Mathematical Sciences: Analysis and Numerical Modeling of Wave Phenomena in Composite Materials

主要研究人员将研究周期性微结构介质中波动现象的长期行为。特别是，这项研究包括两条主要的调查路线。第一种是利用多尺度渐近性的分析方法，得到一种在色散效应很重要的情况下长期有效的介质理论。第二个是构造和实施数值格式，采用较大的网格来精确地描述波的精细结构及其整体性质。这样的研究在许多不同的领域都有应用，包括地震学、石油勘探和结晶学。波浪现象是自然界中常见而又重要的一部分。在过去的一个世纪里，数学家和流体动力学家发展出了相当准确的理论，描述了波在空气或水等均匀介质中的行为，这些理论已经通过广泛的实验室实验得到了验证。相比之下，我们对均匀介质中波的行为的了解非常有限，这在很大程度上是因为很难进行可靠的实验。例如，了解地球内部(地震)波的行为对任何地震理论都是至关重要的，但建立一个实验来测量如此复杂的现象几乎是不可能的。因此，必须求助于非均匀介质中的波的数学模型。在这项提案中，首席研究员将使用解析和数值技术来描述波在周期介质中的传播。

The principal investigator will study the long-time behavior of wave phenomena in media with periodic microstructure. In particular, the research consists of two major lines of inquiry. The first one is the use of analytical methods derived from multi-scale asymptotics to obtain an effective medium theory that is valid for long times when dispersion effects are important. The second one is the construction and implementation of numerical schemes that employ relatively large grids to describe accurately the fine structure of waves as well as their bulk properties. Research such as this has applications to many diverse areas, including seismology, oil exploration and crystallography. Wave phenomena are a familiar and significant part of the natural world. Over the last century mathematicians and fluid dynamicists have developed fairly accurate theories that describe the behavior of waves in homogeneous media such as air or water, theories that have been validated through extensive laboratory experiments. In contrast to this, our understanding of the behavior of waves in homogeneous media is very limited, due in large measure to the difficulty of performing reliable experiments. For example, understanding the behavior of (seismic) waves inside of the earth is crucial to any theory of earthquakes, and yet it is well nigh impossible to set up an experiment that would allow us to measure such complicated phenomena. Thus recourse must be had to mathematical models of waves in inhomogeneous media. In this proposal the principal investigator will use analytical and numerical techniques to describe the propagation of waves in periodic media.