A study of the influence of vorticity, capillarity and slope angle on the properties of shoaling and breaking waves

Abstract

Fluid motion and wave patterns have fascinated humans for centuries if not millennia. Water waves have been studied out of curiosity such as observing the ripples created by a stone dropped in a pond as well as practical need. Designing a breakwater to shield a harbor from incoming waves and predicting sediment erosion at a beach are typical engineering problems which arise in the protection of the coast, and require advanced knowledge of the underlying wave motion.

One fundamental problem in the study of water waves is the transition of the waves from large or intermediate depth to shallower waters, and the subsequent breaking. While this situation has been studied from various angles by a large number of authors, many issues remain unresolved, and there is not a single mathematical theory which has been found to work generally in all cases that arise.

In the present thesis, we focus on the shoaling of waves on plane slopes in different cases. The properties of these waves are described mathematically and compared with measurements and observations. It is found that if an appropriate model equation is used for the description of the wave properties, then it is possible to obtain good comparisons between these models and laboratory experiments, field measurements and observations. In some cases, shear flows are a dominant factor while in other cases, capillarity or the steepness of the bed slope are the most important features.

In the first three papers, the shallow-water equations on a shoaling beach are considered. Following the method of Carrier and Greenspan (1958), exact solutions are found and compared with field observations (Paper C). The method of Carrier and Greenspan is extended to flows which may include background shear flows such as for example caused by wind set-up and the required return flow at the coast (Paper A, Paper B)

A laboratory study of the run-up of a solitary wave is given in Paper D. These experiments were set to produce a collapsing breaker on a slope of moderate steepness (1:20), and the breaking wave was dominated by capillarity. The flow field under the collapsing breaker is studied, and a qualitative analysis of this flow is explained using the Navier-Stokes equations.

In Paper E, breaking of an undular bore is considered. The KdV equation is used in the context of background shear, and the unset of breaking is found by analyzing the underlying flow field. The results are compared with an experiment conducted by Favre (1935), and the comparison suggest that the KdV theory may be used to give an approximate prediction of the incipient of wave breaking of the leading wave in the bore.

In paper F, a field campaign is conducted in order to study shoaling waves in the surf zone. Measured Eulerian and Lagrangian orbital velocities are obtained and the correlated relation to wave-by-wave variations in mean-water level, wave height and incipient wave breaking are considered. By using buoyant traces, Lagrangian mass transport at the free surface is studied and it is shown that the KdV equation gives good predictions of the particle motions relative to the mean-water level.