## 3 Determining the Pressure Wave Speed in the Tested System
### Method #2
This translates into changes in the bulk modulus and density of the fluid, which in turn causes changes in the propagation speed of the pressure wave under these conditions (i.e., greater pressure reduces the volume of air microbubbles in the fluid, which increases the propagation speed of the pressure wave). It is roughly estimated that in the analyzed cases the air content could be less than \(10^{-3}\%\) of the total fluid volume at standard pressure [46]. It is noteworthy that the resultant wave speed, \(a_{r\_\text{meas}}\), is 7-10% higher than the speed value \(a_{2\_\text{meas}}\) and by as much as 10-13% from \(a_{1\_\text{est}}\) determined using Equations (4) and (5), respectively (method #1). Moreover, the value of \(a_{r\_\text{meas}}\) is only 4-7% less than the speed of sound in water (1487.4 m/s under the test conditions). The determination of the value of \(a_{r\_\text{meas}}\) is burdened with a comparable absolute uncertainty as for the determination of \(a_{1\_\text{est}}\) and \(a_{2\_\text{meas}}\) using Equations (4) and (5). This is because in the case of determining the wave speed \(a_{r\_\text{meas}}\) using method #2, the relative uncertainty does not depend only on the number of free oscillations that will be taken into account when determining the average period of the pressure wave. It must also take into account the change in the wave frequency during the phenomenon. Figure 5: The method #2 of determining the resultant speed of the pressure wave (\(a_{r\_\text{meas}}\)) in a system composed of two pipes of different diameters. It is known that friction causes a decrease in the wave frequency as the transient state phenomenon continues. Appendix A presents a rough analysis of the effect of damping on the wave frequency change--the analysis was carried out assuming a linear damping decrement and showed a negligible, approx. 0.05%, impact of such damping on reducing the frequency of linearly damped vibrations relative to undamped vibrations. In the case of a water hammer, friction is non-linear, which means that the frequency of the pressure wave during this phenomenon, and thus also the speed of pressure wave propagation, depends on its duration. In order to estimate this effect, method #2 was used based on the measurement of the time for six subsequent five-wave period time intervals. Additionally, the pressure wave speeds determined on this basis were compared with the value determined for 30 wave periods. The results presented in Figure 7 confirm the decrease in the speed of pressure wave propagation with the duration of the phenomenon. However, one should be careful in interpreting these results due to the observed scattering of the speed values around the trend line. This may be related to the fact that in the analyzed phase of the pressure oscillations, the influence of changes in the diameter of pipes in the flow system on the course of pressure changes can be still present. Nevertheless, it should be emphasized that this scattering is small (reaching a maximum of 2.5 m/s, i.e., approx. 0.15% of the average value) and confirms that the use of method #2 based on the average frequency value from several dozen periods of the pressure wave in its final phase does not significantly affect the results of this method. This should be taken into account when determining the uncertainty of the method, which should be about 0.2% for the analyzed cases, i.e., as for method #1. The appropriate uncertainty band of method #1 determined in accordance with [47], together with the uncertainty band of method #2, were assessed based on the above considerations and are shown in Figure 6.  
Figure 6: Pressure wave speed for pipe #1 and #2 (\(a_{1\_{\text{est}}}\), \(a_{2\_{\text{meas}}}\)) determined using method #1 (based on the measurement of the time of pressure wave passage between two consecutive pipe cross-sections) and the speed (\(a_{r\_{\text{meas}}}\)) of the resultant pressure wave (in the developed phase of oscillations).