Maximum wave run-up over beaches of convex/concave bottom profiles

Abstract

In this communication, we provide exact solution for the wave run-up event owing to impulsively started solitary waves towards beaches of various sloping bottom profiles. A rigorous asymptotic formula is obtained corresponding to the solitary wave maximum wave height depending on concavity feature of the coast bottom. It unifies many well-studied beaches and their run-up predictions in the literature. It also explains how the sloping profiles will contribute to the wave run-up process by the response of the approaching long waves. The main outcome of the work is that for the smaller offshore amplitude to offshore depth ratios, oceans with high sloping seabeds result in the least maximum wave run-ups no matter whether the concave or convex coastline bottom profiles (for profile shapes refer to figure 1) are accounted. Unlike this, higher maximum wave run-ups take place with enhanced amplitude ratios. Shores with a concave bottom shape are shown to delay the run-up process by significantly opposing the moving waves acting like a tsunami barrier. However, convex bottom cross-sectional beaches are assisted by the faster wave climbing feature and with increased maximum wave run-ups. More convex bottom profiles show up peculiar singular structures with larger run ups The presented wave run-up formula with the convex/concave beach profiles may be beneficial in many future coastal applications in terms of coastline processes, despite the fact that the proposed solution is linearized with limited range of validity.