Draw-down and run-up of tsunami waves on sloping beaches

The dynamics of waves and their interaction with a beach depends on whether the leading wave component is elevated or depressed. These differences are explained in this paper using a hydraulic model and the principle of conservation of impulse. Laboratory experiments of depression waves, conducted using a novel wavemaker, are compared with model predictions. Over a sloping beach, these waves have a nearly constant V-shaped depression trailed by a growing Λ-shaped positive wave. The shoreline recedes over a significant distance, caused by shoreward water being drawn into the V-shaped depression. When the trailing Λ-shaped positive wave breaks, an energetic hydraulic bore develops and moves up the beach. The hydraulic model leads to general formulae for wave slopes, draw-down and run-up. The run-up of negative waves can be larger or smaller than that of positive waves, depending on the wave amplitude and beach parameters. The predictions are compared with results from photographs of depression waves taken d...