Mathematical Models Of Wave Propagation In Coastal Region

As surface waves propagate from deep to shallow water, due to the effects of water depth, topography, sea bed friction, obstacles and ambient currents etc, many phenomena occur, such as shoaling transformation, refraction, diffraction, reflection, wave breaking, non-linearity and so on, which are the main physical processes and characteristics during wave propagation. There are several kinds of mathematical models of wave propagation in coastal area now, however, they should be developed and perfected for many deficiencies exist. In this report, mathematical models for combined refraction-diffraction waves in water of slowly varying topography are presented. At the same time, being compared with application of the model for non-linear long waves, the knowledge of characteristics of wave propagation models in near shore area is deepened further.Firstly, under the curvilinear coordinates, mathematical model for wave propagation in water of slowly topography is presented. The model is suitable to arbitrary boundary shapes and overcomes the limitation of other models with algorithm transformation. The mathematical model for wave propagation on non-uniform currents is established also. In the models, the time dependent parabolic equations, deduced from the mild slope equations with currents or not, are used as the governing equations. Based on the general conditions for open and fixed natural boundaries with an arbitrary reflection coefficient and phase shift, the boundary conditions for the present models are treated. The alternative direction implicit method is used to solve the governing equations and the numerical schemes are unconditional stable. The required computer storage is reduced. The computer speed is speeded up. The numerical results of the present models are in agreement with the theoretical solution and those of physical models. Systematical numerical tests show that the present models can reasonably simulate the wave transformation, such as shoaling, refraction, diffraction, reflection, effect of currents and so on. So the present models areable to be used in coastal engineering with complicated boundary shapes extensively.Secondly, a mathematical model suitable to large coastal region is developed, whose governing equations are deduced from the mild slope equation with dissipation terms and discretized with Crank-Nicolson scheme. This model is accurate and easy to be applied. The numerical tests show that the results of numerical solution are consistent with those of corresponding analytical solution and physical models. Being utilized in the wave propagation for Nan Gang of Yangtze River Estuary, this model can give good results of numerical simulation by effectively reflecting the influence of complicated topography which is comprised of shoal-channel spaced in between. So it can be applied to large areas.In the end, in view of the fact that Boussinesq-type equations and the mild slope equations are deduced from different hypothesis conditions and behave differently in simulation of wave propagation, the numerical results of wave propagation effected by strong non-linearity are given by the nonlinear three-dimensional mathematical model which was established for the calculation of 3-D wave particle velocity and wave pressure and suitable to small size waters of arbitrarily varying depth. The model for non-linear long wave and the mild slope equation are respectively applied to simulation wave propagationon a classical topography for small size waters-submerged shoal with concentriccontours. The differences between them in wave propagation are got through comparing the numerical solutions. And the results are accordant with actual cases.