Solution of a mixed problem for the wave equation with a dynamic boundary condition and discontinuous initial data in the first quadrant of the plane

We consider a mixed problem for the one-dimensional wave equation in the first quarter of the plane with discontinuous initial and boundary conditions arising from the theory of longitudinal impact on an elastic semi-infinite rod with elastic fastening at the left end. We propose a new approach to solve the problem, which constructs the solution as a limit of classical solutions. The proposed method does not use any additional conditions that were required in previous methods providing a more direct and rigorous construction. We find the solution in an explicit analytical form as a piecewise continuous function. For the problem under consideration, the uniqueness of the solution is proved and the necessary and sufficient matching conditions are established under which the solution exists. 

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