Nonlinear Wave Resonances in Bounded Media

有界介质中的非线性波共振

• 批准号: RGPIN-2019-06169
• 负责人: Amundsen, David
• 金额: $ 1.24万
• 依托单位: Carleton University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=715954
• 关键词: Nonlinear; Wave; Resonances; Bounded; Media

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.

共振是从流体力学到声学再到光学的环境中物理现象的基本方面。它们通常的特征是外部强迫和天然内部频率之间的建设性相互作用。在某些情况下（例如激光器和乐器）是必需的，而在其他情况下（例如流体运输和发动机设计）则不是。虽然直接的机制已经充分理解，但这些相互作用的长期影响可能会很多样化，并取决于多种因素。 在一系列应用的驱动下，近几十年来出现了一项强大的文献，用于建模并数学研究共振现象。有了新的分析工具，就有新的见解和改进，这反过来促使人们从基础模型和分析技术中获得了进一步的细节需求。这种积极的反馈回路为应用和分析提供了富有成果的见解，并导致了一系列技术的发展。但是，尽管这些为手头应用提供了见解和解决方案描述，但它们可能会错过基本的数学连接。 该提议分为两个主要主题。第一个主题是针对基于基本数学结构的分析方法的发展，并可能将这种方法融合在一起。具体而言，重点将放在较高谐波本身是共鸣的案例之间的过渡上，而不是它们不存在的情况。在声波的情况下，这可以对应于例如谐振器本身几何形状的微小变化。 结合了这一点，该提案的第二个主题将两个先前的工作主体联系在一起，这是两个先前的Nserc Discovery Grants的主持人。它的重点是这些方法在分散性（例如浅流体的晃动）和非分散性环境（例如谐振腔中的气体）中的应用。因此，将在看似不同的物理制度和相应的见解之间建立潜在的强大联系。特别是，将解决与管子中气体在有限速率制度中的共振响应以及强迫中时间调制的影响有关的问题。 这项工作的影响将在于开发数学工具，用于分析广泛的应用以及应用本身的共鸣现象。分散性和非分散性方面都将涉及建模，分析和数值模拟的融合。他们将负担得起，需要充分的机会，从而使学生参与从高级本科到博士和博士后水平。他们还将不仅提供有关基本共振机制和结果的基本见解，而且还将在一系列工业环境中直接应用。