## Water Wave Scattering by a Spherical Structure and an Undulating Bottom Topogrphy in a Two-Layer Fluid

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2009

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This thesis studies (i) the interaction of water waves with spherical geometries in a two-layer fluid of finite depth, which is covered by either a rigid flat structure or a very thin ice shelf; (ii) the scattering of water waves by different types of unevenness on the bottom surface under such situations. To solve the ice-covered problems, the common idealization of ice as a thin elastic plate, which is static in all but its flexural response, is followed. Furthermore, the assumptions of linear and time harmonic motions are considered. Firstly, the problem consisting of wave interaction with a spherical body submerged in either layer of the fluid is divided into two parts: one describing the scattering of waves by the fixed structure and the other describing the radiation of waves by the body into otherwise calm water. The radiation problem is further split into a number of parts, each of which corresponds to the body moving in a separate mode of motion. The physical problem involving radiation or scattering case, is reduced to a boundary value problem governed by a three-dimensional LaplaceDs equation for both the upper and the lower layers. The method of solution for both the fluids is based upon the multipole expansions technique. The solutions help in calculating the hydrodynamic forces acting on the spherical body for different modes of motion such as heave and sway. A number of observations are made for these motions with regard to different submersion depths. The multipole expansion method is found to be an extremely powerful method for solving radiation and scattering problems for submerged spheres. It eliminates the need to use large and cumbersome numerical packages for the solution of such problems. Secondly, the latter part of this thesis is solely devoted to the investigation of the scattering of a train of small amplitude harmonic water waves by small bottom undulation of an ocean, which consists of a two-layer fluid, for both normal and oblique incidences. Moreover, it is assumed that both the fluids are of finite depth and the upper fluid is covered by either a rigidinvolving the shape function which represents the bottom undulation. Different special forms of the shape functions are considered to compute the integrals explicitly for the reflection and transmission coefficients and the results are suitably presented graphically. Out of these shape functions, the particular case of a patch of sinusoidal ripples (with the same wave number or two different wave numbers) has considerable significance due to the ability of an undulating bed to reflect incident wave energy which is important in respect of both coastal protection and of possible ripple growth if the bed is erodable. For this ripple patch, in the case of a channel bed assumed bounded above by a rigid boundary, it is observed that if the bed wave number is twice the interface wave number then there is a resonant Bragg-type interaction between the interface waves and the bed forms as observed earlier in the literature. Moreover, in case of an ocean-bed covered by an ice-cover, it is observed that when the wave is obliquely incident on the ice-cover surface we always find energy transfer to the interface, but for inter- facial incident waves there are parameter ranges for which no energy transfer to the ice-cover surface is possible. Problems related.

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Supervisor: S N Bora

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MATHEMATICS