In this paper an optimal control problem with initial-boundary conditions is considered, where the relations between the control and the state vector are determined by the wave equation, which is linear with respect to the phase variable. Also functions that define terminal and integral parts of the objective functional are linear with respect to the phase variable. The statement of the problem allows any combinations of boundary conditions of the first, second and third kinds on both left and right borders of the domain. The wave equation does not contain the first derivatives of the phase variable in the right side. In addition, the differential operator of the equation is of a special form. These two factors make it possible to build integrated equivalent of the original problem in the form of one integral equation with respect to the solution function. The resulting integral equivalent serves as the basis for the definition of a generalized solution of the problem for an arbitrary fixed admissible control. We derive two equal increments of the target functional formula that do not contain residual terms and are therefore accurate. On this basis effective methods of improving the functionality of the target can be built in a similar way as was done for optimal control of processes describable by ordinary differential equations.