Nonlinear Hyperbolic Waves

非线性双曲波

• 批准号: 0607355
• 负责人: John Hunter
• 金额: --
• 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: 2006
• 资助国家: 美国
• 起止时间: 2006-07-01 至 2010-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0607355&HistoricalAwards=false
• 关键词: Nonlinear; Hyperbolic; Waves

Waves are a phenomenon of fundamental physical and technological importance. For example, the observation of waves from distant sources (whether gravitational waves generated by the collision of black holes or sound waves generated by a submarine) allows one to make useful conclusions about the nature of those sources, and the manipulation of waves has many technological applications (such as the use of ultrasonic surface acoustic waves in signal processing devices). brbrNonlinear hyperbolic waves are an especially interesting and important class of waves. They include the shock waves generated by aircraft in transonic or supersonic flight, the seismic waves (and possible Tsunamis) generated by earthquakes, and the gravitational waves predicted by Einstein's general theory of relativity. brbrThe dominant properties of hyperbolic waves are described mathematically by hyperbolic systems of partial differential equations. In a first approximation, hyperbolic waves are non-dispersive, meaning that the velocity at which they propagate, and their other properties, do not depend on the wavelength of the waves. This contrasts with dispersive waves, such as light waves in an optical fiber or Rossby waves in the atmosphere. The effects of nonlinearity on hyperbolic waves have a unique character; for example, the formation of singularities, such as shock waves, is a ubiquitous phenomenon, and one with significant physical implications. brbrThe goal of the proposed research is to study nonlinear hyperbolic waves from a unified mathematical perspective in close connection with various physical applications. Specific topics include the reflection of shock waves and related questions in transonic fluid flows; the propagation of nonlinear hyperbolic surface waves, including Rayleigh (or surface acoustic) waves in elasticity; and the effect of nonlinearity on gravitational waves in general relativity.

波是一种具有基本物理和技术重要性的现象。例如，对来自遥远来源的波（无论是黑洞碰撞产生的引力波还是潜艇产生的声波）的观察使人们能够对这些来源的性质做出有用的结论，并且波的操纵具有许多技术应用（例如在信号处理设备中使用超声表面声波）。brbr非线性双曲波是一类特别有趣和重要的波。它们包括飞机在跨音速或超音速飞行中产生的冲击波，地震产生的地震波（和可能的海啸），以及爱因斯坦广义相对论预测的引力波。brbr双曲波的主要特性在数学上由双曲型偏微分方程组描述。在第一近似中，双曲波是非色散的，这意味着它们传播的速度以及它们的其他属性不取决于波的波长。这与色散波形成对比，例如光纤中的光波或大气中的Rossby波。非线性对双曲波的影响具有独特的性质;例如，奇点的形成，如冲击波，是一种普遍存在的现象，并且具有重要的物理意义。brbr拟议研究的目标是从与各种物理应用密切相关的统一数学角度研究非线性双曲波。具体的主题包括反射的冲击波和相关问题在跨音速流体流动;传播的非线性双曲表面波，包括瑞利（或表面声波）波在弹性;和非线性效应引力波在广义相对论。