Thesis (Ph. D.)--University of Adelaide, Dept. of Applied Mathematics, 1993. / Includes bibliographical references (leaves 117-126).

Thesis (PhD)--Montana State University--Bozeman, 2008. / Typescript. Chairperson, Graduate Committee: Jeff Adams. Includes bibliographical references (leaves 72-75).

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

ABSTRACT Under stationary atmosphere and on uniform depth, this paper treats the standing waves in real fluids formed by two progressive waves possessing same properties but in opposite direction. Being different from the preceding scholars who usually treated the waves in real fluids with boundary layer theory, the author uses complete Navior-Stokes Equ. to analyze the entire flow field. When dealing with the free surface dynamical boundary condition, under the equilibrium of forces, the author takes account of atmosphere pressure, shear stress and surface tension. As for the bottom condition, at first consider the perfect smooth, then no-sliding and sliding condition. After constructing the boundary conditions and the governing equation, perturbation method is used to get those of second order, and the second order solution can be derived. In addition to relative depth , the bottom-adherence affects the bottom boundary effect. No matter in progressive or standing wave fields, we can see the variation of over-shot height, the asymmetric diagrams of fluid particle¡¦s horizontal velocity with phases, the phase difference between the second and first order bottom shear. Besides, in standing wave field, the existence of second-order interaction term not only affects the flow field in the boundary layer but also the field outside it.

The resonance of stationary waves in a mid-latitude (beta)-plane model is examined. In a series of linear experiments it is found that a large, quasi-resonant response occurs even in the presence of realistic damping. With forcing of reasonable amplitude the resonant growth rates are found to be such that a period of several weeks is required for the resonant mode to achieve large amplitude in the troposphere. When nonlinear effects are included the growing resonant wave is found to cause a stratospheric warming and zonal wind reversal and thus drives the system off resonance. We therefore examine flow configurations for which the nonlinear terms vanish. It is found that these flow configurations can resemble realistic blocking patterns. When diabatically forced, the resonant growth of these patterns does not give rise to any nonlinear interactions, although such interactions do occur if other models, which do not satisfy the non-interaction conditions, are present. We conclude that resonant growth can lead to the establishment of large-amplitude, stationary, long waves even when realistic damping and non-linear effects are operating.

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Thesis (Ph. D.)--University of Washington, 2005. / Vita. Includes bibliographical references (p. 95-98).

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Bibliography: leaves 117-126 / vii, 126 leaves : ill ; 30 cm. / Title page, contents and abstract only. The complete thesis in print form is available from the University Library. / Thesis (Ph.D.)--University of Adelaide, Dept. of Applied Mathematics, 1993