The article is concerned with studying the classical solution of the Cauchy problem and the boundary problem for the one-dimensional nonhomogeneous wave equation. Equations in the problems under consideration are defined in the half-plane and in the half-band of two independent variables. The Cauchy nonsmooth conditions are assigned at the bottom of the region. First-kind smooth conditions are defined at the side boundaries of the region. Analytical solutions of problems are obtained using the method of characteristics. The uniqueness of the solution is proved, and the conditions, under which the piecewise smooth solution exists, are determined.