Quasi-linear hyperbolic and surface waves

准线性双曲波和表面波

• 批准号: 1312342
• 负责人: John Hunter
• 金额: $ 26.63万
• 依托单位: University of California-Davis
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-08-01 至 2016-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1312342&HistoricalAwards=false
• 关键词: Quasi; linear; hyperbolic; surface; waves

This project addresses the mathematical modeling and analysis of nonlinear wave propagation in a variety of physical systems. It focuses on nondispersive waves, especially surface waves that propagate on interfaces such as discontinuities in vorticity, vortex sheets, material boundaries, water waves, and shock waves. Many of the wave motions considered in the proposal have constant, nonzero frequency in the linearized limit. These waves form a comparatively little studied class of nondispersive waves, and the proposed research aims to develop an understanding of their nonlinear dynamics, which is qualitatively different from that of dispersive waves or nondispersive hyperbolic waves. The proposed research will derive and study asymptotic descriptions of these waves and will also develop normal form transformations for quasi-linear wave equations. Hamiltonian dynamics provides unifying framework for most of the nonlinear wave motions to be studied in the proposed research. For small-amplitude waves, this description is more easily carried out in spectral form, which is particularly appropriate for the surface waves considered in the proposed research because of the spatial nonlocality of their interactions. The issue of understanding the relationship between the spectral and spatial descriptions of the resulting nonlinear dynamics is a fundamental one and one that is relevant to many other problems. A further topic of the proposed research is a study of the glancing Mach reflection of shock waves. Shock reflection is one of the most important multi-dimensional problems for hyperbolic conservation laws, leading to remarkably interesting and complex phenomena.These results should also shed light on related problems in transonic aerodynamics.Surface waves are waves that propagate along a boundary or interface. The most familiar example consists of the water waves on the surface of a body of water, like an ocean. Another type of surface wave consists of the Rayleigh waves on a solid interface. These waves are generated by earthquakes, and they are also used in technological applications, such as ultrasonic surface acoustic wave devices in cell phones. A further example consists of the electromagnetic surface waves, or surface plasmons, on the interface between a metal and an insulator, which find applications in photonics. Small-amplitude waves are well-described by linear equations, but at larger amplitudes nonlinear effects become important. These effects lead to qualitatively new phenomena such as wave-breaking, the formation of shock waves or other singularities, and the generation of new waves by nonlinear wave-interactions. Nonlinearity, and the possibility of a free surface that moves with the wave, makes the mathematical analysis of these problems very challenging. An additional feature of surface waves is that the effects of nonlinearity may be nonlocal because what happens at one point on the surface can influence what happens elsewhere on the surface through the bulk medium. The principal investigator plans to study the fundamental qualitative properties of such surface waves in the context of a wide variety of physical problems. The results will have potential applications in fluid dynamics, transonic flow, elasticity, magnetohydrodynamics, geophysics, and condensed matter physics.

该项目致力于对各种物理系统中的非线性波传播进行数学建模和分析。它侧重于非色散波，特别是在界面上传播的表面波，如涡度、涡片、材料边界、水波和激波中的不连续性。建议中考虑的许多波动在线性化极限中具有恒定的、非零的频率。这些波形成了一类研究相对较少的非色散波，所提出的研究目的是加深对其非线性动力学的理解，这与色散波或非色散双曲波有质的不同。拟议的研究将推导和研究这些波的渐近描述，并将发展准线性波动方程的规范型变换。哈密顿动力学为拟研究的大多数非线性波动问题提供了统一的框架。对于小幅度波，这种描述更容易以频谱形式进行，这特别适合于拟议研究中考虑的表面波，因为它们相互作用的空间非定域性。理解由此产生的非线性动力学的频谱和空间描述之间的关系是一个基本的问题，也是一个与许多其他问题相关的问题。这项研究的另一个主题是激波掠过马赫反射的研究。激波反射是双曲型守恒律中最重要的多维问题之一，导致了非常有趣和复杂的现象。这些结果也应该有助于揭示跨音速空气动力学中的相关问题。表面波是沿着边界或界面传播的波。最常见的例子是水面上的水波，就像海洋一样。另一种面波由固体界面上的瑞利波组成。这些波是由地震产生的，它们也被用于技术应用，例如手机中的超声波表面声波设备。另一个例子是金属和绝缘体之间界面上的电磁表面波，或表面等离子激元，它在光子学中有应用。小幅度波可以用线性方程很好地描述，但在较大幅度时，非线性效应变得重要。这些效应导致了质的新现象，如波破裂、激波或其他奇点的形成，以及通过非线性波相互作用产生新的波。非线性，以及自由表面随波移动的可能性，使得对这些问题的数学分析非常具有挑战性。面波的另一个特征是，非线性的影响可能是非局部的，因为表面上某一点发生的事情会通过主体介质影响到表面上其他地方发生的事情。首席研究员计划在各种物理问题的背景下研究这种表面波的基本性质。这些结果将在流体动力学、跨音速流动、弹性力学、磁流体力学、地球物理和凝聚态物理中有潜在的应用。