Green Functions of Thermal-Wave Fields in Cylindrical Coordinates

Abstract

The importance of the cylindrical coordinate system in all kinds of diffusion-wave fields and their mathematical formulations is paramount due to the wide use of lasers as modulated sources which possess cylindrical spatial symmetry when emitting in the TEM₀₀ Gaussian mode. For thermal waves, the chapter introduces convenient operational theorems, which allow the extension of one-dimensional Green functions to cylindrical representations, in the case of azimuthal symmetries. Throughout the chapter, Green-function formulations are naturally separated into two groups: those for laterally infinite domains and those for geometries with finite cylindrical boundaries. The latter geometries include the thermal-wave Green functions of thin disks, and semi-infinite and finite-height cylinders with a variety of spatially impulsive sources allowed by the cylindrical geometry (points, rings, shells). These cases are followed by hollow cylinder geometries of finite height and their natural extension to composite concentric cylindrical domains of different thermophysical properties. Here, equivalence relations between the heat-transfer coefficient of homogeneous boundary conditions of the third kind and inhomogeneous conditions of continuity of thermal-wave field and flux will be derived and presented in a form of a Theorem (5.4) and a Lemma (5.1), as is done in Chapters 1 and 3. This chapter closes with the derivation of Green functions for cylindrical sectors and wedges as a separate family of boundary-value problems which require the introduction of special eigenvalue and eigenfunction sets.