## 3 Determining the Pressure Wave Speed in the Tested System
### Method #2  
The direct effect of the configuration of the hydraulic pipeline system is the resultant frequency of pressure oscillations (free vibrations) observed at the end of the transient phenomenon--Figure 5. Based on the analysis of the pressure wave period for this phase of the phenomenon (averaged value from 20 selected final cycles of pressure oscillation in its developed phase, for which the impact of the change in diameter is insignificant and their shape is much more regular than for oscillations in the initial phase of the water hammer), the resultant wave speed \(a_{r\_\_\text{meas}}\) was determined and its values are summarized in Table 2. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{3}{*}{**Test No**} & \multirow{3}{*}{\begin{tabular}{c} **Pressure in the** \\ **Upstream Tank** \\ \end{tabular} } & \multirow{3}{*}{**Initial Flow**} & \multirow{3}{*}{\
\begin{tabular}{c} **Period of the Pressure Wave** \\ **(Calculated from 20 Final** \\ **Periods of Pressure** \\ **Oscillations)** \\ \end{tabular} } & \multicolumn{1}{c}{
\begin{tabular}{c} **The Resultant Pressure Wave** \\ **Speed in the Pipeline System** \\ **Composed of 2 Pipes with** \\ **Different Diameters** \\ \end{tabular} } \\ \cline{2-5}  & & \(H_{0}\) & \(Q_{0}\) & \(T\) & \(a_{r\_\text{meas}}\) \\ \hline - & **m wc** & **L/h** & **s** & **m/s** \\ \hline
1 & 26.4 & 113.7 & 0.3120 & 1387 \\ \hline
2 & 51.6 & 240.8 & 0.3090 & 1400 \\ \hline
3 & 83.3 & 361.1 & 0.3065 & 1411 \\ \hline
4 & 119.0 & 530.1 & 0.3043 & 1422 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameters of selected measurement runs tested on a laboratory stand together with data for determining the resultant pressure wave speed of the pipeline composed of two pipes with different diameters. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{3}{*}{**Test No**} & \multirow{3}{*}{\begin{tabular}{c} **Pressure in the** \\ **Upstream Tank** \\ \end{tabular} } & \multirow{3}{*}{**Initial Flow**} & \multirow{3}{*}{\begin{tabular}{c} **Time of Wave Passage** \\ **between the Measurement** \\ **Sections of Pipe \#2 Spaced** \\ **\(\Delta l\) = 29.392 m Apart** \\ \end{tabular} } & \multicolumn{1}{c}{\
\begin{tabular}{c} **The Pressure Wave Speed** \\ **in the Pipe \#2** \\ **Determined on the Basis of** \\ **of Equation (4)** \\ \end{tabular} } & \multicolumn{1}{c}{
\begin{tabular}{c} **The Pressure Wave Speed in the** \\ **Pipe \#1 Calculated** \\ **on the Basis of** \\ **Equation (5)** \\ \end{tabular} } \\ \cline{2-5}  & \(H_{0}\) & \(Q_{0}\) & \(\Delta t\) & \(a_{2\_\_\text{meas}}\) & \(a_{1\_\_\text{est}}\) \\ \hline
**-** & **m wc** & **L/h** & **ms** & **m/s** & **m/s** \\ \hline
1 & 26.4 & 113.7 & 24.05 & 1222 & 1192 \\ \hline
2 & 51.6 & 240.8 & 23.77 & 1236 & 1206 \\ \hline
3 & 83.3 & 361.1 & 23.67 & 1242 & 1211 \\ \hline
4 & 119.0 & 530.1 & 23.49 & 1251 & 1221 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of selected test runs with pressure wave speed determined using method #1. The comparison of the results summarized in Tables 1 and 2 and presented in Figure 6 shows that the determined resultant wave speed \(a_{r\_\text{meas}}\), similarly to the speed in individual sections determined using method #1, also depends on the level of equilibrium pressure (pressure in the upstream tank), \(H_{0}\). This effect is widely known and has been repeatedly confirmed in scientific research (e.g., in [46]). It results from the fact that in the liquid (in this case, water) there is a certain volume of air undissolved in it, mainly in the form of microbubbles. This volume, although usually small, can be responsible for a significant change in the bulk modulus as well as the density of the water-air fluid. Both of these quantities affect the speed of pressure wave propagation, reducing it in relation to the speed in the liquid without air. Moreover, due to the considerable compressibility of air, a change in the pressure of the fluid causes a change in the proportion of the volume of air to the volume of the liquid.