Executive Summary : | The scattering of surface water waves is crucial in science and engineering, particularly in Ocean Engineering and Naval Architecture. Traditional mathematical methods address these problems using linear theories of water waves. However, nonlinear differential equations for boundary value problems are more difficult to solve than linear ones. Liao developed the Homotopy Analysis Method (HAM), which is independent of small/large physical parameters. The study involving nonlinear wave propagation in finite depth of water and steady state resonant water waves is interesting due to their applications in ocean engineering. With the progress of HAM, the study aims to study three problems: 1. Studying surface water waves over an undulated topography using HAM instead of perturbation theory. This will investigate the behavior of the amplitude of the resonant wave and the convergence of the solution.
2. Investigating the resonance of finite amplitude steady state wave groups in finite depth water. This will examine the nonlinear interactions between wave groups and analyze their behavior in finite depth water. The resonance of finite amplitude steady state wave groups using HAM will be examined, and equilibrium steady state solutions will be derived.
3. Studying the diffraction of nonlinear water waves traveling beneath an elastic plate in deep water. This study aims to consider nonlinear interactions of water waves floating over an incompressible and inviscid fluid of arbitrary depth. The square residual error for the solution will be computed to confirm the validity of our results. |