## 3 Result
### Characteristics of the Initial Waves  
As mentioned above, the initial waves appear and grow on top of the \"thick\" initial film, prior to the stabilization of film thickness. These waves can be approximately described by the equation of a running sinusoidal wave with exponentially growing amplitude, namely:  
\[h(x,t)=h_{0}(x)+A_{0}*\exp(ikx-ikct+\alpha x) \tag{2}\]  
where \(h_{0}\) is local time-averaged film thickness, \(A_{0}\) is wave amplitude at \(x=0\), \(k\) is the wavenumber, \(c\) is the phase velocity and \(\alpha\) is the spatial increment of amplitude growth. Three main parameters of the waves which can be measured to characterize them are the frequency (\(f=kc/2\pi\)), velocity, and increment. The frequency is measured by a direct counting algorithm applied to the temporal records of film thickness within the distance range corresponding to the appearance and growth of initial waves on the thick film. The algorithm identifies parts of the record when \(h(x_{i}t)\) is larger or lower than \(h_{0}(x_{i})\). The median value of the duration of adjacent pairs of such \"crest\" and \"trough\" parts of waves is taken as a wave period and used to calculate the frequency. The total frequency was estimated as the median over all the \"local\" values within the area of existence of initial waves. The results of frequency measurements are presented in Figure 5 in the same manner as used previously for \(X\) in Figure 4. Frequency shows a prominent, approximately linear, growth with gas velocity and a weaker growth with the liquid flow rate (Figure 5a,b). There is also a minor effect of liquid viscosity: larger viscosity results in slightly higher frequencies at the same liquid flow rate. The viscosity effect is stronger in the case of upward flow and larger liquid flow rates. Figure 5: \(Cont.\)  
The frequency values are larger in downward flow if all the other conditions are the same (Figure 5c). In this Figure, the largest frequency values correspond to the largest gas and liquid flow rates; in this area, the strongest deviations from the average line are observed. At moderate gas velocities, the frequency shows linear growth with a liquid flow rate, and the data for the liquids with different viscosities lie on quite the same straight lines (Figure 5d). The difference between the frequencies of the initial waves in the upward and downward flows increases with the liquid flow rate. Velocities of the waves were measured based on the time delay between the temporal records obtained at different downstream coordinates determined by the position of maximum in cross-correlation function between the two records. The measurement results are shown in Figure 6. The wave velocity is nearly independent on gas velocity (Figure 6a,b), which is, again, unusual for waves on gas-sheared liquid films. It grows with the liquid flow rate, but there is no noticeable effect of viscosity or flow orientation. The absence of any strong orientation effect is also demonstrated in Figure 6c,d. The velocity grows linearly with a liquid flow rate and, as in the case of stabilization length and frequency, the data for different viscosity lie on the same line. The amplitude of the waves at a fixed downstream distance is identified as the standard deviation of a temporal record. This value is several times smaller than the peak-to-peak amplitude of the signal, but it can be used for the estimation of the spatial increment of amplitude growth. The latter is defined as the logarithm of the ratio of amplitudes, and it is assumed that the shape of the waves does not change significantly during the short distance so that the ratio of peak-to-peak amplitude to standard deviation remains the same. In general, the measured values of increment show a much larger uncertainty than those of frequency or velocity. This could be expected since a relatively small change of amplitude has to be detected over a very short length. The discretized character of the film thickness values increases uncertainty even further.