We calculate suspended sediment transport, erosion and deposition under reflected long waves on a plane beach. Our method employs the shallow-water equations, and is based on the calculation of the concentration field in Lagrangian coordinates: this allows results to be obtained readily throughout the whole domain, including the often troublesome shoreline region. These results suggest that the tendency of wave-dominated muddy shores to export sediment and erode over long time scales is not due to wave forcing at infragravity frequencies, since the sediment transport under such waves is localized close to the shoreline, and is principally directed shorewards. They also provide easily reproduced test cases against which to validate the numerical methods used in more detailed studies of coastal sediment transport and morphodynamics.