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Faraday waves arise on the surface of a liquid in a container that is undergoing vertical periodic oscillations. We investigate two-dimensional Faraday waves in a long rectangular container, both theoretically and experimentally. Hysteresis occurs when both finite amplitude solutions and the flat surface solution are available. We derive a nonlinear model of a standing wave, extending the Lagrangian method of Miles (1976). The model is used to investigate hysteresis. It is found necessary to retain cubic damping, cubic forcing and the fifth-order conservative term in order to achieve agreement with experiments. The fifth-order conservative term was omitted from all previous studies of Faraday waves. Stable limit cycles are found to arise from this single-mode equation. We examine the structure of this new solution in detail, both analytically and numerically. We describe local bifurcations using a multiple time scales analysis and global bifurcations using Melnikov's method. The coefficients of linear and cubic damping are derived for a standing wave in a rectangular container by considering energy dissipation in the main body of the fluid (due to potential flow and streaming) and in boundary layers at the sidewalls and at the surface. Surface contamination, due to the presence of a thin viscoelastic surface film, creates a boundary layer at the surface which causes enhanced dissipation comparable to, or greater than, that caused by the boundary layers at the walls of the container. Three-mode interaction equations are used to model intermittency and complex modulations which are found to arise from a sideband instability mechanism similar to that of Eckhaus (1963) and Benjamin & Feir (1967). The role of cubic and fifth-order nonlinear terms on this instability mechanism is examined. Theoretical results are found to compare quite favourably with experimental data.

Thesis (Ph. D.)--University of Texas at Austin, 2003. / Vita. Includes bibliographical references. Available also from UMI Company.

Thesis (Ph. D.)--West Virginia University, 2003. / Title from document title page. Document formatted into pages; contains xii, 156 p. : ill. (some col.) + MPEG video files. Includes MPEG video files. Includes abstract. Includes bibliographical references.

Several significant wave propagation problems in the fields of nonlinear optics and hydrodynamics are studied in this thesis. In optics, the physical model considered is the two-core optical fiber (TCF), which is an essential component of lightwave technology. In hydrodynamics, the motion of a wave packet on the free surface of water of finite depth allowing modulations from two mutually perpendicular and horizontal directions, governed by the famous Davey-Stewartson (DS) equations, is taken into account. The main contributions of this thesis are: In optics, the effects of the intermodal dispersion (IMD) and the birefringence induced effects, both of which always exist in the TCFs, have been ignored in the previous studies of the modulation instability (MI) of continuous waves (CWs) in the TCFs. In this thesis, a detailed analysis of these effects on the MI spectra has been done. It is found that IMD does not seriously affect the MI spectra of the symmetric/antisymmetric CW states, but can significantly modify the MI spectra of the asymmetric CW states. In exploring the birefringence induced effects, a particular class of asymmetric CW states, which admits analytical solutions and has no counterpart in the single-core fibers, is focused on. It is found that the MI spectra of a birefringent TCF in the normal dispersion regime can be distinctively different from those of a zero-birefringence TCF especially for the circular-birefringence TCF. All the findings of MI analysis can be well verified by the wave propagation dynamics. Another contribution of this thesis is that we find the dramatic pulse distortion and even pulse splitting phenomenon due to IMD in TCFs, which is unexpected in many situations, can be effectively suppressed and even avoided by Kerr nonlinearity, which has never been reported in the literatures in the studies of TCFs. In hydrodynamics, DS equations describe the evolution of weakly nonlinear, weakly dispersive wavepackets with slow spanwise dependences on a fluid of finite depth. Generally, DS equations are divided into two types e, i.e. DSI and DSII equations, depending on the specific fluid configurations (fluid depth, wavelength of the water wave, surface tension etc). Due to the importance of DS equations, many exact solutions have been derived by different nonlinear wave methods over the years in the literature. In this thesis, two new exact doubly periodic wave patterns of DS equations are derived by the use of properties of the theta functions, or equivalently, the Jacobi elliptic functions, and the corresponding solitary waves are also deduced in the long wave limits. The new feature of the two wave patterns found is that they can be applied to both DSI and DSII systems at the same time. / published_or_final_version / Mechanical Engineering / Doctoral / Doctor of Philosophy

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