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

首席研究员将研究长期行为 具有周期性微观结构的介质中的波动现象。 在 特别是，这项研究包括两个主要的调查路线。 第一种是使用分析方法， 多尺度渐近性，以获得有效介质理论， 在色散效应很重要的情况下长期有效。 二是构建和实施 采用相对较大的网格来描述 精确地描述了波的精细结构和体积 特性. 像这样的研究可以应用于许多 不同的领域，包括地震学，石油勘探和 结晶学 波动现象是电磁学中一个熟悉而重要的部分， 自然界 在过去的世纪，数学家和流体 动力学家已经发展出相当精确的理论， 波在均匀介质如空气或水中的行为， 经过大量实验室验证的理论 实验 与此相反，我们对 波在均匀介质中的行为非常有限， 很大程度上是由于执行可靠的 实验 例如，了解 地球内部的（地震）波对任何理论都是至关重要的。 地震，但几乎不可能建立一个 这个实验可以让我们测量如此复杂的 现象。 因此，必须求助于数学模型， 非均匀介质中的波 在这一建议中，主要 调查人员将使用分析和数字技术， 描述波在周期性介质中的传播。

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.