Summary: This paper is concerned with the equation of a nonhomogeneous string of length one with one end fixed and the other one damped with a parameter \(h\in \mathbb{C}\). This problem can be rewritten as an abstract Cauchy problem for a densely defined closed operator i\(A_h\) acting on an appropriate energy Hilbert space \(H\). Under assumptions that the density function of the string \(\rho \in W_2^1 [0,1]\) is strictly positive and has \(\rho(1)\neq 2\) (if \(h\in \mathbb{R}\)), we prove that the set of root vectors of \(A_h\) form a basis with parentheses in \(H\). We show that with the additional condition \[ \int_0^1 \frac{\omega_1^2 (\rho^\prime , \tau)}{\tau^2} d\tau <\infty, \] where \(\omega_1\) is the integral modulus of continuity, the root vectors of the operator \(A_h\) form a Riesz basis in \(H\).