# Solitary-like Wave Dynamics in Thin Liquid Films over a Vibrated Inclined Plane
## 3 Experimental Results  
Figure 2 shows the representative images of the solitary-like surface waves propagating over a downstream section of the inclined plate, when the vibration is turned off (Figure 2a) and when the plate is vibrated with the frequency of \(48\,\mathrm{Hz}\) and the peak amplitude is \(1\,\mathrm{g}\) (Figure 2b), g being the gravitational acceleration. The frequency of the forcing of the solitary-like waves is \(2\,\mathrm{Hz}\) in both panels of Figure 2. The images were obtained from the selected individual fluorescence frames of the recorded videos of the propagating waves. Without vibration (Figure 2a), we can observe a train of downstream-propagating solitary pulses. A closer inspection also reveals the existence of periodic waves with an amplitude that is much smaller than that of solitary-like waves. When the plate is subjected to vibration (Figure 2b), we continue observing a train of solitary pulses with approximately the same pulse periodicity as in Figure 2a. However, the peak amplitude of the pulses is  
Figure 1: Sketch of the experimental setup used to observe the solitary-like surface waves. For the sake of clarity, only the main constructive features are shown, including the inclined plate, where the waves are observed, the pathway for recycling of the used liquid, and the vibrating table. The dimensions and relative positions of the components in this sketch are not to scale. lower than in the case without vibration. Yet, in agreement with the relevant theory [42; 44] and experiment on the vertical plane [45], we also observe the short wavelength ripples arising due to the onset of the Faraday instability. Using our fluorescence intensity analysis software, we register the profiles of the waves at the points located on the centre line of the inclined plate along the downstream direction, and we plot the so-obtained data as a function of time. The resulting spatiotemporal false-colour maps are plotted in Figure 3 with the observation period of 2 s for the scenario of no vibration (Figure 3a) and with the 48 Hz vibration (Figure 3b). In those figures, we can see the traces of several solitary-like waves that propagate in the downstream direction. The traces are more distinguishable and have higher false-colour amplitudes in the case of no vibration than with the vibration, which confirms our observation of a decrease in the peak amplitude of the solitary pulses due to the vibration in Figure 2. The ripple waves caused by the vibration-induced Faraday instabilities can also be seen in Figure 3b. It is noteworthy that the separation between the traces and the relative position of the traces in the time-downstream coordinate space are very similar with and without the vibration. This indicates that, even though the peak amplitude of the solitary-like waves is affected by the vibration, in general the vibration does not change the shape of the soliton pulses. Then, we apply a two-dimensional Fourier transformation to the spatiotemporal data to obtain the dispersion maps as a function of frequency \(f\) and wavevector \(k\). Since the speed of a wave is given by \(u=\omega/k=2\pi f/k\), using the resulting dispersion maps we can identify the bands of constant \(f/k\) ratio that correspond to waves travelling along the incline plate at a constant speed. However, we apply the standard f-k filtering procedures to remove noise from the dispersion characteristics [55]. Figure 4 shows the dispersion maps for the case of no vibration (Figure 4a) and with the 48 Hz vibration (Figure 4b). While the negative frequency regions of the dispersion maps originate from the mathematical properties of the Fourier transformation, the sign of the wavevector has the physical meaning as it determines the direction of the wave propagation. Figure 2: Instantaneous profiles of the surface waves in a liquid film flowing over the inclined plate. The profiles were obtained from the fluorescence images. The frequency of the flow forcing resulting in the formation of solitary-like waves is 2 Hz.