In this work, we develop a unified strategy which allows to derive observability results for waves in an annulus when the observation is done on the external boundary and under various boundary conditions at the internal boundary. Our approach is based on the fact that the observability of a linear abstract conservative system is equivalent to a suitable resolvent estimate. We develop a full machinery to derive suitable resolvent estimates under some weak assumptions on the boundary condition on the internal sphere, which are shown to be close to sharp. In fact, our strategy allows to exclude the existence of trapped rays close to the internal sphere when the boundary conditions satisfy a suitable assumption. As an application, we apply our results to several wave models.