## 3 Determining the Pressure Wave Speed in the Tested System
### Verification of the Numerical Method Using the Models of Unsteady Friction Losses  
The main scope of this work was focused on the verification of unsteady friction models used to simulate the water hammer in the analyzed system consisting of pipes of different diameters connected in series. The expected better compliance of the pressure amplitudes between measurements and calculations with unsteady friction models has been validated for the efficient version of the Vardy and Brown model (Vardy and Brown efficient) as well as the Brunone et al. model in version 1 (with the \(k_{3}\) coefficient calculated from Equation (A18)--[PERSON] et al. ver.1) and version 2 (with a constant value of \(k_{3}\) coefficient adjusted to the reference run--in the case of the test run No. 4, the assumed value of this coefficient was \(k_{3}=0.045\) --[PERSON] et al. ver. 2). These models were also subjected to experimental verification in the authors' previous studies for a homogeneous pipeline with constant parameters [3].  
The results of the calculations for the above-described Vardy and Brown efficient and [PERSON] et al. ver. 1 models are shown in Figure 9. In terms of quality, the pressure wave simulated using the Vardy and Brown efficient model represents much better agreement with the measurements than the calculations carried out using the [PERSON] et al. ver. 1 model. The Vardy and Brown efficient model generates a wave that in the developed phase of oscillations is regular, very similar to the measured wave. In the case of the [PERSON] et al. ver. 1 model, the pressure wave is much more irregular, which indicates that the combination of pipes with a stepwise change in diameter affects the calculated pressure wave over the entire analyzed time of simulation. This means that the shape of this wave is significantly different from the reference wave (measured), especially in the developed phase of pressure oscillation. In addition, attention should be paid to a better compliance of the pressure wave simulated using the Vardy and Brown efficient model with the measurements mainly in terms of wave attenuation.  
Comparison of the calculation results with the measurement results showed that the period of computational pressure wave, \(T\), depending on the model used, is as follows:  
* approx. 0.309 s for the Vardy and Brown efficient model,
* approx. 0.310 s for the [PERSON] et al. ver. 1 model.  
This means that the frequency of the wave simulated with the use of these models of friction losses is lower by almost 1.7% and 1.8%, respectively, than the frequency of the wave measured during the tests. The resultant wave speed \(a_{r}\) in the case of calculations was 1398 m/s for the [PERSON] and Brown efficient model and 1397 m/s for the [PERSON] et al. ver. 1 model. These values are less than the resultant speed value for the measured wave (1422 m/s). This discrepancy cannot be the result of pressure wave damping and its insignificant impact on the pressure wave frequency as was shown in the analysis presented in Figure 7 and in Appendix A. Taking into account the results of these analyses, it can be stated that the friction-induced damping effect of the pressure wave, because it is vanishingly small, cannot be considered the cause of the observed disagreement and it should be assumed that these differences most likely result from the properties of the models of friction losses used. However, confirmation of this thesis requires additional research and analyses. Nevertheless, it should be emphasized that there is a very high agreement with the measurements considering the damping ratio achieved using the Vardy and Brown efficient model and that the differences in the pressure wave propagation speed are relatively small. This means that this model, apart from its theoretical quality, seems to have great potential for application in engineering practice.