Integral equation and allied methods for wave interaction with ocean structures in the presence of seabed undulation

Abstract

The study presented in this thesis is solely concerned with the solutions of the bound- ary value problems arising in a natural way while modelling a class of wave structure interaction problems in the areas of coastal and marine engineering. The objective of the present study is to solve the problems associated with surface wave interaction with

oating structures in the presence of bottom undulation. The seabed undulations such as concave, convex, parabolic, trapezoidal, triangular and trench type pro les are con- sidered. The oating structures considered in this thesis are rigid and elastic in nature. The emphasis is being given for (i) developing di erent mathematical techniques such as integral equation method based on Havelock's expansion formula, eigenfunction expan- sion method along with the method of algebraic least squares, method of singular value decomposition and method of step approximation for solving the problems and (ii) inves- tigating the role of various system and structural parameters involved in the scattering problems. In the rst part of this thesis, the problems of propagation of oblique incident surface water waves over a single trench as well as a pair of trenches in a channel of nite depth are examined for their approximate solutions. In the case of trench type bottom topography, the singularity in the ow near each edge of each trench is considered. In the later part of the thesis, the problems involving di raction of surface water waves by oating structures especially rigid as well as elastic in the presence of arbitrary bottom topography are studied. By assuming the uid is inviscid and incompressible and the ow is irrotational, mixed boundary value problems (bvps) arise based on the linear and small amplitude wave the- ory. Due to the oblique incidence of waves, the governing partial di erential equation happens to be Helmholtz equation with mixed boundary condition at the free surface, condition at the bottom topography and conditions on the structures. As the uid region extends to in nity, one more condition arises namely, the far- eld condition to ensure the uniqueness of the problem. The solutions of bvps are utilized to determine the physi- cal quantities, namely, the re ection and transmission coe cients in each problem. The variation of these coe cients against the various system and structural parameters are analyzed and depicted through di erent graphs and tables. Also, the free surface eleva- tions are plotted for various values of system parameters. An important relation, namely, the energy balance relation is derived with the help of the Green's integral theorem. This relation ensures the correctness of the numerical results for re ection and transmission coe cients. Also, the behaviour of hydrodynamic force on the structure is investigated and depicted graphically. In some of the problems, the force experienced by seawall which is situated at a nite distance from the oating structure, is calculated. The results of the present study are validated with the known results available in the literature for particular cases.