Forced Capillary-Gravity Waves in a 2D Rectangular Basin
Brunnhofer, Harald Michael
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This dissertation concerns capillary-gravity surface waves in a two-dimensional rectangular basin that is partially filled with water. To generate the surface waves, a harmonic forcing is applied to the vertical side walls of the basin. The dissertation consists of four parts which work with different assumptions on the frequencies of the forcing. The first part discusses the linearized model with Hockingâ s edge condition and gives an eigenvalue equation and an asymptotic expansion for the eigenvalues. Then, for the nonlinear problem, it is assumed that the frequency of the forcing is close to an eigenfrequency and the solution has an asymptotic expansion using a two time-scales approach. Under an edge condition, the first- and second-order approximations of the solution and a solvability condition from the third-order equations yield an ordinary differential equation for the amplitude of the solution. In part two, it is assumed that the frequency of the forcing applied to the boundary is close to the sum of two eigenfrequencies. In this case, the solvability conditions give a system of two differential equations for the complex valued amplitudes of the two eigenmodes. The system can be reduced to one real-valued differential equation. Its solutions yield the solutions of the original system and their properties. A condition for the existence of homoclinic orbits connecting the trivial equilibrium is obtained. These results are confirmed by numerical experiments. The third part is based on the results in the second part. Here, one of the eigenfrequencies is chosen to be much larger than the other one, and different orders of the amplitudes of the eigenmodes are assumed. The orders of the coefficients of the system found in the second part are obtained, and the resulting special case is discussed in detail. In particular, numerical examples of orbits that can be associated with homoclinic orbits connecting nontrivial equilibria are given. The behavior of solutions close to those orbits is demonstrated. In the fourth part, an additional frequency for the forcing terms given in parts two and three is introduced. In each situation, the modified systems are presented and discussed.
- Doctoral Dissertations