Mathematical techniques for water waves scattering by structures in the presence of uneven bottom

Abstract

The study presented in this thesis is solely concerned with the solutions of the boundary value problems arising in a natural way while modeling the problems involving scattering of surface water waves. The objective of this thesis is to deal with a class of wave structure interaction problems arising in costal and marine science engineering with significance being given for (i) developing different mathematical techniques for the problems involving interaction of surface water wave with the rigid or porous structures in presence of uneven bottom, and (ii) investigating the role of various system parameters involved in the scattering problems. It is assumed that the fluid is inviscid, incompressible and the motion of the fluid is irrotational. Furthermore, the assumption of linear and time harmonic motion is considered. Firstly, the problems consisting of wave interaction with a fixed rigid or porous vertical barrier over undulating bottom is considered for their solution. Due to the angle of incidence, the problem is one of these kinds: one describing the scattering of water waves when the progressive wave incident normally to the vertical barrier and undulating bottom and the other when the progressive wave incident obliquely to the vertical barrier and undulating bottom. In each situation, the physical problem is formulated as a mixed boundary value problem (bvp) to determine the velocity potential and the physical parameters namely, the reflection and transmission coefficients. The governing partial differential equation happens to be the Laplace’s equation in two-dimensions in case of normal incidence where as the Helmholtz equation in threedimensions in case of oblique incidence with mixed boundary conditions at the free surface, conditions on the barrier, conditions in the gap and a condition at the bottom. As the fluid region extends to infinity, one more condition arises namely, the far-field condition to ensure the uniqueness of the problem. Various mathematical techniques such as perturbation analysis, eigenfunction expansion method along with matching technique, Green’s function technique, approach based on Green’s integral theorem, least-squares method and boundary element method are applied to solve different bvps arising in the study. The velocity potential, the solution of bvp, is utilized to determine the physical quantities, namely, reflection and transmission coefficients in each problem. The variation of these coefficients against the various physical parameters are analyzed and depicted graphically. In some problems, the energy identity, an important relation in the study of the water wave scattering problem, is derived with the help of the Green’s integral theorem. This identity ensures the correctness of the numerical results obtained for the physical quantities. The other important factors of the study namely, hydrodynamic forces and moments are also investigated and shown graphically. Secondly, the later part of this thesis is solely devoted to the investigation of the scattering of water waves by floating rigid structure placed horizontally on the free surface over an abrupt change in the bottom topography. The solution of these kind of scattering problems is determined analytically with the aid of matched eigenfunction expansion method and numerically using the boundary element method. The numerical results obtained in both methods are also compared and a good agreement is achieved. Here also, various physical phenomena associated with the wave scattering are analyzed. Furthermore, energy identity is derived and checked. For most of the physical problems studied in the thesis, the results of the present study are validated with the known results available in the literature. The present study is an endeavor to take the matters to another step forward towards a real and practical situation occurring in other areas of mathematical physics.