## 5 Discussion
This effect is not compatible with some equivalent models assuming that for systems composed of many pipes with different diameters and/or material characteristics, the equivalent pressure wave speed (equal to the resultant wave speed) follows from the  
Figure 10: Pressure wave speed change in relation to gas fraction in the fluid. principle of maintaining the total wave transit time in individual pipes. This means that according to such models, the equivalent pressure wave speed should have an intermediate value between the speed values for individual serially-connected pipes. The conducted research presented in this paper showed that for the developed phase of the water hammer, these results were not confirmed--the measured resultant wave speed values for the final stage of the pressure course in the tested cases were significantly higher than wave speed values characteristics for each individual pipe. The results of the calculations made using the method of characteristics with the special discretization procedure of the solution domain that allows to eliminate the numerical damping confirmed that for the laboratory test stand configuration the computational resultant pressure wave is also characterized by a wave speed exceeding each of the wave speeds calculated for the individual pipes. This paper proposes a procedure for determining the pressure wave speed in individual pipes of a pipeline system on the basis of the resultant pressure wave speed read from the measurements and with the use of appropriate theoretical considerations regarding the equivalent model presented in [33; 34]. The proposed method was verified on the basis of the results obtained using the method of determining the pressure wave propagation speed in individual pipes of the flow system, which uses the time of passage of the pressure wave between adjacent measurement sections. It has been shown that the uncertainties of both methods are comparable, while the proposed method requires only the knowledge of one course of pressure changes, which makes it attractive especially in terms of its practical use. Thanks to this procedure and the use of unsteady friction loss models, the numerical simulation method obtains results that are highly compatible with the measurement results. This paper proposes two methods, the results of which are very similar. Method #2, which uses the measurement of the final phase of free pressure oscillations, is, however, easier to use from a practical point of view. This is because it allows for determining the pressure wave speed in individual pipelines of the flow system based on a single pressure measurement in any chosen cross-section of the flow system (the closer to the shut-off valve causing the water hammer the better, but this is not a necessary condition for this method). Method #1 requires measuring the pressure in two cross-sections of one pipeline, most preferably the pipeline in which the flow is cut off by a valve. While in practice flow systems are usually not equipped with conveniently installed measurement equipment, method #1 may be more problematic to use than method #2. The increase in the complexity of the system construction increases the level of complexity when using the proposed methods for determining the wave velocity in individual pipeline elements. Method #2 can be used for systems consisting of a larger number of pipelines connected in series using the more general version of Equation (9) in the following form [33]:  
\[\sum_{i=1}^{n-1}\left[\frac{a_{i}}{A_{i}}\cdot\tan\frac{\omega L_{i}}{a_{i}} \sum_{j=i+1}^{n}\left(\frac{A_{j}}{a_{j}}\cdot\tan\frac{\omega L_{j}}{a_{j}} \right)\right]=1 \tag{11}\]  
where \(n\) is the number of pipelines connected in series (\(n>1\)) and \(i\), \(j\) are the indices of the individual pipelines. In the case of very complex flow systems, solving Equation (11) and determining the pressure wave speed for individual pipelines can be quite problematic. Therefore, in such a situation, using method #1 can be an effective alternative.