## 3 Determining the Pressure Wave Speed in the Tested System
### Method #1  
Equipping the laboratory stand with pressure measurements in several cross-sections of the measuring pipeline opened the possibility of determining the speed of pressure wave propagation in the system using the measurement of time of the pressure wave passage between two sections in which the pressure measurements were carried out. After detailed analyses of the measurement signals, it was decided to use for this purpose the measurement signals from the transducer installed directly at the valve (transducer no. 1) and the transducer installed in about half of the pipe #2 (transducer no. 2)--Figure 1. In the case of the other signals, there was too high signal distortion manifested by pressure fluctuations originating mainly from wave reflections from the cross-section of diameter changes, but also from FSI effects, prevented their use in this method. Due to the fact thatas the water hammer phenomenon continues, these distortions also become present in the signals from transducers no. 1 and 2, the analysis could be carried out only for the first pressure increase after closing the valve--Figure 4.  
The distance between Sections 1 and 2 of the pressure measurement was \(\Delta l=29.392\) m. Using the following basic relationship:  
\[a=\frac{\Delta l}{\Delta t} \tag{4}\]  
the value of the pressure wave propagation speed in the pipe #2 \(a=a_{\text{2\_meas}}\) can be determined on the basis of the value \(\Delta t\) standing for the time of wave passage between the measurement sections. To estimate the pressure wave speed in pipe #1 (\(a_{\text{1\_est}}\)), which is connected to the upstream tank, the value of the pressure wave speed in pipe #2 (\(a_{\text{2\_meas}}\)) can be used, determined on the basis of the analysis shown in Figure 4. For this purpose, the following formula can be used:  
\[\frac{a_{\text{1\_est}}}{a_{\text{2\_meas}}}=\sqrt{\left(1+\frac{D_{\text{2 \_meas}}}{e_{\text{2\_meas}}}.\frac{K}{E_{\text{2\_2}}}c_{\text{1\_2}}\right) /\left(1+\frac{D_{\text{1\_1}}}{e_{\text{1\_1}}}.\frac{K}{E_{\text{1\_1}}}c_{ \text{1\_1}}\right)} \tag{5}\]  
Formula (5) is based on Equation (1) and takes into account the fact that the pipelines used during the tests are made of the same material (copper) and are supported in the same way, and the underlying assumptions should be considered justified. For the considered case the pipe #1 (\(D_{\text{1\_1}}=0.02\) m, \(e_{\text{1\_1}}=0.001\) m) and pipe #2 (\(D_{\text{2\_2}}=0.016\) m, \(e_{\text{2\_2}}=0.001\) m) are made of the same material (\(E_{\text{1\_1}}=E_{\text{2\_2}}=E=120\cdot\)10\({}^{9}\) for copper) and are rigidly fixed to the foundation in a similar way (\(c_{\text{1\_1}}=c_{\text{1\_2}}=c_{\text{1\_1}}=1-\text{v}^{\text{2}}\) is equal 0.8775 using [PERSON]'s ratio for copper pipes v = 0.35). The wave speed ratio defined by formula (5), calculated for \(K=K_{\text{Water}}=2.21\cdot\)10\({}^{9}\) Pa (at water at 23 \({}^{\circ}\)C and under pressure approx. 100 m wc [49]), is ca. 0.975. In Table 1, the values of the pressure wave speed determined using Equations (4) and (5) are shown for both pipes of the measuring system. These values apply to four test runs differing in initial conditions (values of \(Q_{\text{0}}\) and \(H_{\text{0}}\)). The uncertainty of these quantities is mainly related to the time resolution of the recorded pressure oscillations. Therefore, the relative uncertainty will depend on the time interval that is being measured.