M.Sc Thesis
M.Sc Student Avni Ronen High Order Mild Slope Equation Department of Applied Mathematics Professor Yehuda Agnon

Abstract

When an incident wave travels over and interacts with a bottom of non-uniform depth, the wave is modified and may be partially reflected, although for mild bottom variations the reflection is generally weak. However, when the bottom contains periodic undulations and the incident wave and bottom ripple wave numbers satisfy the so-called Bragg conditions, the scattered wave becomes resonant and can be greatly amplified. Such resonant wave interactions with bottom ripples play a significant role in the evolution of near-shore surface waves.

The areas near the shore are usually characterized by shallow or intermediate depth water, and a mild-slope of the bottom. The mild slope theory deals, therefore, with areas where many human activities take place, hence the need for an accurate model. For this reason efforts have been made to study this issue and several mild slope equations have been derived. These models, usually eliminating the vertical coordinate, try to predict the wave field by an approximation of the problem to the first order. This approximation is related to the dominant resonance phenomena, an assumption which is not always valid far away from the resonance conditions, as can be clearly seen from experiments.

The motivation for this work is to provide more accurate prediction of the linear wave field under a special case of resonance conditions. The equations derived here are based on an operational expansion, which incorporates an infinite number of derivatives and may therefore be considered as the full equation for this case. Existing models may be obtained by an expansion of this equation with a small parameter, which reflects the physical nature of the problem. Different versions of the models have different mathematical forms, i.e. different degrees of the derivatives and different coefficients. The link between the existing models and the operational equation sheds light on the assumption implicitly used in these models as well as their accuracy.

The accuracy in most of the models is explained to be of first order, which provides motivation for a derivation accurate to the second order, obtained in this work by using an operational expansion.