## 1 Introduction  
The vast majority of water hammer laboratory tests concern pipelines with constant cross-sections. There are many examples in the literature of different kinds of simulation models based on the measurement results of pressure wave propagation in such pipelines. These models concern friction loss [1; 2; 3; 4], column separation and transient cavitation [5; 6; 7; 8; 9], FSI and visco-elasticity of pipeline materials [10; 11; 12; 13; 14], as well as transient flow of liquids in pipelines with air content [15; 16; 17; 18]. The use of pipelines with constant diameters in these examples allowed the avoidance of, among others, considering different pressure wave parameters characteristic of individual pipes with different diameters connected in series. Flow conditions in such pipes that make up a complex system are characterized with different liquid flow velocities, as well as different pressure wave propagation speeds. These parameters significantly affect the parameters of the numerical model and thus the obtained calculation results. This is a particularly important issue when using the method of characteristics (MOC), in which the selection of the numerical grid requires a special approach to eliminate the influence of numerical damping on the calculation results that is crucial when the purpose is to verify the friction models used in the calculations. In such a situation, the aim should be to obtain calculation results free of e.g., numerical damping,which requires an appropriate approach to discretization of the solution area together with precise knowledge of the wave propagation speed in all elements of a complex flow system. Since this type of issue is often analyzed in the setting of engineering practice, in some research it is recommended to replace pipelines composed of different pipes connected in series with one pipe with a constant, equivalent diameter with respect to wave speed, inertia forces and pressure losses [19; 20; 21; 22; 23]. This approach is particularly useful in the simulation of transient phenomena in complex transmission networks, where connections of pipes of different diameters are usually very common. Therefore, both from the theoretical and practical point of view, there is a need to know how reliable such an approach is. The available literature shows that in recent decades, numerous researchers have paid attention to the study of transients in pipelines composed of pipes of variable diameters and different materials [24; 25; 26; 27; 28; 29; 30; 31; 32; 33]. In the publications of [25; 26; 27; 28; 29], the results of studies of unsteady liquid flow in the viscoelastic pipeline system with variable diameters and materials are considered. In [32; 33] a method based on the analysis in the frequency domain of determining the equivalent speed in the pipeline system consisting of different pipes connected in series has been introduced. This method can also be found in the [34] monograph. Authors of [24] investigated numerically the effect of a narrowed section of an elastic pipeline on a water hammer using the MOC method. Transient friction losses were taken into account using the Zielke friction model applicable for laminar flow, which is very rare in engineering practice. In [30], the MacCormack scheme for a system of elastic pipes connected in series is used to solve the equations describing the water hammer. The Brunone model was used in the friction loss calculations. The obtained results were compared with the results obtained using the MOC, while the comparison with the results from the experiment was made only for a single pipeline with constant parameters (concerning geometry and wave) along its length. In [31], a mathematical model for calculating the course of a water hammer in a system of elastic (steel) pipes connected in series is presented. In the numerical simulations, the MacCormack scheme was also used, assuming a constant pressure wave speed equivalent to a given pipe configuration, and the calculation of friction losses was based on the Brunone model.