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: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=759233
• 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.

共振是物理现象的一个基本方面，从流体力学到声学再到光学。它们通常以外部强迫和内部固有频率之间的建设性相互作用为特征。在某些情况下（如激光和乐器）需要这种效果，而在其他情况下（如流体输送和发动机设计）则不需要。虽然直接的机制很好理解，但这些相互作用的长期影响可能是多种多样的，并取决于多种因素。在一系列应用的推动下，近几十年来出现了大量的文献来对共振现象进行建模和数学研究。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基础模型和分析技术的进一步细节的需求。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，尽管这些方法为手头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。这个建议分为两个主题。第一个主题致力于分析方法的发展，这些方法基于潜在的数学结构，因此可能将不同的制度联系在一起。具体来说，重点将放在高次谐波本身共振的情况和它们不共振的情况之间的过渡。在声波的情况下，这可以对应于，例如，谐振器本身几何形状的微小变化。与此相结合，本提案的第二个主题将两个先前在NSERC发现资助下的两个先前的工作机构联系在一起。它侧重于这些方法在色散（例如浅流体的晃动）和非色散环境（例如谐振腔中的气体）中的应用。在这样做的过程中，看似不同的物理机制之间的潜在强大联系，以及相应的见解，将被建立起来。特别地，有关在有限速率下管内气体的共振响应的问题，以及在强迫中的时间调制的影响将被解决。这项工作的影响将在于数学工具的发展，用于分析共振现象在广泛的应用范围内，以及应用本身。色散和非色散方面都将涉及建模、分析和数值模拟的混合。他们将为学生提供丰富的参与机会，从高级本科到博士和博士后水平。它们不仅将提供对潜在共振机制和结果的基本见解，而且还将提供在一系列工业环境中直接应用的机会。