Linear And Nonlinear Wave Models With Accurate Dispersion

Wave is one of the key dynamic factors in ocean or in coastal and offshore region, and its propagation from deep water to offshore region is the research focus. The present study studies the mild slope model and the nonlinear wave equations with accurate dispersion. The advantage of these two kinds of equations is that they have no limitation in water depth and so can be used to simulate the whole process when waves propagation form deep ocean region to nearshore region.Through introducing the nonlinear terms related to amplitude dispersion, Copeland's (1985) classical hyperbolic mild-slope equations are modified to take the amplitude dispersion into account. To achieve this goal, wave frequency is modified by amplitude variation while wave number is kept unchanged, which is in consistency with Stokes wave theory, i.e., wave frequency depends on wave amplitude. Under the assumption of mild slope, the modified equations approximately satisfy energy conservative equations. To extend the model's application range, the energy dissipative term is incorporated to mimic wave breaking and also the bottom friction is considered by employing empirical friction formula. Besides, alternative modification method, i.e., replacing the celerity and group celerity in classical mild-slope equations with the corresponding nonlinear ones, is described and the corresponding model is verified by simulating wave propagation on plane beaches with mild slopes (1:40,1:100).The above models are extended for irregular. wave case using carrier frequency perturbation method and linear superposition theory. Correspondingly, a novel approach is presented, i.e., replacing the carrier amplitude with representative amplitude. This new method is also applied to Smith and Sprinks' equations for comparison. Both models are validated and inter-compared by simulating irregular wave propagation over elliptical shoal and over plane beach.Though simple and effective, the above methods used to improve the models' amplitude properties are approximate, and hence a precise way is desired to remove this unsatisfactory. Through introducing potential function on free surface and using Fourier integral transformation, a new set of equations is derived based on the Laplace equations and corresponding boundary conditions. The equations are fully dispersive and hence have no limitation on water depth. The nonlinearity of the model is investigated using 1D model, in which, the 1st and 2nd order nonlinearity is tested against Stokes wave theory while the 3rd one is checked by simulating random waves propagation over constant water depth. The 2D model is validated tested against experimental data of wave propagation over complex topography. The good agreements between the numerical results and experimental data demonstrated that the present model possesses good amplitude dispersion property and is applicable for predicting wave elevations in near-shore region.