Solutions of problems with discontinuous conditions for the wave equation

In this paper, we consider various approaches to solving of mixed problems with discontinuous conditions based on functional and classical methods. We show differences in solutions, which correspond to different techniques (the Laplace transform and the method of characteristics) and definitions. We demonstrate the results on the case of one boundary-value problem from the theory of mechanical impact about longitudinal oscillations of a semi-infinite elastic rod with discontinuous initial and boundary conditions. A model example is the problem of vibrations of the rod after a longitudinal impact on the end (e. g., shooting a plasticine bullet sticking to the end of a rod).