Analytic Simulations For Linear Ocean Waves Propagating Over Three-dimensional And Un-idealized Topographies

The main work presented in this thesis is to extend the newly established analytical tech-nique for solving two-dimensional wave scattering to construct analytical solutions for wavescattering by three-dimensional bathymetries, including submerged cylindrical sill with a sourpit and a cylindrical island mounted on an un-idealized paraboloidal shoal.Firstly, by using the variable separation technique, the two-dimensional partial diferen-tial equations MSE (mild-slope equation) and MMSE (modified MSE) are transformed intoordinary diferential equations with respect to radial variable r. Then, by using Taylor seriesexpansion technique together with several recurrence formulae for calculating various orderderivatives of implicit wave number, analytical solutions to the two-dimensional long-waveequation, the two-dimensional MSE and MMSE in the Taylor series form are constructed,which converge in the whole physical domain. Based on these analytical solutions, the influ-ence of the depth and width of the scour pit on wave amplification is intensively analyzed,which shows that the deeper and wider the scour pit is, the weaker the wave amplificationfactor is, due to more reflection and more energy scattering laterally. It is also found our ana-lytical solution includes Longuet-Higgins’ classical analytical solution for wave scattering bya cylindrical sill as a special case.Secondly, wave scattering by a cylindrical island mounted on an un-idealized paraboloidalshoal is considered. Since the shoal is assumed to be un-idealized, that is the water depth inthe shoal region is a power function of the radial variable plus a constant, the problem be-comes much more difcult. To remedy this, a variable transform from the radial variable r toa new variable t is employed. Under this variable transform, the original water depth is clearlychanged. If we see the composite function of the water depth function as a new topography, itis found that the dispersion relation and all the recurrence relation for calculating derivativesof implicit coefcients still hold, thus the analytical technique used in Chapters3-5can bestill employed and analytical solutions can be constructed for waves in whole spectrum prop-agating a cylindrical island mounted on an un-idealized paraboloidal shoal are constructed.