Summary: We consider the stability of the inverse problem consisting of the determination of a coefficient of order zero \(q\), appearing in the Dirichlet initial-boundary value problem for a wave equation \(\partial_t^2u-\Delta u+q(x)u=0\) in \((0,T)\times\Omega\), with \(\Omega=\omega\times\mathbb{R}\) an unbounded cylindrical waveguide and \(\omega\) a bounded smooth domain of \(\mathbb{R}^2\), from boundary observations. The observation is given by the Dirichlet to Neumann map associated to the wave equation. Using suitable geometric optics solutions, we prove a Hölder stability estimate in the determination of \(q\) from the Dirichlet to Neumann map. Moreover, provided that the coefficient \(q\) is lying in a set of functions \(\mathcal A\), where, for any \(q_1,q_2\in\mathcal A\), \(|q_1-q_2|\) attains its maximum in a fixed bounded subset of \(\overline{\Omega}\), we extend this result to the same inverse problem with measurements on a bounded subset of the lateral boundary \((0,T)\times\partial\Omega\).