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The fractal dimension of measured ocean wave profiles is found to be in the range of 1.5-1.8. This non-integer dimension indicates the fractal nature of the waves. Standard formulations to analyze waves are based on a differential approach. Since fractals are non-differentiable, this formulation fails for waves with fractal characteristics. Integral solutions for long waves that are valid for a non-differentiable fractal surfaces are developed. Field observations show a positive correlation between the fractal dimension and the degree of nonlinearity of the waves, wave steepness, and breaking waves. Solutions are developed for a variety of linear cases. As waves propagate shoreward and become more nonlinear, the fractal dimension increases. The linear solutions are unable to reproduce the change in fractal dimension evident in the ocean data. However, the linear solutions do demonstrate a finite speed of propagation. The correlation of the fractal dimension with the nonlinearity of the waves suggests using a nonlinear wave equation. We first confirm the nonlinear behavior of the waves using the finite difference method with continuous function as the initial condition. Next, we solve the system using a Runge-Kutta method to integrate the characteristics of the nonlinear wave equation. For small times, the finite difference and Runge-Kutta solutions are similar. At longer times, however, the Runge-Kutta solution shows the leading edge of the wave extending beyond the base of the wave corresponding to over-steepening and breaking. A simple long wave solution on multi-step bottom is developed in order to calculate the reflection coefficient for a sloping beach. Multiple reflections and transmissions are allowed at each step, and the resulting reflection coefficient is calculated. The reflection coefficient is also calculated for model with thousands of small steps where the waves are reflected and transmitted once over each step. The effect of depth-limited breaking waves is also considered. / Graduation date: 2003

Surprisingly, Fourier series on certain fractals can have better convergence properties than classical Fourier series. This is a result of the existence of gaps in the spectrum of the Laplacian. In this work we prove a general criterion for the existence of gaps. Most of the known examples on which the Laplacians admit spectral decimation satisfy the criterion. Then we analyze the infinite family of Vicsek sets, finding an explicit formula for the spectral decimation functions in terms of Chebyshev polynomials. The Laplacians on this infinite family of fractals are also shown to satisfy our criterion and thus have gaps in their spectrum.

Surprisingly, Fourier series on certain fractals can have better convergence properties than classical Fourier series. This is a result of the existence of gaps in the spectrum of the Laplacian. In this work we prove a general criterion for the existence of gaps. Most of the known examples on which the Laplacians admit spectral decimation satisfy the criterion. Then we analyze the infinite family of Vicsek sets, finding an explicit formula for the spectral decimation functions in terms of Chebyshev polynomials. The Laplacians on this infinite family of fractals are also shown to satisfy our criterion and thus have gaps in their spectrum.

Thesis (Ph. D.)--University of Missouri-Columbia, 2000. / Typescript. Vita. Includes bibliographical references (leaves 87-91). Also available on the Internet.

Thesis (M.S. in E.S.M.)--School of Civil and Environmental Engineering, Georgia Institute of Technology, 2004. Directed by Donald R. Webster. / Includes bibliographical references (leaves 118-121).

Thesis (M. Phil.)--University of Hong Kong, 2001. / Includes bibliographical references (leaves 119-122).

published_or_final_version / Computer Science and Information Systems / Master / Master of Philosophy

One of the images compression methods – fractal image compression is analyzed in the work. After work carried out, it is possible to state, that selecting parameters of method of fractal compression depends on user’s demands.

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