Correction method of test solutions of the general wave equation in the first quarter of the plane for minimal smoothness of its right-hand side

A method is proposed for correcting of test classical solutions of the general inhomogeneous factorized oscillation equation for a semibounded string in order that they have minimal (necessary) smoothness requirements on its righthand side. The idea of the method is to calculate the correction to some of its trial (test) classical solutions, which may require an overestimated smoothness from the right-hand side of the equation. To this end, the correcting Goursat problem for the canonical form of this oscillation equation of a string is formulated and solved. Then, in the resulting solution, the smoothness of the test solution is analyzed and, if necessary, it is corrected by the corresponding solution of the homogeneous oscillation equation of the string. We find new classical solutions and the previously unknown necessary smoothness of the right-hand side.