Abstract
The object of the research described in this dissertation is the development of a theory of transient waves propagating on a non-uniform bathymetry. The simultaneous linear processes of refraction, shoaling and frequency spreading are considered in such a way that nonlinear processes (e.g., nonlinear wave damping and shoaling) can be approximately included in the computations.The transformation of linear periodic waves over non-uniform bathymetry is well studied. The phenomena considered include wave shoaling, refraction, reflection (from the bottom) and wave diffraction. In most cases of practical importance the first two phenomena are most important. The method of geometrical optics is commonly used to determine linear shoaling and refraction. This method is popular since along wave rays, wave action is conserved which allows inclusion of nonlinear effects, such as wave damping due to interaction of the wave with the seafloor and nonlinear shoaling and breaking, into the computation of wave height predicted at a shallow water location. In this investigation, the method of geometrical optics is generalized to include frequency spreading, which must be considered in a study of transient wave since wave frequency varies continuously with time and location. The solution, in general, must be computed numerically but in the case of a bathymetry dependent on 1 spatial coordinate only, analytic solutions can be computed by quadrature. These solutions are explicitly computed and presented in graphical form for a plane beach having constant slope. Analytical and numerical comparisons with results and methods of previous investigators are of transient waves propagating from deep to shallow water are provided. A numerical model was written to predict the transformation of transient initial circular waves (a specified type) propagating over a two dimensional bathymetry. The model accounts for linear refraction, shoaling, and frequency spreading as well as approximate nonlinear shoaling and nonlinear damping due to (mostly) turbulent boundary layer effects at the ocean-seafloor interface. Also provided are samples output from the model showing propagation on a hypothetical one dimensional continental shelf.