Surface and flexural gravity wave scattering with BRAGG resonance: linear and nonlinear analyses

Abstract

This thesis presents a comprehensive study of a class of water wave scattering problems relevant to Ocean and Polar Engineering, with a focus on wave interaction with bottom topographies and floating elastic plates. The findings have direct applications in the design of resilient coastal structures, optimization of offshore platforms, and the assessment of wave-induced stresses on ice sheets. These investigations are particularly motivated by the need to understand wave behavior in regions with complex seabed geometry, stratified fluids, and ice-covered oceans, with applications to coastal protection, offshore infrastructure, and climate-responsive engineering. The fluid is assumed to be inviscid, incompressible, and undergoing irrotational motion. Further, the flow is assumed to be harmonic in time. Assumption of small amplitude waves allows for the use of linearized water wave theory, and gravity acts as the primary restoring force. In certain problems involving ice sheets, the nonlinear effects are considered through the Homotopy analysis method. The objective of this thesis is to examine a broad class of wave-structure interaction problems relevant to ocean and polar environments. The study begins with an investigation of wave scattering by an asymmetric trench in a homogeneous fluid, followed by an analysis of wave interaction with periodic bottom topography in a two-layer stratified fluid, accounting for surface and interfacial tension as well as a uniform background current. It then explores hydroelastic wave interaction with sinusoidally varying elastic plates representing spatially non-uniform ice sheets. Further, the thesis analyzes Bragg resonance phenomena arising from non-periodic ice geometries such as Gaussian and Gaussian oscillatory profiles in the presence of current. Finally, it examines the nonlinear interaction of waves with a variable ice sheet and the formation of class-I Bragg resonant waves in ice-covered fluid using the Homotopy Analysis Method (HAM). Each physical configuration is modeled by formulating appropriate boundary value problems. These involve Laplace’s equation or modified Helmholtz’s equation as the governing equation for the velocity potential, subject to linearized free surface and interface conditions, impermeable bottom conditions, and hydroelastic boundary conditions based on thin plate theory for floating elastic plates. In stratified media, continuity of pressure and velocity is enforced at the fluid interface. The Homotopy Analysis Method is used in the last problem to capture nonlinear equilibrium-state resonant waves beyond the reach of perturbation techniques. The analytical and semi-analytical methods employed include Takano’s approach, eigenfunction expansions, the Fourier transform technique, the asymptotic method, and HAM. Numerical solutions are obtained using MATLAB and Mathematica. The study presents detailed numerical results for reflection and transmission coefficients, free surface elevation, velocity potentials, plate deflection, and wave energy distributions. The variations of these physical quantities with system parameters, such as depth ratios, current speed, plate geometry, wavenumber, etc. are analyzed and graphically illustrated. The findings offer insight into standing waves, resonance amplification, and the impact of geometric and environmental factors on wave behavior. This work contributes meaningfully to the understanding and mitigation of wave effects on ice-covered and coastal environments, and has direct applications in the development of climate-resilient infrastructure in polar and oceanic regions.