## 1 Introduction
According to them:  
* The influence of FSI on the pressure wave propagation speed change is not taken into account. Only the quasi-static response of the pipeline structure to the pressure wave is taken into account;
* The bulk modulus of the liquid is constant, regardless of pressure changes;
* Changes in liquid density due to pressure are not taken into account--e.g., it is assumed that the liquid is not a water-gas mixture;
* A two-dimensional state of stress in the pipeline shell is assumed (thin wall pipe);
* The manner and influence of the pipeline support and fastening on the pressure wave speed is taken into account by using an approximate value of the \(c_{1}\) coefficient. With the appropriate configuration of the flow system and rigid mounting of the pipelines, the influence of FSI on the pressure wave speed can be eliminated or at least significantly reduced. The variability of the compressibility modulus and density from pressure for a fluid consisting of a mixture of water and a small amount of air may also be an insignificant concern for the pressure wave speed. The assumption of treating the pipeline as thin walled depends on the \(D_{outer}/e\) ratio. It is usually assumed that pipelines with a \(D_{outer}/e\) ratio not lower than 20 can be treated as thin walled, which means that assuming a two-dimensional state of stress in such cases affects the determination of the value of the pressure wave speed insignificantly. The last mentioned factor related to the fastening of the pipeline and the corresponding correct value of the \(c_{1}\) coefficient most significantly affects the uncertainty of \(a\) estimated using Equation (1). In connection with the assumptions underlying Equation (1), in order to make the most reliable verification of the developed numerical calculation method in this work, the determination of the pressure wave speed for a system consisting of pipe series of different diameters was based on the measured and recorded pressure changes induced by a sudden flow cut-off. Two methods for determining pressure wave speed \(a_{i}\), characteristic for the particular pipes forming the system, were proposed and used in the numerical analysis of the water hammer phenomenon. The following parts of the paper describe the numerical method used to simulate the water hammer phenomenon in complex systems, which takes into account the quasi-steady and unsteady models of friction losses. This method uses an original procedure of discretization of the solution domain, which allows for avoiding the artificial damping generated, usually by the numerical scheme. This is particularly important for the accurate verification of the friction loss models used in the calculations. The mathematical model forming the basis of the numerical method was verified using qualitative comparative analysis between the measurement and calculation results. The conclusions provided a basis for evaluating the friction loss models used in the calculations according to the agreement of the calculation and experimental results. The simplified approach to calculating the course of pressure changes during a water hammer, involving the use of equivalent quantities, was also evaluated. The problem related to the occurrence of the water hammer phenomenon concerns most technological systems encountered in practice. Wherever these systems are built from pressure pipelines, this phenomenon should be absolutely taken into account at the conceptual and design stage. Additionally, special attention should be paid to it during the current operation of such systems by designing appropriate operating procedures to prevent the adverse effects of the occurrence of a water hammer and to ensure the work safety of these systems. Pipeline systems cooperating with pumps, penstocks of hydropower plants, and long transport pipelines in which numerous regulating and shut-off fittings are installed, which may cause this phenomenon, are particularly vulnerable to the formation of a water hammer.