Mathematical techniques for scattering of water waves by horizontal/vertical barrier(s) over different bottom topographies

Abstract

In this thesis, a detailed analysis of a class of water wave problems arising in Ocean and Marine Engineering due to water waves interaction with the barrier(s) over di erent type of bottom topographies is carried out. The physical problems associated with water wave propagation are modelled mathematically by utilizing the assumptions that the uid under consideration is homogeneous, inviscid, incompressible, and the motion of the uid is irrotational and harmonic in time. Further, the motion of the uid which is under gravity and the free surface deviation from its horizontal position are assumed to be small in the sense that the linearized theory of water waves can be utilized. The objective of the thesis is to give emphasis for a class of wave-structure interaction problems with signi cance being given for i) developing various numerical techniques for a class of physical problems associated with surface wave interaction with rigid barriers in presence of uneven bottom topography, and ii) investigating the inuence of various system parameters associated with the physical problems. Both the cases of horizontal and vertical barriers are considered in this thesis. On formulating the physical problems, the governing partial di erential equation comes about Laplace equation for the case of normal incidence of surface waves while it is Helmholtz equation for the case of oblique incidence of water waves. The boundary condition at the free surface i.e. at the air-water interface is of the Robin type and the impermeable boundary condition at the bottom is of Neumann type. In addition to this, far- eld conditions are imposed at innite uid boundaries to ensure the uniqueness of the solution. The boundary value problem involving the scattering of water waves by nite dock over stepped-type bottom pro le is solved by using eigenfunction expansions in conjunction with orthogonality of eigenfunctions. The problems of vertical barrier(s) over step type or shelf type bottom are solved by utilizing the eigenfunction expansions in conjunction with least-square method. The physical quantities, namely, re ection and transmission coe cients, force on the barriers and free surface elevation are calculated. The variation of these physical quantities against the various system parameters is presented and depicted through di erent graphs and tabular data. In the last part of the thesis, a di erent approach namely, the nite element based technique is used to solve the boundary value problem involving arbitrary topography at the bottom. The nite domain is constructed by truncating the radiation boundary conditions at some nite distance. The nite element formulation is done using weighted residual method. The solution of boundary value problem, the scattered velocity potential, is further utilized to determine the various physical quantities, namely, re ection and transmission coe cients, the force on the barriers. For each of the above physical problems, the energy balance relation is derived with the aid of Greens integral theorem and the veri cation of this identity ensures the accuracy of the present numerical results carried out for the physical quantities. In addition, the convergence of number of evanescent modes in the series expansions is performed numerically. Also, the convergence of the nite element analysis is computationally carried out. The present numerical results are also compared with the results available in the literature for validating the model. The present study is of immense importance in the eld of ocean and marine engineering towards the application of breakwaters.