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等领域中物理现象的一个基本方面。它们的一般特征是外部强迫和自然内部频率之间的建设性相互作用。在某些情况下（例如激光和乐器）需要这种效果，而在其他情况下（例如流体输送和发动机设计）则不需要。虽然直接机制已被充分理解，但这些相互作用的长期影响可能差异很大，并且取决于多种因素。 在一系列应用的推动下，近几十年来出现了大量用于对谐振现象进行建模和数学研究的文献。新的分析工具带来了新的见解和改进，这反过来又推动了对底层模型和分析技术的更多细节的需求。这种正反馈循环为应用和分析提供了富有成效的见解，并导致了一系列技术的发展。然而，虽然这些为当前的应用程序提供了见解和解决方案描述，但它们可能会错过潜在的数学联系。 该提案分为两个主题。第一个主题致力于分析方法论的发展，这些方法论基于底层的数学结构，因此可能将不同的体系联系在一起。具体来说，重点将放在高次谐波本身谐振的情况与不谐振的情况之间的过渡。在声波的情况下，这可以对应于例如谐振器本身的几何形状的轻微变化。 与此相结合的是，该提案的第二个主题将在之前两项 NSERC 发现拨款资助下的两项先前工作联系在一起。它重点关注这些方法在色散（例如浅层流体的晃动）和非色散环境（例如谐振腔中的气体）中的应用。通过这样做，看似不同的物理状态和相应的见解之间将建立潜在的强大联系。特别是，将解决与管中气体在有限速率范围内的共振响应以及强迫中的时间调制的影响相关的问题。 这项工作的影响将在于开发用于分析广泛应用中的谐振现象的数学工具，以及应用本身。色散和非色散方面都将涉及建模、分析和数值模拟的结合。他们将为从高年级本科生到博士和博士后级别的学生提供充足的参与机会。它们不仅可以提供对潜在共振机制和结果的基本见解，而且还提供了在一系列工业环境中直接应用的机会。