# Solitary-like Wave Dynamics in Thin Liquid Films over a Vibrated Inclined Plane
## 3 Experimental Results
(**a**) No vibration. (**b**) The inclined plate is vibrated with the frequency of 48 Hz and the peak vibration amplitude of 1 g.  
We first analyse the dispersion map in Figure 4a and its magnified image presented in Figure 5a, where we can see a set of high-magnitude discrete bands that are superimposed on a broader continuum band of a lower magnitude. The frequencies of the discrete bands correspond to the forcing frequency of the solitary-like waves of 2 Hz and its higher-order harmonics of 4, 6 and 8 Hz and so forth. The origin of the harmonics is due to the nonlinear effects, as discussed below. The spectrum of the discrete bands changes as the frequency of forcing of the solitary-like waves is changed. When the modulation of the pump flow was turned off, i.e., with no wave forcing, the discrete bands completely disappeared. However, a continuum band was always observed independently of whether the forcing was turned on or off. Subsequently, we associate the continuum band with natural periodic rolling waves propagating on the surface of the liquid film flowing over the inclined surface. According to the frequency-wavevector spectral analysis theory [55], a fit of the observed bands with a straight line produces the velocity of the solitary-like wave of 0.27 \(\pm\) 0.02 m/s. When the plate is vibrated (Figure 4b), in addition to the dispersion bands discussed in Figure 4a we observe two new isolated bands that can be associated with the Faraday  
Figure 4: Dispersion maps of the solitary-like waves forced at the frequency of 2 Hz. (**a**) No vibration. (**b**) The inclined plate is vibrated with the frequency of 48 Hz and the peak vibration amplitude of 1 g. The plots are slightly oversaturated for the sake of a better visual presentation. Figure 3: False-colour maps showing the spatiotemporal traces of solitary-like waves forced at the frequency of 2 Hz. (**a**) No vibration. (**b**) The inclined plate is vibrated with the frequency of 48 Hz and the peak vibration amplitude of 1 g.  
instability. Moreover, the close-up of the dispersion map (Figure 5b) shows that the magnitude of the discrete modes decreased due to the vibration, which is an observation that is consistent with our conclusions made earlier in the text. Yet, the bands in Figure 5b can also be fitted with a straight line that corresponds to the wave velocity of \(0.27\pm 0.02\) m/s. Empirically, the presence of the discrete bands at the forcing frequency of \(2\,\mathrm{Hz}\) and its higher-order harmonics can be explained using the well-established analogy between rolling waves and acoustic waves [56]. Indeed, the solitary-like surface waves in Figure 2 can be regarded as large-amplitude shock-like disturbances (in the sub-field of physically similar roll waves in open channel such a discontinuity is called the hydraulic jump [14; 21; 34]). Shock waves are also well-known in the field of nonlinear acoustics, where their formation is accompanied by strong nonlinear effects such as the generation of higher-order harmonic frequencies [50]. Considering longitudinal acoustic waves that can be described as alternating areas of compression and rarefaction in the medium, we can show that the points of the crests of an acoustic wave travel faster than the speed of sound in the medium, but the points of the wave troughs travel slower [50]. This physical process underpins the formation of an acoustic shock wave [56]. In turn, in the field of rolling waves, the crest of a large-amplitude solitary-like wave is connected to its trough by a discontinuity, where the flow regime abruptly changes from a supercritical condition and where the fluid moves faster than the wave, to a subcritical one, where the fluid moves slower [14; 34]. As a result, the spectrum of the wave becomes enriched by the higher-order harmonic of the frequency of forcing.