Green Functions in Three- and Two- Dimensional Cartesian Thermal-Wave Fields
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.
KeywordsHeat Transfer Coefficient Green Function Homogeneous Boundary Condition Homogeneous Dirichlet Boundary Condition Infinite Domain
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- J. V. Beck, K. D. Cole, A. Haji-Sheikh, and B. Litkouthi, Heat Conduction Using Green’s Functions (Hemisphere, Washington, DC, 1992).Google Scholar
- H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids (Oxford University Press, Oxford, 1959).Google Scholar
- L. D. Favro, P-K. Kuo, and R. L. Thomas, in Photoacoustic and Thermal Wave Phenomena in Semiconductors (A. Mandelis, ed.), (North-Holland, New York, 1987), Chap. 4.Google Scholar
- I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (English Translation) (A. Jeffrey, ed.), (Academic, Orlando, Fl, 1980).Google Scholar
- P. M. Morse and K. U. Ingard, Theoretical Acoustics (McGraw-Hill, New York, 1968).Google Scholar
- W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Methods in C, 2nd Ed. (Cambridge University Press, Cambridge, 1992).Google Scholar