# Solitary-like Wave Dynamics in Thin Liquid Films over a Vibrated Inclined Plane
## 5 Conclusions  
In conclusion, our experiments with a sub-millimetre-thick water layer on a slightly inclined vertically vibrated plate demonstrate that low-frequency vibration in the range between 30 and 50 Hz suppresses the development of long rolling surface waves propagating downstream. These surface waves appear as the result of the long-scale gravitational instability of the base flow in the absence of vibration [15, 16] and may also be excited by mechanically perturbing the flow at the inlet. A relatively small thickness of the water layer (under 1 mm) is required to suppress the three-dimensional instability of the rolling waves that is known to develop at large flow rates. Experimental findings are verified using a boundary-layer hydrodynamic model [52, 53] obtained from the Navier-Stokes equation by assuming a self-similar parabolic longitudinal flow velocity. Linear stability and nonlinear dynamics of the surface waves obtained with the model qualitatively confirm the main experimental findings. Without vibration, the Fourier content of surface waves is represented by a broad band of unstable wavevectors \(k<k_{c}\), where \(k_{c}\) is a critical cut-off wavevector of the gravitational instability (see Equation (3)). As the instability unfolds, the wavelength of the dominant wave quickly increases until it develops into a solitary-like wave [24]. For fluids with a relatively small viscosity, such as water, the characteristic time required for solitary rolling waves to develop on a \(3^{\circ}\) incline is in the order of several seconds. In the nonlinear regime, the Fourier content of the surface waves is dominated by solitary-like waves characterised by a small wavevector with a background of smaller amplitude shorter waves, which is shown in Figures 4a and 8a. We observe that the properties of the surface waves change dramatically when the layer is vibrated. Thus, a relatively weak vibration (the vibration amplitude \(a<g\)) leads to the onset of the secondary Faraday instability in the form of short waves with a wavelength  
Figure 8: (**a**) Dispersion map obtained from the solution of Equation (1) in the absence of vibration for a 0.6 mm water film on a \(\theta=3^{\circ}\) incline. (**b**) Dispersion map of the solution of Equation (1) when the inclined plane vibrated at \(f=48\) Hz with the amplitude \(a=0.8\). The scaling for the vertical axis is in arbitrary logarithmic units. of \(\lambda\approx 5\ldots 10\,\)mm when vibrated at \(f=48\,\)Hz. In agreement with the earlier theoretical studies [42; 43; 44], the temporal frequency of the Faraday waves is shifted away from the harmonic (48 Hz) and subharmonic (24 Hz) response that is typical of Faraday instability in horizontal liquid layers. In fact, the inclination angle of the plate acts as a wave filter, splitting a standing Faraday wave into two plane waves: one propagating upstream and one propagating downstream. Similarly to the Doppler effect, the wave that propagates downstream increases its speed and, therefore, increases its temporal frequency, while the wave that propagates upstream decreases its speed and frequency. For water layers vibrated at 48 Hz, we observe the following shifts in frequency away from the subharmonic response: from 24 Hz to approximately 40 Hz for the downstream wave and from 24 Hz to approximately 8 Hz for the upstream wave. In the nonlinear regime, the interaction between shorter Faraday waves and longer gravitational waves leads to the broadening of their respective bands in the f-k dispersion map. Most importantly, we find that the average and peak amplitudes of the long-scale gravitational waves are significantly reduced when vibration is applied. This result is rather intriguing since the total influx of energy is larger in the vibrated system when both gravity and vibration together drive the flow, unlike in the non-vibrated case, where the only source of energy is due to gravity.