Within the framework of the non-conservative form of quasi-one-dimensional linearized equations N.E. Zhukovsky formulated the problem of a transient process on an elementary relief section of a pipeline when the boundary velocities of an incompressible fluid change over time. The incomplete telegraph equation, compiled for the averaged flow velocity, is solved by the method of separation of variables. The pressure value was obtained by integrating the original equations using the resulting velocity solution. Numerical results concerning the case of constant values of functions in initial and boundary conditions are presented. Variants of constant, increasing and decreasing average pressure values over the area are identified. Features of the behavior of shock waves with and without taking into account the friction force are revealed.