# Solitary-like Wave Dynamics in Thin Liquid Films over a Vibrated Inclined Plane
## 1 Introduction
Subsequently, high-amplitude vibrations result in an increase in the film thickness and a concomitant increase in the speed of the waves. However, the opposite conclusions were drawn in [40], which is, most likely, a result of the differences in the system (a liquid film under two-phase flow conditions) investigated in that paper. It is also well-known that in horizontal liquid layers a harmonic vibration excites two different types of standing surface waves: harmonic waves that oscillate at the vibration frequency and subharmonic waves that oscillate at half of the vibration frequency [51]. However, the presence of a mean flow across the layer changes the response frequency of the excited waves [42; 43; 44]. Surface waves excited by harmonic vibration in a liquid film flowing over a vertical plane were investigated experimentally in [45] and the results obtained in that work validated the linear theory developed in [42; 43; 44]. Thus, the experimental work [45] mostly represents an attempt to systematically study the physics of wave motion on a vibrated plane. However, in general, building a setup involving liquids flowing down a vibrated vertical surface requires non-standard equipment built according to demanding technical specifications. In particular, the liquid should be supplied to the inlet located at the upper part of the plane so that the flow rate is not affected by the vibration. This is because the thickness of the liquid film is known to be very sensitive to external disturbances, including vibrations caused by the pump used to deliver the liquid from a reservoir to the inlet [22]. Moreover, the shaker producing the vibration should be connected to the vertically positioned surface via a vibration transmission structure. Some of the engineering challenges of creating such a structure are the need to move a considerable total mass of the supporting structure and liquid with high precision, and to ensure that the amplitude of the vibration across the plane area is uniform [45]. To resolve the problem of non-uniform vibration amplitude, in [45] it is was suggested that qualitatively similar results could be obtained vibrating just one side of the plane, i.e., vibrating just a portion of the liquid, thus also significantly reducing the total mass that needs to be moved by the shaker. In this paper, we present and discuss a technically simple and compact experimental setup for the investigation of solitary-like surface waves on a slightly inclined plane positioned on top of a vibrating table and equipped with an auxiliary channel that recyclesthe liquid used in the experiment, thus decreasing the chance of spills of the liquid and its unwanted contact with the measurement and imaging equipment, and also decreasing the total mass that needs to be moved by the shaker. We employ this setup to demonstrate that the instabilities of the thin liquid film caused by the vibrations result in a decrease in the peak amplitude of the solitary-like surface waves. We conclude that, despite these changes, the speed of the solitary-like waves does not appreciably change due to vibration. As a result, these waves can propagate for long distances without changing their shape and, therefore, can be used in the practical applications discussed in this work. We also demonstrate the advantage of using frequency-wavevector dispersion maps for the analysis of the properties of rolling waves, thus extending the toolbox of researchers working on this class of wave motion phenomena. Our experimental results are validated using the Shkadov model [52; 53]: a boundary-layer hydrodynamic model derived from the Navier-Stokes equation under the assumption of self-similar parabolic longitudinal velocity flow field across the layer.