Application of various methods in studying wave propagation over different bathymetries

Abstract

Scattering and resonance in water waves are fundamentally important for both scienti c understanding and practical applications in oceanography, coastal engineering, and environmental uid mechanics. This thesis investigates the scattering and resonance behavior of surface and internal water waves interacting with non-uniform bathymetries. Studying wave scattering over non-uniform bathymetries is not only scienti cally rich but also practically indispensable. Few problems involves the presence of oating structures both in the absence and presence of bathymtery. This allows for employing a range of mathematical techniques in order to solve the boundary value problems arising in various problems considered here. The analysis of oating structures in conjunction with non-uniform bathymetries provide a comprehensive framework to understand and design advanced coastal protection systems, e cient wave energy harvesting setups, realistic models for wave-induced transport and predictive tools for o shore structural stability. Focusing on non-uniform bottom con gurations, as well as single, two, and three-layer uid systems, the study presents a uni ed approach to understanding wave dynamics in realistic oceanographic environments. These wave interactions are crucial for understanding wave energy dissipation, re ection, and transmission patterns, which directly impact coastal erosion, sediment transport, and the stability of maritime structures. Under the umbrella of linearised water wave theory along with certain assumptions, the problems in each chapter results into a mixed boundary value problem. The complexity in these boundary value problems requires sophisticated mathematical and computational techniques, and this study employs several advanced methods like Multi-scale analysis, Boundary element method, Eigenfunction expansion method and Integral equation method to analyze and solve these mixed boundary value problems arising from the water wave scattering problems. The thesis begins with the analytical modeling of Bragg resonance phenomena using multi-scale perturbation analysis for non-uniform seabeds. In another work, we consider the Class II Bragg resonance which requires the bottom to be the combination of two sinusoids resembling a uneven bathymetry. The study then transitions to scattering by array of non-uniform rigid barriers over

at seabed, utilizing Eigenfunction expansion method and Boundary element method. Various arrangements of the barriers with unequal draft are investigated to optimize wave re ection, revealing that monotonically varying bar con gurations signi cantly enhance wave attenuation through Bragg scattering. We also examine another study when a rigid barrier is present over non-uniform oscillatory seabed in order to see their combined e ect of wave re ection and transmission. A further extension to step-type bathymetries which represents the continental shelves is taken. Integral equation approach followed by Galerkin techniques are employed which incorporate the e ect of cube root singularity at the edge of the corner and then further approximated by general polynomial whose weight function can be chosen suitably depending upon the above singularity present in the system. Same technique is employed to the problem involving a rigid thin barrier over a step type of bottom where both cube root singularity at the edge of the step and square root singularity at the edge of barrier are taken into account. Re ection and transmission coe cients are obtained for obliquely incident waves. The thesis also addresses internal wave dynamics and Bragg resonance in strati ed two-layer systems, analyzing weakly nonlinear e ects and interfacial mode behavior over non-uniform oscillatory bottoms using the method of perturbation and followed by Fourier transform technique. Lastly, in a shift from scattering to transport phenomena, uid particle trajectories in a three-layer system are studied under the in uence of interfacial tension and density strati cation. The analysis reveals non-monotonic trajectory behavior across layers, in uenced by varying density ratios and interface interactions. The closed trajectories are analysed using small-excursion principle. In all problems, results are validated by comparing with available results as far as possible which proves the e ectiveness of the present models. Also, in all the problem, the numerical results are plotted through graphs to analyze the e ect of di erent system and wave parameters.