## 3 Determining the Pressure Wave Speed in the Tested System
### Method #2
where \(\omega_{e}\) is the circular frequency of pressure oscillation (index \(e\) stands for \"equivalent system\"):  
\[\omega_{e}=\frac{2\pi}{T_{e}}\, \tag{8}\]  
which in the case of two pipes connected in series fulfills the following equation [33; 34]:  
\[\frac{a_{1}A_{2}}{a_{2}A_{1}}\cdot\tan\frac{\omega_{e}L_{1}}{a_{1}}\cdot\tan \frac{\omega_{e}L_{2}}{a_{2}}=1 \tag{9}\]  
Other symbols in Equations (7)-(9) stand for: \(a_{1}\) or \(a_{2}\) is the actual pressure wave speed characteristic for each individual pipe [m/s]; \(A_{1}\) or \(A_{2}\) is the area of cross-section of each individual pipe [m\({}^{2}\)] (\(A=0.25\)\(\pi\cdot\)D\({}^{2}\)); \(L_{1}\) or \(L_{2}\) is the length of the individual pipe [m]; \(L_{e}\) is the equivalent length of a series connected pipe system, being the sum of the lengths of the system components [m]; and \(T_{e}\) is the wave period [s]. It should be emphasized that the above analysis does not take into account the full impact of friction losses on the parameters of water hammer. Omitting the so-called unit resistance of the pipe does not mean that the losses are completely excluded from the analysis, but only their impact is reduced, which is manifested, inter alia, in a wave frequency change. However, as it is shown in the further analysis, such an assumption has a negligible effect on the wave speed in the tested system. When the values of \(a_{1}\) and \(a_{2}\) are known, the Equation (9) enables the determination of the value of \(\omega_{e}\), and consequently, the value of the equivalent wave propagation speed, \(a_{e}\). On the other hand, with the known geometric parameters of the pipes, i.e., \(A_{1}\), \(A_{2}\) and \(L_{1}\) and \(L_{2}\), and the ratio \(a_{1}/a_{2}\) given by Equation (5), the above procedure can be used to determine numerically the value of the pressure propagation speed in individual sections of the pipeline \(a_{1}\) and \(a_{2}\) in accordance with Equation (9) for a given value of \(\omega_{e}\). The values of the pressure wave speed in individual pipes in the pipeline system, calculated in accordance with the described procedure for the analyzed test runs are presented in Table 3. These values were compared with the values of the pressure wave speed in individual pipes determined using method #1 based on Equations (4) and (5). The differences between the compared values are on average approx. 0.08% and max. 0.1% (this is within the uncertainty bands estimated for method #1 and #2). The comparison of the results obtained with the two analyzed methods confirms the positive verification of method #2. It should be noted that in this method, unlike method #1, it is sufficient to know the pressure changes in one measurement cross-section only, which makes method #2 much more attractive to use than method #1 in the context of analyzing and determining the  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{4}{*}{Test No} & \multicolumn{3}{c}{**Pressure Wave Speed in Individual Pipes**} & \multicolumn{3}{c}{**Differences between the Values of**} \\  & \multicolumn{3}{c}{**The Velocity of Propagation of**} & \multicolumn{3}{c}{**The Resulting Pressure Wave in a**} \\  & \multicolumn{3}{c}{**System of 2 Pipelines with**} & \multicolumn{3}{c}{**Methed \#1 (on the Basis**} \\  & \multicolumn{3}{c}{**Variable Diameters, Calculated**} & \multicolumn{3}{c}{**Ingations (4) and (5)**} \\  & \multicolumn{3}{c}{**The Results of Equation (9)**} \\  & \multicolumn{3}{c}{**The Results of Equation (9)**} \\  & \multicolumn{3}{c}{**The Results of Equation (4)**} \\speed of pressure wave propagation in systems composed of pipes connected in series, at least when it comes to less complex flow systems consisting of a few pipelines connected in series.