We consider the inverse problem that consists in the determination of characteristic sources, in the modified and classical Helmholtz equations, based on external boundary measurements. We identify the location of the barycenter establishing a simple formula for symmetric shapes, which also holds for the determination of a single source point. We use this for the reconstruction of the characteristic source, based on the Method of Fundamental Solutions (MFS). The MFS is also applied as a solver for the direct problem, using an equivalent formulation as a jump or transmission problem. As a solver for the inverse problem, we may apply minimization using an equivalent reciprocity functional formulation. Numerical experiments with the barycenter and the boundary reconstructions are presented.