Graduation date: 1987 / Best scan available for figures.

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

The Melnikov method is applied to a model of parametrically generated cross-waves in a long rectangular channel in order to determine if these cross-waves are chaotic. A great deal of preparation is involved in order to obtain a suitable form for the application of the Melnikov method. The Lagrangian for water waves, which consists of the volume integrals of the kinetic energy density, potential energy density, and a dynamic pressure component, is transformed to surface integrals in order to avoid constant conjugate momenta. The Lagrangian is simplified by subtracting the zero variation integrals and written in terms of generalized coordinates, the time dependent components of the crosswave and progressive wave velocity potentials. The conjugate momenta are calculated after expanding the Lagrangian in a Taylor series. The Hamiltonian is then determined by a Legendre transformation of the Lagrangian. Ordinarily, the first order evolution equations obtained from derivatives of the Hamiltonian are suitable for applications of the Melnikov method. However, the crosswave model results in extremely complicated evolution equations which must be simplified before a Melnikov analysis is possible. A sequence of seven canonical transformations are applied and yield a final set of evolution equations in fairly simple form. The unperturbed system is analyzed to determine hyperbolic fixed points and the equations describing the heteroclinic orbits for near resonance cases. The Melnikov function is calculated for the perturbed system which must also satisfy KAM conditions. The Melnikov results indicate the system is chaotic near resonance. Furthermore, the heteroclinic orbits, about which chaotic motions occur, are transformed back to the original set of variables and found to be extremely complicated; this orbit would be impossible to determine analytically without the canonical transformations. The theoretical results were verified by experiments. Poincare maps obtained from measurements of the free surface displacement indicate both quasi-periodic and chaotic motions of the water surface. Power spectra and time series of the water surface displacement are also analyzed for chaotic behavior, with less conclusive results. Stability diagrams of cross-wave generation confirm behavior consistent with parametric excitation. / Graduation date: 1994

This report details the derivation of an analytical model for a flexible membrane wave barrier. The wave barrier consists of a thin flexible membrane suspended in the water column by a moored cylindrical buoy on the free surface and fixed to a hinge at the seafloor. The analytical model combines the three-degree of freedom rigid body motion of the cylindrical buoy with the two-dimensional analog of a vibrating string for the response of the flexible membrane. Theoretical results for reflection and transmission coefficients, dynamic mooring line tension, horizontal hinge force, horizontal and vertical displacements and rotation of the cylindrical buoy are compared with measured results presented by Bender(1989). In general, the theoretical results compare favorably with measured results for moored systems. However, additional studies are required to more precisely quantify the added mass and radiation damping properties of flexible membranes in oscillating flows. / Graduation date: 1991

A general convolutional version of perfectly matched layer (PML) formulation for second-order wave equations with displacement as the only unknown based on the coordinate stretching is proposed in this study, which overcomes the limitation of classical PML in splitting the displacement field and requires only minor modifications to existing finite element programs. The first contribution concerns the development of a robust and efficient finite element program QUAD-CPML based on QUAD4M capable of simulating wave propagation in an unbounded domain. The more efficient hybrid-stress finite element was incorporated into the program to reduce the number of iterations for the equivalent linear dynamic analysis and the total time for the direct time integration. The incorporation of new element types was verified with the QUAD4M solutions to problems of dynamic soil response and the efficiency of hybrid-stress finite element was demonstrated compared to the classical finite elements. The second development involves the implementation of a general convolutional perfectly matched layer (CPML) as an absorbing boundary condition for the modeling of the radiation of wave energy in an unbounded domain. The proposed non-split CPML formulation is displacement-based, which shows great compatibility with the direct time integration. This CPML formulation treats the convolutional terms as external forces and includes an updating scheme to calculate the temporal convolution terms arising from the Fourier transform. In addition, the performance of the CPML has been examined by various problems including a parametric study on a number of key coefficients that control the absorbing ability of the CPML boundary. The final task of this thesis is to apply the developed CPML models to the dynamic analyses of soil-structure interaction (SSI) problems. Typical loading conditions including external load on the structure and underground wave excitation on the medium has been considered. Practical applications of CPML models include the numerical study on the effectiveness of the rubber-soil mixture (RSM) as an earthquake protection material and the report of vibrations induced by the passage of a high-speed train. The former investigates the effectiveness of the CPML models for the evaluation of the performance of RSM subject to seismic excitation and the latter tests the boundary effects on the accuracy of the results for train induced vibrations. Both studies show that CPML as an absorbing boundary condition is theoretically sound and effective for the analysis of soil-structure dynamic response. / published_or_final_version / Civil Engineering / Doctoral / Doctor of Philosophy

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published_or_final_version / abstract / toc / Mechanical Engineering / Doctoral / Doctor of Philosophy

published_or_final_version / abstract / toc / Mechanical Engineering / Doctoral / Doctor of Philosophy

published_or_final_version / Electrical and Electronic Engineering / Master / Master of Philosophy