Degree Name

Doctor of Philosophy


School of Mathematics and Applied Statistics


Studies about ocean waves have been evolving over a period of time. Re¬cently, there has been renewed interest in problems of refraction, diffraction and radiation of ocean waves around structures. In this thesis, the analytic solutions for linear waves propagating in an ocean with variable bottom to¬pography and their applications in renewable wave energy are presented. In the first part, we present an analytic solution to the shallow water wave equa¬tion for long waves propagating over a circular hump. As a useful tool in coastal engineering, the solution may be used to study the refraction of long waves around a circular hump. It may also be used as a validation tool for any numerical model developed for coastal wave refraction. To validate the new analytic solution, we have compared our new analytical solution with a numerical solution obtained by using the finite difference method. The agreement between these two solutions is excellent. By using the analytic solution, the effect of the hump dimensions on wave refraction over the circular hump are examined.

In the second part of this thesis, based on the mild-slope equation derived by Smith and Sprinks [1] and the extended refraction-diffraction equation developed by Massel [2], we have constructed a two-layer mild-slope equation for interfacial waves propagating on the interface of a two-layer ocean model. First, we follow Smith and Sprinks’s [1] approach to derive the mild-slope equation for the propagation of interfacial waves, with the higher-order terms proportional to the bottom slope and bottom curvature all being neglected. We then derived the extended version of the mild-slope equation with the higher-order terms included. While we were able to solve the first equation analytically, we presented a numerical solution for the second equation. As a part of the verification process, both solutions were compared with each other and also with the single-layer mild-slope equation when the density of the upper layer goes to zero. We then used the new solution to study the effect of the hump dimensions on the refraction of the interfacial waves over a circular hump.

Finally, in the final section of this thesis, we have used what we have developed before to construct the two-layer mild-slope equation with free surface on top. By utilizing this equation, we then derived an analytic solu¬tion for long waves propagating over a circular hump with a hollow circular cylinder floating in the free surface. In order to validate our new analytic solution, we have compared our problem with Mac Camy and Fuchs [3] solu¬tion, because our solution has reduced to their solution when the lower water depth, h2, goes to zero. We have also compared our solution with the flat bottom case in order to further verified our solution. Finally, by using the new solution, both diffraction and refraction effects from the hollow cylinder and hump dimensions are examined and discussed.

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