset such that the response remains regular. Inspection of the crests (labelled 1, 2 and 3) reveals that the wave is not periodic from crest to crest as was observed for a similar single-frequency excitation in figure 17. In contrast, here both the maximum wave amplitudes and reflection patterns vary from crest to crest. Addition of the $2{\it\omega}$ component breaks the subharmonic periodicity of the $(0,1)$ mode with the primary $3{\it\omega}$ component and the resulting wave is instead periodic with respect to the basic frequency, ${\it\omega}$ . We suspect that ${\it\phi}$ contributes to the observed crest-to-crest fluctuations, but this was not studied.

Figure 21.

Images taken from two cell periods ( $2T=4{\rm\pi}/{\it\omega}$ ) of the simultaneous excitation of a $(0,1)$ and a $(1,1)$ mode with $[l/m,f,A,{\it\chi}]=[3/2,2.55~\text{Hz},1.40~\text{mm},48.0^{\circ }]$ . These images of the maximum $(0,1)$ wave deflections are taken at an interval $T/3$ and highlight its subharmonic periodicity with respect to the $3{\it\omega}$ excitation. The central deflections of these images are labelled on the time series data of figure 23. Also see supplementary movie 1.

Figure 22.

Images taken from the same two-mode excitation as figure 21. These images of the maximum $(1,1)$ wave deflections are taken at an interval $T/2$ and highlight its subharmonic periodicity with respect to the $2{\it\omega}$ excitation. The $(1,1)$ component of the wave is parallel to the single-headed arrows drawn in panels (a) and (b). The sidewall deflection time series of figure 23 is built from data marked by the two-headed arrow of panel (a).

8.2. Multi-mode subharmonic $3/2$ excitation

To illustrate a unique aspect of the two-frequency single-mode system, we present images and time series of a two-mode nonlinear response in figures 21–23. Also see supplementary movie 1 available at http://dx.doi.org/10.1017/jfm.2014.724. The images and data have been collected from the saturated response for two periods of a $3/2$ excitation with $f=2.55~\text{Hz}$ and ${\it\chi}=48.0^{\circ }$ , conditions near those of figure 20. This excitation was performed at a higher amplitude such that a $(1,1)$ mode (see figure 4) was excited by the $2{\it\omega}$ component, in addition to the $3{\it\omega}$ excitation of the $(0,1)$ mode.

Figure 23.

Time series data of (a) the central deflection and (c) the sidewall deflection, and (b) cell motion for the multi-mode excitation of figures 21 and 22.

The images of figures 21 and 22 have been selected to highlight the different periodicities of the $(0,1)$ and $(1,1)$ components of this wave. The times during the excitation at which both sets of images are taken are denoted by the numbered slices in the time series data. The images of figure 21, taken at an interval of $T/3$ , depict the maximum deflections of the $(0,1)$ mode component of the wave. The images of figure 22 were taken at an interval of $T/2$ in order to highlight the maximum deflections of the $(1,1)$ mode of the excitation. The arrow in images 7 and 8 is drawn parallel to this component. Time series data for the interface motion along the cell sidewall are also presented (bottom, figure 23) by taking data from the far left of the cell images, viz. the double arrow in image 7. It is evident that an ${\it\omega}$ modulation by the $(1,1)$ mode appears on top of the $3{\it\omega}/2$ response due to the $(0,1)$ mode. In contrast, the ${\it\omega}$ modulation is not noticeable in the time series of the central deflection (top, figure 23), which is suggested by the linear spatial form of the $(1,1)$ mode.

In summary, this excitation produces a wave whose $(0,1)$ and $(1,1)$ components execute three and two oscillations, respectively, over two periods of the basic frequency ${\it\omega}$ . It is similar to excitations used to create superlattice patterns in the high-frequency experiment; however, here only two linearly unstable modes participate instead of two continuous bands of wavenumbers (cf. Kudrolli, Pier & Gollub Reference Kudrolli, Pier and Gollub1998). Multi-mode waves are also possible in the discretized regime when only a single frequency is used, under the condition that the participating modes are close in wavenumber. This constraint is relaxed in two-frequency excitation, where any two modes can be simultaneously excited by correct tuning of the frequency ratio.

8.3. The $3/4$ subharmonic excitation of the $(0,1)$ mode

We have just considered a case where the $(0,1)$ mode is primarily excited by a $3{\it\omega}$ frequency, and is modified by a slower $2{\it\omega}$ frequency. To reveal potential behaviour when the modifying frequency is faster, we present results of the $(0,1)$ mode onset and bifurcation for $l/m=3/4$ with an average ${\it\chi}=23.1^{\circ }$ in figure 24. Little difference is found, as model agreement for the variable frequency and variable ${\it\chi}$ onsets in figure 24(a,b) is quite similar to what was seen for $3/2$ excitation in figure 18(a,b). Again, ${\it\phi}$ was not observed to have any effect on either the calculated curves or the experimental onsets. The $(0,1)$ mode bifurcation in figure 25(a,b) shows the same scatter as was observed for $3/2$ excitation. The crest-to-crest variance of the time series data of figure 26 supports the same explanation. Hence, within the scope of the experimental study, we do not find a difference between the addition of a greater or lesser second frequency.

Figure 24.

Comparison between the experiment and the model for $l/m=3/4$ excitation for (a) average ${\it\chi}=23.1^{\circ }$ and (b) fixed $f=2.45~\text{Hz}$ . The experimental data points are defined as was done for figure 15.

Figure 25.

Saturated wave response of the central $(0,1)$ mode deflection for $l/m=3/4$ excitation at (a) 2.45 Hz and (b) 2.55 Hz, as marked in figure 24. Parameters for the fit (7.1) are $[{\it\alpha},{\it\beta},A_{c}]=[-1.5\times 10^{-2},3.9\times 10^{-4},0.794]$ in (a) and $[1.1\times 10^{-2},0,0.74]$ in (b). Both fitting parameters $A_{c}$ are labelled on figure 24.

Figure 26.

(a) Time series data for a $(0,1)$ mode excited with $[l/m,f,A,{\it\chi}]=[3/4,2.55~\text{Hz},1.10~\text{mm},26.9^{\circ }]$ and (b) the input parametric signal. Two periods of the $f=2.55~\text{Hz}$ excitation are depicted. These parametric conditions are marked by the star in figure 24.

8.4. The $1/2$ harmonic excitation of the $(0,1)$ mode

Finally, we turn to the study of harmonic excitation of the $(0,1)$ mode with $l/m=1/2$ . Here, as discussed in § 5.3, the mode for finite values of ${\it\chi}$ is simultaneously excited by both the ${\it\omega}$ and $2{\it\omega}$ components. The most dramatic aspect of this excitation is that the phase ${\it\phi}$ can significantly alter the $(0,1)$ mode threshold or suppress the mode entirely (see figure 8). Before confirming this prediction, we first review the experimental onset of single-frequency excitation of the $(0,1)_{h}$ mode in figure 27, and its bifurcation in figure 28. It was observed by Batson et al. (Reference Batson, Zoueshtiagh and Narayanan2013) that the instability appeared early for this mode, which was explained as the interaction between the harmonic meniscus waves and the harmonic excitation of the instability. The outcome of this interaction is that the $(0,1)_{h}$ mode is the only mode in the single-frequency experiment whose onset lies below the model prediction. Three data points in figure 28(a) exhibit a finite inhomogeneous deflection before the subcritical jump to Faraday resonance. This erroneous detection of resonance is indicative of the inhomogeneous excitation, which is not observed in any of the $(0,1)_{sh}$ bifurcation data.

Figure 27.

Comparison between the experiment and the model for single-frequency harmonic excitation of the $(0,1)$ mode.

Figure 28.

Saturated wave response of the $(0,1)_{h}$ mode at (a) 3.6 Hz and (b) 3.75 Hz. Parameters for the fit (7.1) are $[{\it\alpha},{\it\beta},A_{c}]=[-0.13,4.15\times 10^{-3},8.7]$ in (a) and [0.17, 0, 9.2] in (b). Both fitting parameters $A_{c}$ are labelled on figure 27.

The fluctuations in the Faraday response of figure 28(a,b) are a separate phenomenon unique to harmonic excitation of the $(0,1)$ mode. Recalling that stray components can enter into the parametric signal produced by the shaker, figure 29 treats the intended single-frequency signal as a $l/m=1/2$ signal, and juxtaposes the resulting values of ${\it\chi}$ against a zoom of the wave response in figure 28(a). Here it appears that fluctuations in the wave response correspond to those in the angle ${\it\chi}$ , most apparent at excitation amplitudes of 9.2 and 10.2 mm. This kind of interaction was not observed in the single-frequency wave response for the subharmonic $(0,1)$ mode (see figure 16), where undesired signal harmonics were also present. These are the only results we report that exhibit a high sensitivity to stray frequencies in the parametric input, and, along with the phase dependence highlighted by the model predictions, motivate us to further study this $1/2$ excitation.

Figure 29.

Juxtaposition of the wave response of figure 27 with the angle ${\it\chi}$ . The parametric signals producing the response data, intended to contain only a single frequency, inevitably contain undesired harmonics, and the values of ${\it\chi}$ have been calculated using the $2{\it\omega}$ content in each signal.

Figure 30(a–c) compares experimental onset data with the predictions that were initially given in figure 8. Up to this point, phase has not been observed to affect the onset behaviour for either $l/m=3/2$ or $3/4$ excitation. The phases for these input signals have been calculated by Fourier analysis and are presented in figure 30(d). Although the phases clearly fluctuate about the set points 0°, 90° and 135°, they show the desired trends when moving from one data set to another, especially at low values of ${\it\chi}$ , where the phase average is 0, 90 and 135°, respectively. As a guide, the model predictions have been calculated using these averages for each figure. The threshold data at ${\it\chi}=0$ for each figure are identical, as expected. Inspecting the first unstable point for a finite ${\it\chi}$ in each figure, the calculated ${\it\phi}$ for these signals increases from 6.6° in figure 30(a) to 82.8° in figure 30(b) and then to 147.4° in figure 30(c). The associated threshold amplitudes decrease, in accordance with the model predictions, where the upper point in figure 30(a) excites the $(3,1)$ mode as predicted. To map more precisely this region where the divergence in the onset occurs, the saturated response is presented in figure 31(a) for a fixed amplitude $A$ for values of ${\it\chi}$ where the gap occurs. The amplitude for these data was exceptional and held between 9.8 and 9.9 mm, and the phases are given in figure 31(b). Both figures clearly show that parametric resonance is restored as the phase of the $2{\it\omega}$ component is increased.

Figure 30.

Comparison between the experiment and the model for $l/m=1/2$ excitation at a fixed frequency of 3.6 Hz for average ${\it\phi}$ of (a) 0°, (b) 90° and (c) 135°. (d) The actual phases for the signals in each data set. The $(0,1)$ mode was excited for all these data, except for the point at ${\it\chi}=4.1^{\circ }$ and $A=20.4~\text{mm}$ in (a), where the $(3,1)$ mode was excited.

Figure 31.

Saturated wave response of the $(0,1)_{h}$ mode for $1/2$ excitation, where measurements were taken for a fixed frequency of 3.6 Hz, amplitude $A$ of 10 mm, for input ${\it\phi}$ of 0, 45 and 90°. The calculated values of ${\it\phi}$ for the signals producing the data in (a) are given in (b).