# Solitary-like Wave Dynamics in Thin Liquid Films over a Vibrated Inclined Plane
## 2 Background and Experimental Methods  
When a single-layer liquid film flows down an inclined plane with a no-slip boundary, the resulting Nusselt flat film flow profile assumes a parabolic longitudinal velocity shape, having the largest velocity at the free surface [13; 14; 24]. In this flow regime, a long-wavelength surface instability develops when the average flow rate exceeds a certain critical value [15]. When the disturbances are excited naturally, in general four regimes of different wave behaviour can be observed in the downstream regions of the inclined plane [13]. The first regime is observed in a section of the plane that is adjacent to the inlet of the liquid, where small disturbances caused by the inlet structure are amplified while moving downstream and forming predominantly monochromatic waves. The second regime is observed in the following downstream region, where the monochromatic waves grow in amplitude and then develop higher-order frequency harmonics due to nonlinear effects. Then, as a result of complex nonlinear interactions, two-dimensional solitary-like waves are formed, and then they propagate further downstream exhibiting unique properties that, in part, coincide with those of other known solitary waves but, in general, are unique [24]. Finally, three-dimensional waves start to form due to transverse variations [13; 14]. It is noteworthy that not all aforementioned regimes can necessarily be observed in practice [13]. Yet, it is well-known that when the initial natural disturbance at the inlet is nearly monochromatic, the waves emerging in the region located immediately after the inlet can first inherit the frequency of the disturbance and then evolve into a solitary-like wave far downstream [13; 14; 24]. However, when either the thickness of the liquid film or the fluid flow is periodically modulated at the inlet, solitary-like surface waves develop almost immediately after leaving the inlet area [14; 24], which indicates that the nonlinear evolution of the flow over an inclined plane is dominated by solitary-like waves independently of whether their formation was deliberately forced or resulted naturally. Figure 1 shows a sketch of the setup that enables observing the formation of both forced and natural (unforced) solitary-like surface waves. The setup is assembled on a vibrating table that is driven by a shaker (35 W, 20-80 Hz response, Dayton Audio, OH, USA). The electric signal applied to the shaker is produced using a digital signal generator that has the frequency tuning accuracy of \(\pm 1\,\upmu\)Hz. The vibration amplitude is measured by an analog accelerometer (ADXL326, Analog Devices, OH, USA) that has the accuracy of \(\pm 0.08\) g. The inclined plane, where the waves are observed, is a 5-cm-wide and 50-cm-long rigid aluminium plate. The surface of the plate was chemically treated to improve the formation of the liquid film. In general, the inclination angle \(\theta\) must be relatively small to suppress the three-dimensional instabilities of the rolling waves that are known to develop at relatively large flow rates [54]. Subsequently, for the fluid parameters used in this work, we choose \(\theta=3^{o}\) and experimentally verify that in the absence of vibration the observed two-dimensional rolling waves remain stable during the entire observation period. The inclined plate was mounted on top of wider open channel used to recycle the  
liquid by redirecting it to the main reservoir. A low-vibration DC voltage pump driven via a customised electronic circuit was used to supply water from the reservoir to the inlet. The electronic modulation of the pump flow rate enabled controlling the thickness of the liquid film and the creation of solitary-like waves. Based on the pump flow rate, the width of the channel and the average speed of the flow close to the inlet, the estimated thickness \(h\) of the liquid layer varies in the range \(0.5<h<1\,\mathrm{mm}\).