## 1 Introduction
The calculations were experimentally verified by adjusting the parameters of the adopted friction model. In the context of pipelines of variable diameter, attention should also be paid to a number of works whose main purpose is to detect extended blockages in pipelines using the nature of transient pressure wave waveforms (i.e., transient test-based techniques). Most often, the physical model in this issue is a section with a smaller diameter inserted into the pipeline, and in order to detect blockages, an analysis of pressure changes is carried out in the frequency domain [35; 36; 37; 38; 39; 40; 41] or in the time domain [42; 43; 44; 25]. Techniques for detecting blockages in pipelines are also proposed based on the coupling of these two approaches, i.e., on transient tests analyzed in the frequency domain and time domain [45]. Both experimental and numerical tests are used to validate the detection techniques. Numerical tests are prepared using MOC or other methods of solving equations describing the water hammer in pipelines with constrictions simulating extended blockages. On the basis of the available publications, it should be noted that the results of the numerical simulation of the water hammer with the use of various models of friction losses for elastic pipelines connected in series and the comparison of these results with the results of experimental tests are not very frequent. Examples of articles that present such results analyze the issues of numerical errors generated by the calculation scheme in a superficial and insufficient way. Also, the issue of determining the pressure wave speed in individual pipes of a complex flow system is not the subject of thorough analysis. Typically, the proposed solutions are based on classical methods of estimating the pressure wave speed and ignore the impact of these methods on the calculation results. The main purpose of work presented in this paper was to conduct laboratory tests of a water hammer in a relatively simple pipeline system composed of two copper pipesconnected in series with different diameters and relative wall thicknesses. The results of these tests were used to verify the developed numerical method for simulating a water hammer, which takes into account different models of friction losses. In order to ensure the appropriate conditions for such verification, the method, based on MOC, includes a special procedure allowing for the avoidance of numerical damping. The basis of this method is the precise determination of the pressure wave speed for each of the pipes in the complex flow system. This is necessary to select the appropriate size of the numerical grid, which for the determined wave speeds will ensure the calculation results without numerical attenuation. Considering the pipes in the flow system as thin walled, the calculation of the pressure wave speed in its individual pipes can be based on the following formula [19]:  
\[a_{i\_\text{calc}}=\sqrt{\frac{K}{\rho\left(1+\frac{K}{E_{i}}\frac{D_{i}}{e_{i }}c_{1\_i}\right)}} \tag{1}\]  
where \(a_{i\_\text{calc}}\) is the computational pressure wave speed in the \(i\)-th pipe [m/s], \(K\) is the liquid elasticity modulus [Pa], \(E_{i}\) is the Young's modulus for the material of the \(i\)-th pipe [Pa], \(\rho\) is the liquid density [kg/m\({}^{3}\)], \(D_{i}\) is the internal diameter of the \(i\)-th pipe [m] and \(e_{i}\) is the wall thickness of the \(i\)-th pipe [m]. The coefficient \(c_{1\_i}\) [-] depends on the method of fixing the \(i\)-th pipe:  
* \(c_{1\_i}=1-0.5\
u_{i}\) for a pipe anchored at its one end only;
* \(c_{1\_i}=1-\
u_{i}{}^{2}\) for a pipe anchored throughout from axial movement;
* \(c_{1\_i}=1\) for a pipe anchored with expansion joints throughout. where \(v_{i}\) [-] is the Poisson's ratio of the \(i\)-th pipe material. Calculation of the pressure wave speed using Equation (1) may be associated with even a few (or more) percent inaccuracy, influenced by the underlying assumptions [46].