Off the beaten path: dispersive wave-current interactions on shear flows, and path-following methods
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The focus of this work has been the influence of depth-dependent currents on the properties of dispersive water waves, in furtherance of recent work by Ellingsen, et al. This has primarily comprised the development of: efficient numerical techniques, in particular a path-following strategy to calculate the numerical range of a matrix and the dispersion relation for dispersive water waves; extending previous results for free surface waves on an inviscid fluid to waves on a laminar viscous flow with arbitrary current profile; a preliminary derivation for higher-order expansions for free surface waves on inviscid fluids with arbitrary current profile; and, an analysis using stochastic numerical experiments to quantify the influence of measurement error on the dispersion relation arising from field measurements such as those using acoustic Doppler current profilometry equipment. The path-following method for calculating the dispersion relation is a notable result from this work and was predicated on the path-following method for calculating the numerical range of a matrix. In most cases, the path-following method is the fastest algorithm currently available for calculating inviscid linear wave dispersion relations for arbitrary depth-dependent velocity profiles. The key principle underpinning this approach is computationally treating the solution as a parametric algebraic curve instead of a collection of independent problems. Calculating for a single point on the curve using a pseudospectral collocation method yields a quadratic eigenproblem, which is amenable to solution using QZ decomposition on a size 2N companion matrix. The path-following algorithm can swap most of these QZ decompositions for linear solves on a size N matrix, representing a significant performance improvement. The other notable result is the analysis of the influence of measurement error on the dispersion relation, aimed at providing guidance to those undertaking field measurements. Subsurface current sensors produce results as discrete data points subject to error whereas all the numerical schemes for calculating the dispersion relation assume an exact smooth function: therefore, an approximation scheme must be used to obtain a smooth function from noisy data. The influence of Gaussian measurement error on several approximation schemes was simulated for various arrangements of sensors. One of the key results of this work is that, after a certain threshold, it is preferable to improve sensor accuracy than increase the number of sensors. Asymmetry in the results was also demonstrated. This thesis draws together the work, presents some new results, and provides context from the literature.