Abstract
The rainbow reflection effect describes the gradual slowing down of a wave and the simultaneous spatial separation of its spectral components, which occurs in spatially graded arrays of resonators. Although it has been studied more frequently in the context of electromagnetic and acoustic metamaterials, the potential applications of rainbow reflection to water waves are promising. In particular, the underlying principles could be used to design smart coastal technologies, such as wave energy parks for electricity generation or breakwaters for coastal protection.
Part I of this thesis investigates the rainbow reflection of water waves in a two-dimensional fluid (one horizontal and one vertical) by arrays of surface-piercing vertical barriers, in which the submergence depth of the barriers is graded. The problem is studied using linear water wave theory, and time-harmonic fluid motions are assumed. The rainbow reflection effect arises naturally in the graded array of vertical barriers, as wave energy becomes amplified at different locations depending on frequency. A local Bloch wave approximation (LBWA), which assumes that the wave can be locally represented as a superposition of propagating wave solutions of the cognate infinite periodic media (the so-called Bloch waves), is developed and implemented numerically. The LBWA accurately predicts the free surface elevation at most frequencies, and large errors occurring at certain frequencies are shown to be a consequence of our implementation of the LBWA, rather than a consequence of the underlying assumptions. This indicates that it is valid to locally describe the wave as a superposition of propagating Bloch waves. Band diagram calculations are used to show that the local Bloch waves gradually slow down throughout the array, leading to local energy amplification.
In order to demonstrate the potential applications of rainbow reflection to water wave energy conversion, Part I also introduces a model of a wave energy converting device, in which heave-restricted, rectangular floating pistons equipped with a linear power take-off mechanism are positioned between each adjacent pair of vertical barriers in a graded array. This model was chosen in light of the low-frequency resonant mode of a pair of vertical barriers without a piston, which consists of a vertical fluid motion. Constrained optimisation techniques are used to determine the parameters of the wave energy converting device so that near-perfect energy absorption can be achieved (i) over a discrete set of frequencies and (ii) over a continuous frequency interval.
Part II of this thesis pivots to time-domain wave scattering. Two methods for solving time-domain problems are considered, namely the generalised eigenfunction expansion method (GEM) and the singularity expansion method (SEM). The GEM expresses the time-domain solution in terms of the frequency-domain solutions, whereas the SEM expands the time-domain solution over the discrete set of unforced, complex resonant modes of the scatterer. The normalisation of the complex resonant modes is achieved by regularising divergent integrals (i.e.\ by analytic continuation). The GEM and SEM are implemented numerically and applied to canonical problems in one and two dimensions. In particular, we consider the scattering of waves on a stretched string by a point mass, and the scattering of two-dimensional acoustic waves by bounded scatterers. The results show that while the SEM is usually inaccurate at t=0, it converges rapidly to the GEM solution at all spatial points in the computational domain, with the most rapid convergence occurring inside the resonant cavity.