The wave propagation behaviour of centered difference schemes on one-dimensional nonuniform staggered grids is investigated. Previous results for the linear advection equation are extended to the case of the shallow water equations on staggered grids. For waves of a given frequency, the wave field is decomposed into right-and left-propagating components, and a wave energy conservation law is derived in terms of these components. For slowly varying grids, separate evolution equations for the right-and left-propagating components are derived, leading to the result that there is asymptotically no reflection in the limit of a slowly varying grid, provided that waves of that frequency are resolvable. However, there will be reflection from any location at which the wave group velocity goes to zero. The possibility for wave energy to tunnel through a narrow region of the grid too coarse for propagation is noted. Grids with an abrupt jump in resolution are also investigated. It is possible to tailor the scheme at the jump to minimize spurious wave reflection over a range of frequencies provided the waves are resolvable on both sides of the jump. However, it does not appear possible to avoid complete reflection, except by introducing extra dissipation terms, if the waves are not resolvable on one side of the jump. An example is presented of a second-order accurate scheme that spontaneously radiates waves from the resolution jump.