The paper concentrates on the exponential stability of the linear \[ u_{tt} - u_{xx} +a(t)u_t +b(x)u =0 \] and nonlinear \[ u_{tt} -f(u_x)x +a(x)u_t + b(x)u =0 \] wave equations, the solution satisfying initial and Dirichlet boundary conditions on a bounded interval \((0,L)\). The case where the damping coefficient \(a(x)\) changes its sign is investigated. One shows that the result of \textit{J. E. Muñoz Rivera} and \textit{R. Racke} in [Nonlinear Anal., Theory Methods Appl. 68, No.~9 (A), 2531--2551 (2008; Zbl 1140.35355)], can be generalized to the case \(b\neq 0\). Under certain assumptions regarding the functions \(a, b\) and \(f\) one proves sufficient conditions for the existence of a unique global solution for the nonlinear wave equation.