Nonlinear Wave Resonances in Continuous Media

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Resonances are a fundamental aspect of physical phenomena in settings ranging from fluid mechanics to acoustics to optics. They are generally characterized by a constructive interaction between external forcing and natural internal frequencies. While the immediate mechanism is well understood, the long term effects of such interactions can be quite varied and depend on a multitude of factors. A robust literature has emerged over recent decades to model and investigate resonant phenomena, but numerous fundamental questions remain. Given the distinct mathematical features which arise when dispersive effects are present, this proposal has two primary and parallel themes, one where dispersive effects are present (e.g. nonlinear sloshing) and one where they are not (e.g. acoustic resonators). The focus of this proposal will be on resonances in gas and fluid settings as these represent both tractable and broadly applicable regimes. For the non-dispersive case, based on the conservation principles embodied by the Euler Equations for inviscid flow, various systems will be modelled and studied. Of primary interest will be how the qualitative nature of the long term response depends on the underlying geometric and material (density) profiles. It is well known that in some cases (e.g. shock tubes) a discontinuous response arises, while in other cases (e.g. spherical resonators) the profile, while amplified, remains continuous. The nature of the transition between these cases is not well understood. A key objective of the present proposal will be to address this fundamental question by studying and characterising the transition regime linking them. This will involve detailed analysis of the solution structure and asymptotic scalings, leading to further insights into the underlying mechanisms. Given that each regime requires a distinct mathematical treatment, this will also involve development and extension of asymptotic techniques in order to establish this connection. Once these fundamental questions are resolved, the models will be extended to include additional effects such as control and combustion, with immediate application to design strategies in various industrial settings such as engines, pumps and compressors.On the dispersive side, the primary focus of the present proposal will be on nonlinear sloshing of shallow fluids. Prior work and subsequent generalizations provided an asymptotic framework for the steady state periodic response in the limit that dispersive effects are small. However experiments have shown that while one-dimensional effects do persist in two and three dimensional settings, there are significant limitations. Therefore the first objective of the present study will be to extend the previous results to higher dimensions with a view towards understanding of resonant response in more general geometric settings. This in turn will involve further refinement of the layer based asymptotic methodology with particular attention to the underlying bifurcation structure and potential break-down near bifurcation points. Based on these fundamental investigations, the final objective will be to extend this work to allow for more generalized pitching and surging motions, with important applications, for example, to the design and operation of cargo vessels and tankers. Both the dispersive and non-dispersive aspects will involve a blend of modelling, analysis and numerical simulation. They will afford and necessitate abundant opportunity for student involvement ranging from senior undergraduate to doctoral and postdoctoral levels. They will also not only provide fundamental insights into the underlying resonant mechanisms and outcomes, but also the opportunity for direct application in an array of industrial settings.