We consider a one-dimensional wave equation, which governs the vibrations of a damped string with spatially nonhomogeneous density and damping coefficients. We introduce a family of boundary conditions depending on a complex parameter
. Corresponding to different values of
, the problem describes either vibrations of a finite string or propagation of elastic waves on an infinite string. Our main object of interest is the family of non-selfadjoint operators
in the energy space of two-component initial data. These operators are the generators of the dynamical semigroups corresponding to the above boundary-value problems. We show that the operators
are dissipative, simple, maximal operators, which differ from each other by rank-one perturbations. We also prove that the operator
coincides with the generator of the Lax-Phillips semigroup, which plays an important role in the aforementioned scattering problem. The results of this work are applied in our two forthcoming papers both to the proof of the Riesz basis property of the eigenvectors and associated vectors of the operators
and to establishing the exact and approximate controllability of the system governed by the damped wave equation.