Nonlinear Wave Resonances in Bounded 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. In some cases (such as lasers, and musical instruments) such effects are desired, and in other cases (such as fluid transport and engine design) they are not. While the immediate mechanism is well understood, the long term effects of these interactions can be quite varied and depend on a multitude of factors. Driven by a range of applications, a robust literature has emerged over recent decades to model and mathematically investigate resonant phenomena. With new analytic tools has come new insights and refinements which in turn has driven demand for further detail from the underlying models and techniques of analysis. This positive feedback loop has provided fruitful insights for both application and analysis and it has lead to the development of an array of techniques. However while these provide insights and solution descriptions for the applications at hand, they can miss underlying mathematical connections. This proposal is divided into two main themes. The first theme is dedicated to the development of analytic methodologies which are based on the underlying mathematical structure and as such potentially tie together disparate regimes. Specifically the focus will be on the transition between cases where the higher harmonics are themselves resonant, to cases where they are not. In the case of acoustic waves this can correspond to, for example, slight variations in geometry of the resonator itself. In conjunction with this, the second theme of this proposal ties together two two prior bodies of work under the auspices of two previous NSERC Discovery Grants. It is focused on the application of these methodologies in both dispersive (e.g sloshing of a shallow fluid) and non-dispersive contexts (e.g. gas in a resonant cavity). In so doing potentially powerful connections between seemingly disparate physical regimes, and corresponding insights, will be made. In particular, questions related to the resonant response of a gas in a tube in the finite rate regime, as well as the impact of temporal modulation in the forcing will be addressed. The impacts of this work will lie both in the development of mathematical tools for analysis of resonant phenomena across a broad range of applications, as well as for the applications themselves. 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.