Green Functions of One-Dimensional Thermal-Wave Fields

Abstract

In this chapter, the formalism developed in the Introduction will be applied to the derivation of one-dimensional Green functions for a number of fundamental geometric configurations, which are of theoretical interest and of experimental relevance. As was discussed in the Introduction, Green functions are field distributions in response to an impulsive source inside, or on the boundaries of, the spatial region of interest. The mathematical analysis will proceed pedagogically in the form of detailed derivations of building-block Green functions, starting with the simplest case of the infinite one-dimensional “solid” or “medium.” The material presentation is intended to familiarize the reader with the mathematical methodologies for such derivations, so that he or she will be able to use them more broadly for his or her own purposes. Besides the pedagogy, the resulting formulas are intended to be of use to researchers in the field of thermal-wave science for the analysis of experimental data and measurements associated with them. The structure of this chapter is such that it forms the background for Chapter 2, where a number of Green functions will be used for the derivation of one-dimensional thermal-wave fields due to distributed sources. It is assumed throughout that the material thermal properties are independent of coordinate. Linearity will also be assumed, in that the temperature rise due to the application of thermal-wave sources is well below the threshold where the material properties may depend on it.