Cross-waves generated by an oscillating wavemaker in a rectangular wave tank are examined when two or more modes are simultaneously unstable. The partial differential equations governing the evolution of the complex amplitude of inviscid cross-waves are shown to be two coupled nonlinear Schrodinger equations with transverse modulations. Energy dissipation in the system is taken into account by the inclusion of a linear viscous damping term into the amplitude equations. A linearized stability analysis is performed on these equations to determine the critical modes, the growth rates and the stability curves. A center manifold analysis is used to reduce the PDE's to a system of ODE's in the neighborhood of a codimension two point where two adjacent spanwise modes are simultaneously nearly marginal. Four possible steady states of the system are found, one of which is a mixed mode state. A Hopf bifurcation from the mixed mode is predicted for a certain region of the parameter plane, suggesting the possibility of energy interchange between the two modes. The stability of the Hopf bifurcation is determined by studying a fifth order problem, where the quintic contributions come from the higher modes as well as the perturbations of damping and detuning.
| Identifer | oai:union.ndltd.org:arizona.edu/oai:arizona.openrepository.com:10150/184872 |
| Date | January 1989 |
| Creators | Ayanle, Hassan Shiekh-Ali. |
| Contributors | Lichter, Dr. S. H., Chen, Dr. C. F., Peterson, R.A. |
| Publisher | The University of Arizona. |
| Source Sets | University of Arizona |
| Language | English |
| Detected Language | English |
| Type | text, Dissertation-Reproduction (electronic) |
| Rights | Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author. |
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