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The classical solution of the mixed problem for the one-dimensional wave equation with the nonsmooth second initial condition

In this article, we study the classical solution of the mixed problem in a quarter of a plane for a one-dimensional wave equation. On the bottom of the boundary, the Cauchy conditions are specified, and the second of them has a discontinuity of the first kind at a point. The smooth boundary condition is set at the side boundary. The solution is built using the method of characteristics in an explicit analytical form. The uniqueness is proved, and the conditions under which a piecewise-smooth solution exists are established. The problem with conjugate conditions is considered