Dr. Alan Whitman

Research

Over the past several years my main effort has been in the following problem area:

Acoustic Wave Propagation in Ducts Containing Random Media

This effort began as a study of how an acoustic pulse injected into a two dimensional duct, and filled with a medium whose speed of sound was randomly varying in space, would evolve as it propagated along the duct. Equations were derived for the modal coherences in the high frequency limit and numerical solutions were produced. In addition, asymptotic results were obtained for long propagation distances. Subsequent developments included extensions to three dimensions and the inclusion of wall absorption. More recently, on accounting for temporal as well as spatial fluctuations, backscattering was included in the model and its effect on the propagation was elicited, especially in the vicinity of a mode cutoff. All this work has been published in a series of papers that have appeared in the journal, Waves in Random Media, and are listed under Publications. At present we are looking at the problem of isotropic scattering and are trying to find solutions.

The following is a short description of a problem I am working on currently:

Asymptotic Theory of an Infectious Disease Process

An infectious disease process in a human body can be modeled as a preditor-prey system between a pathogen population and an immune system. This kind of system of nonlinear differential equations has been studied for many years and a number of things are known about it. However, due to the nonlinearity few analytic solutions are available. We are analyzing a particular system that has been proposed, using asymptotic techniques, in order to obtain an approximate, yet numerically accurate, solution that could be used to generate insights about the system behavior, organize data, and suggest new measurements. A first paper has been published and we are now looking at some extensions.

The following is a description of another recent project, the entire paper is available here as a PDF file:

A Simple Expression for the Equation of Time

The equation of time quantifies the difference between time as measured by a sundial and time as measured by a mechanical clock. That this difference exists has been known since astronomers began making these measurements and comparing them. Moreover, since Newton showed how planetary orbits could be analyzed mathematically, it has been possible to calculate the difference as well. However, the calculation is complex and can not be represented by a simple explicit analytical expression. The purpose of this study is to present some asymptotic approximations to the known basic relations that produce a simple explicit formula for the equation of time that is accurate enough to be useful.