Waves in Collections of Circular Shoals and Bathymetric Depressions

Abstract

New approximate analytic solutions are developed to study wave propagation through collections of coastal features. Solutions are developed for circular shoals formed by a submerged cylinder with water depth shallower than the surrounding sea, and for bathymetric depressions formed by a circular pit deeper than the surrounding sea. Classic solutions for monochromatic wave propagation through a single isolated coastal feature are extended using the analytic element method to achieve analytic solutions for a number of coastal features that collectively shape the wave field. Each element is formulated as a Riemann–Hilbert interface problem, where the wave amplitude and phase are continuous between the element and its surroundings; however, a discontinuity in the normal derivative of the wave field occurs as a result of the change in water depth across the interface. Interface conditions are satisfied nearly exactly for typical problems, as demonstrated by normalized root mean square errors of the order of 10−16. This article contributes new mathematical and computational methods that shed insight into wave amplification and dissipation generated within collections of coastal features, with potential applications including the study of trapped waves and tsunamis near coastal features, and of waves traveling through aquatic vegetation.