A mixed problem for the wave equation in a curvilinear halfstrip with discontinuous initial data

We study a mixed problem for a one-dimensional wave equation in a curvilinear half-strip. The initial conditions have a discontinuity of the first kind at a single point. The mixed problem models the problem of a longitudinal impact on a finite elastic rod with a movable boundary. Using the method of characteristics, we obtain the solution in an explicit analytical form. For the problem in question, we prove the uniqueness of the solution and establish the conditions under which its classical solution exists.