Degree Name

Doctor of Philosophy


School of Mathematics and Applied Statistics - Faculty of Informatics


This thesis considers the refraction and diffraction of both linear and nonlinear waves. In the first part, a linear numerical model based on the dual reciprocity boundary element method (DRBEM) is presented for the study of combined diffraction and refraction of linear waves. This model is more general than that presented by Zhu (1993a) in the sense that areas or coastlines where the water depth is zero can be successfully dealt with. Our comparison study shows that the new model is very accurate for the propagation of long waves such as tsunamis. Moreover, it is numerically very efficient in comparison with models based on finite elements or differences. Using the new model, the interaction between the diffractive and refractive effects is examined.

In the second part, a numerical model is developed by expanding the Boussinesq equations using a perturbation method and the DRBEM. Based on the assumption that the incident waves are harmonic, the time-dependent nonlinear Boussinesq equations are transformed into three time-independent linear equations, where no approximation for the seabed slope is made. Then the first-order solution No is found as the solution of the linear shallow-water equation. The first-order solution is then used in the governing equations at second-order. By employing a transformation, all the third-order and the fourth-order partial derivatives of No in the right-hand sides are removed, resulting in the minimization of any errors which occur in approximating these derivatives. To validate the new model, the wave run-ups of weakly-nonlinear waves scattered by islands are found. Thirteen cases of run-ups around a vertical cylindrical island are considered and it is found that the nonlinear and dispersive contributions of the new model are significant and a much better comparison with experimental results is obtained than for the linear diffraction theory. The combined wave diffraction and refraction by a conical island is also modelled and discussed. Our model is found to be more accurate than other nonlinear models as the dispersive effects have been included, but is also more computationally efficient since there is no time marching and the spatial dimensionality of the numerical calculation has been reduced by one with the adoption of the DRBEM.

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