Green Functions in Three- and Two- Dimensional Cartesian Thermal-Wave Fields

Abstract

This chapter introduces the necessary mathematical formalism for developing expressions for thermal-wave Green functions in Cartesian coordinates for use with three-dimensional (3-D) and two-dimensional (2-D) problems. The presentation starts with the calculation of the Green function for an infinite three-dimensional space. The importance of this one function as the building block for other Green functions in laterally infinite three-dimensional Cartesian spaces is highlighted by a double derivation of the Green function in terms of a temporal Fourier transform and by means of a complex contour formalism. Subsequently, Green-function formulations are naturally separated into two groups: those for laterally infinite geometries and those for geometries with finite orthogonal boundaries. Two-dimensional Green functions are treated as separate cases. Besides the mathematically unique behavior of two-dimensional thermal-wave Green functions (reminiscent of the distinctive behavior of propagating wave fields in even dimensions as compared to those in odd dimensions [Morse and Ingard, 1968, Chap. 7]), their study here also reflects their practical importance in thin-film and thin-layer thermal-wave physics. This chapter closes with the derivation of Green functions in three-dimensional geometries with edges or corners, an important family of boundary-value problems for applications, which cannot be treated directly by the methods advanced for laterally infinite or finite geometries.