The author studies the problem of small vibrations of a smooth inhomogeneous string damped at an interior point and fixed at the endpoints; the problem is reduced to a three-point Sturm-Liouville boundary problem. This is considered as an eigenvalue problem for a nonmonic quadratic operator pencil of special type with the spectrum located in the upper half-plane of the spectral parameter. Concerning the corresponding inverse problem, it is shown that the spectrum does not determine the potential of the Sturm-Liouville problem uniquely. In order to recover the potential uniquely, the spectrum of the “truncated” Sturm-Liouville problem is chosen as an additional information; the self-consistency of the two spectra is discussed.