This article provides an intercomparison of the dispersive and diffusive properties of several standard numerical methods applied to the 1D linearized shallow-water equations without the Coriolis term, including upwind and central finite-volume, spectral finite-volume, discontinuous Galerkin, spectral element, and staggered finite-volume. All methods are studied up to tenth-order accuracy, where possible. A consistent framework is developed which allows for direct intercomparison of the ability of these methods to capture the behaviour of linear gravity waves. The Courant–Friedrichs–Lewy (CFL) condition is also computed, which is important for gauging the stability of these methods, and leads to a measure of approximate equal error cost. The goal of this work is threefold: first, to determine the shortest wavelength which can be considered ‘resolved’ for a particular method; second, to determine the effect of increasing the order of accuracy on the ability of a method to capture wave-like motion; and third, to determine which numerical methods offer the best treatment of wave-like motion.