Nonlinear Wave Resonances in Continuous Media

连续介质中的非线性波谐振

• 批准号: RGPIN-2014-05401
• 负责人: Amundsen, David
• 金额: $ 1.02万
• 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: 2016
• 资助国家: 加拿大
• 起止时间: 2016-01-01 至 2017-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=611620
• 关键词: Nonlinear; Wave; Resonances; Continuous; 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. 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.

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。虽然直接的机制已被充分理解，但这种相互作用的长期影响可能会很多样化，并取决于多种因素。近几十年来，出现了强大的文献，以模拟和调查共鸣现象，但仍然存在许多基本问题。鉴于存在分散效应时出现的独特数学特征，该建议具有两个主要和平行的主题，一个主题存在分散效应（例如非线性荡妇），而一个不存在的分散效应（例如，声音谐振器）。 该提案的重点将放在气体和流体环境中的共振上，因为这些代表了可拖动和广泛适用的制度。对于非分散案例，基于Euler方程无关流的保护原理，将对各种系统进行建模和研究。主要兴趣的将是长期响应的定性性质如何取决于潜在的几何和材料（密度）曲线。众所周知，在某些情况下（例如电击管）会产生不连续的响应，而在其他情况下（例如球形谐振器）谱图虽然放大，但仍保持连续。这些情况之间过渡的性质尚不清楚。本提案的一个关键目标是通过研究和表征将其联系起来的过渡制度来解决这个基本问题。这将涉及对溶液结构和渐近尺度的详细分析，从而进一步了解基本机制。鉴于每个制度都需要独特的数学处理，这还将涉及渐近技术的发展和扩展，以建立这种联系。一旦解决了这些基本问题，将扩展模型，包括控制和燃烧等其他效果，并立即应用在各种工业环境中的设计策略，例如发动机，泵和压缩机。 在分散侧，本提案的主要重点将放在浅流体的非线性晃动上。先前的工作和随后的概括为稳态周期性反应提供了一个渐近框架，即分散效应很小。但是，实验表明，尽管一维效应确实在两个和三维设置中持续存在，但仍有很大的局限性。因此，本研究的第一个目标是将先前的结果扩展到更高的维度，以了解在更一般的几何环境中了解共振反应。反过来，这将涉及基于层的渐近方法进一步改进，特别注意基础分叉结构和附近分叉点附近的潜在分解。基于这些基本调查，最终的目标将是扩展这项工作，以允许更广泛的投球和飙升动作，例如，重要的应用于货船和油轮的设计和操作。 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。