Graduate Thesis Or Dissertation
 

Nonlinear diffraction theory by an eigenfunction expansion of the Green's function

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  • A second-order nonlinear diffraction theory is developed for a large diameter circular cylinder under the action of surface gravity waves of finite amplitude in water of-finite depth. Boundary value problems are derived by using a perturbation expansion and the solutions for the diffracted waves are given in terms of Fredholm integral equations in which the resolvent kernel is a Green's function. Green's functions are obtained by the eigenfunction expansion method and the velocity potentials for the diffracted waves are recovered from Fredholm integral equations which incorporate the prescribed inhomogeneous boundary conditions and the Green's functions. Green's functions are extended to the second-order boundary value problem by an expansion in a complete set of orthonormal eigenfunctions. The well-posed Sturm-Liouville problems at second-order yield two sets of eigenfunctions which result in weakly dispersive scattered waves. The results for the second-order velocity potential are shown to satisfy all the prescribed boundary conditions. The dimensionless hydrodynamic forces and moments on the cylinder are presented. Graphs of the force and moment coefficients and the phase angles for both the first- and second-order solutions are presented. The results show that the second-order contribution may be significant near the free surface and indicate that the second-order effect of the hydrodynamic forces and moments are important in finite depth. Also, a comparison with limited model test results indicates that the nonlinear forces are in better agreement than the linear forces.
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