A Mathematical Model Of Wave Propagation Over Uneven Bottoms

Based on the zero-order and first-order approximate solutions of the wave potential function and wave amplitude equation in Liu and Shi (2008), this thesis elucidates variations of the wave potential function, wave profiles, wave vectors, wave rays and wave heights for one-dimensional and two-dimensional horizontal wave propagations over uneven bottoms. Calculated results are validated by comparison with calculated and measured results available in the literature.For the one-dimensional wave propagation problem, (i) under the constant slope bottom condition, the zero-order and first-order approximate solutions of the present model are comparable with those of Biesel (1952). The breaking positions of waves propagating over different slopes are estimated by Miche's (1944) empirical equation. Waves over a small slope seem to break earlier than over a large slope. To clearly depict the undulating motion of water particles, calculated solutions of the present model are transformed from the Euler coordinate system into the Lagrange coordinate system. Details of successive wave profiles prior to breaking are plotted within the Lagrange coordinate system. (ii) Under the varying slope bottom condition, the wave amplitude equation is solved by the fourth-order Runge-Kutta method to obtain the analytical solution of the wave potential function. Wave field equipotential lines over variable bathymetry regions calculated by the present model are plotted against Athanassoulis and Belibassakis'(1999) results. They are in fairly good agreement. The present results are even more efficient than the latter obtained by retaining six evanescent modes.For the two-dimensional horizontal wave propagation problem, refraction and diffraction of waves must be taken into account. Given the water depth, incident wave direction and height, (i) firstly, the wave number is solved by the dispersion relation; (ii) secondly, the irrotational wave number and eikonal equations are solved to obtain variations in the direction of wave propagation; and (iii) finally, calculated wave angles are used to obtain the wave amplitude from the wave amplitude equation. The wave amplitude equation and boundary conditions are discretized and solved by the finite difference method, in order to gain insight into variations in wave amplitude and direction of wave propagation caused by refraction and diffraction of waves. Calculated relative wave heights are compared with experimental results of the Berkhoff et al. (1982) elliptical shoal test. Results suggest that the present model can be applied to two-dimensional horizontal wave propagation over uneven bottoms.