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.
