Dynamical Models for the Interaction of Shocks with Dispersive Waves

冲击与色散波相互作用的动力学模型

• 批准号: 9800797
• 负责人: Paul Newton
• 金额: $ 6.19万
• 依托单位: University of Southern California
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1998
• 资助国家: 美国
• 起止时间: 1998-05-01 至 2001-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9800797&HistoricalAwards=false
• 关键词: Dynamical; Models; Interaction; Shocks; Dispersive

In this work, the awardee will study models for the propagation of shock waves through a dispersive environment. These kinds of nonlinear, multi-mode interactions occur in a wide range of natural settings. One example arises when acoustic shocks from an underwater explosion propagate through the ocean and interact with internal dispersive waves or surface gravity waves. Another example arises in medical applications where acoustic waves and shocks are used in the destruction of undesirable objects such as kidney stones. A third arises when shock waves due to supernova explosions propagate through the interstellar plasma media. Despite the fact that these three physical settings are characterized by vastly different spatial scales and disparate physical mechanisms, there are important underlying mathematical similarities that unite them. Each is characterized by a multi-component amplitude equation governing a fast-moving hyperbolic wave (shock), interacting nonlinearly with a slowly-moving dispersive wave. This wide difference of timescales, characterized by the ratio of dispersive group velocity to shock speed, forms a small parameter which allows one to perform a multi-scale asymptotic analysis of the interaction process. The physical problem that the awardee will focus on is an example of a shock-dispersive wave system that arises when a supersonic object travels over the ocean surface. The shock wave associated with a sonic boom leaves a pressure `footprint' on the water's surface. The interest is in modelling the interaction of this shock with the nonlinear ocean surface waves in order to understand the penetration pressure below the ocean surface. It is suspected that the finite amplitude surface waves can act as focusing `acoustic lenses', which in principle can increase the noise level below the surface and create `hot spots' of more intense acoustic energy. Such problems are of interest to oceanographers and marine biologists who are concerned with the amount of noise propagating below the ocean surface and its influence on marine life. The model equations that will be studied are interesting in their own right and capture important physical mechanisms requiring new mathematical tools to analyse and understand them.

在这项工作中，获奖者将研究冲击波在弥散环境中传播的模型。这些非线性的、多模式的相互作用发生在广泛的自然环境中。当水下爆炸产生的声波冲击在海洋中传播并与内部色散波或表面重力波相互作用时，就会出现一个例子。另一个例子出现在医疗应用中，声波和冲击被用于破坏不需要的物体，如肾结石。第三种是由于超新星爆炸产生的冲击波在星际等离子体介质中传播时产生的。尽管这三种物理环境具有截然不同的空间尺度和不同的物理机制，但它们之间存在重要的潜在数学相似性。每一个特征都是一个多分量振幅方程，它控制着一个快速移动的双曲波（激波），与一个缓慢移动的色散波非线性相互作用。这种以色散群速度与激波速度之比为特征的时间尺度差异形成了一个小参数，使人们能够对相互作用过程进行多尺度渐近分析。获奖者将关注的物理问题是当超音速物体在海面上飞行时产生的冲击波色散波系统的一个例子。与音爆相关的冲击波会在水面上留下压力“足迹”。我们感兴趣的是模拟这种冲击与非线性海洋表面波的相互作用，以便了解海洋表面以下的穿透压力。人们怀疑，有限振幅的表面波可以作为聚焦的“声透镜”，原则上可以增加表面以下的噪音水平，并产生更强声能的“热点”。海洋学家和海洋生物学家对这些问题很感兴趣，他们关注在海洋表面以下传播的噪音量及其对海洋生物的影响。将要研究的模型方程本身就很有趣，并且捕获了需要新的数学工具来分析和理解它们的重要物理机制。