Nonlinear Wave Interactions

非线性波相互作用

• 批准号: 9802713
• 负责人: Rodolfo Rosales
• 金额: $ 11.1万
• 依托单位: Massachusetts Institute of Technology
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1998
• 资助国家: 美国
• 起止时间: 1998-07-01 至 2002-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9802713&HistoricalAwards=false
• 关键词: Nonlinear; Wave; Interactions

The work supported under this grant will address problems in Nonlinear Wave Dynamics, which will be studied using a combination of analytic and numerical methods. For example: (I) Nonlinear interactions of large scale atmospheric waves in the Equator, which play a fundamental role in the global weather. A new mechanism for nonlinear interactions of the waves via topography and convected properties (e.g.: potential vorticity, temperature and humidity) will be investigated. (II) Interaction of nonlinear hyperbolic waves with media nonuniformities. These waves distort as they propagate, leading to the formation of shocks, which are strongly dissipative. Past research shows that nonuniformities or interactions with other waves can stop this process. For example, a large class of new solutions to the Euler equations of Gas Dynamics was found and will be studied. These solutions are attracting for the time evolution and can achieve large pressure variations without breaking and forming shocks. Their existence gives rise to interesting mathematical issues. A better understanding of this phenomenon could provide means to control (or at least diminish) the influence of shocks in some gas flows. (III) Focusing, reflection and refraction of weak shocks. Strong acoustical waves and weak shocks are ubiquitous. A better understanding of their behavior has considerable scientific significance and will shed light into many important practical problems. Many puzzling problems remain in this area. For example, in many situations experiments show three weak shock waves joining at a point in a nearly flat configuration, but theoretical calculations indicate that this is not possible, leading to a contradiction with the experiments. A second related problem occurs with the prediction of infinities in the amplitudes of weak shock waves as they undergo focusing, reflection or refraction under appropriate conditions.Waves occur in many fields --- such as Acoustics, Atmosphere and Ocean science, Gas Dynamics, Optics, Combustion Theory, etc. --- where they fundamentally affect the dynamics. Many advances have been made in the last few decades, but many open questions remain,both concerning the applications and the mathematics that is used. This award will support research on mathematical models for wave-like phenomena in the global weather, processes in gas dynamics that avoid the formation of shock waves, and the mathematical modeling of complex shock wave interaction.

该基金支持的工作将解决非线性波动动力学问题，将使用解析和数值方法相结合的方法进行研究。例如：(1)赤道大尺度大气波的非线性相互作用，它在全球天气中起着重要作用。一个新的机制的非线性相互作用的波通过地形和对流性质（如：位涡，温度和湿度）将被研究。（II）非线性双曲波与介质不均匀性的相互作用。这些波在传播过程中会发生扭曲，从而形成强烈耗散的冲击。过去的研究表明，不均匀性或与其他波的相互作用可以阻止这一过程。例如，气体动力学欧拉方程的一大类新解被发现并将被研究。这些解决方案在时间演变方面具有吸引力，并且可以在不破坏和形成冲击的情况下实现大的压力变化。它们的存在引起了有趣的数学问题。更好地理解这一现象可以提供控制（或至少减少）某些气体流动中冲击的影响的方法。（三）弱冲击的聚焦、反射和折射。强声波和弱冲击无处不在。更好地了解它们的行为具有相当大的科学意义，并将有助于解决许多重要的实际问题。这个领域还有许多令人费解的问题。例如，在许多情况下，实验表明三个弱激波在一个几乎平坦的结构中连接在一起，但理论计算表明这是不可能的，这导致了与实验的矛盾。第二个相关的问题发生在预测弱冲击波在适当条件下经过聚焦、反射或折射时振幅的无穷大。波浪出现在许多领域，如声学、大气和海洋科学、气体动力学、光学、燃烧理论等，它们从根本上影响动力学。在过去的几十年里已经取得了许多进步，但仍然存在许多悬而未决的问题，包括应用和所使用的数学。该奖项将支持对全球天气中类波现象的数学模型、避免激波形成的气体动力学过程以及复杂激波相互作用的数学模型的研究。