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

  • Andreas Mandelis


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.


Heat Transfer Coefficient Green Function Homogeneous Boundary Condition Homogeneous Dirichlet Boundary Condition Infinite Domain 
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  1. L. C. Aamodt and J. C. Murphy, Appl. Opt. 21, 111 (1982).ADSCrossRefGoogle Scholar
  2. V. S. Arpaci, Conduction Heat Transfer (Addison-Wesley, Reading, MA, 1966).zbMATHGoogle Scholar
  3. J. V. Beck, K. D. Cole, A. Haji-Sheikh, and B. Litkouthi, Heat Conduction Using Green’s Functions (Hemisphere, Washington, DC, 1992).Google Scholar
  4. R. Bellman, R. E. Marshak, and G. M. Wing, Phil. Mag. 40, 297 (1949).MathSciNetzbMATHGoogle Scholar
  5. R. Bracewell, The Fourier Transform and its Applications (McGraw-Hill, New York, 1965).zbMATHGoogle Scholar
  6. H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids (Oxford University Press, Oxford, 1959).Google Scholar
  7. 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
  8. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (English Translation) (A. Jeffrey, ed.), (Academic, Orlando, Fl, 1980).Google Scholar
  9. F. B. Hildebrand, Advanced Calculus for Engineers (Prentice-Hall, Englewood Cliffs, NJ, 1949].zbMATHGoogle Scholar
  10. E. Kreyszig, Advanced Engineering Mathematics, 7th Ed. (Wiley, New York, 1993).zbMATHGoogle Scholar
  11. E. MacCormack, A. Mandelis, M. Munidasa, B. Farahbakhsh, and H. Sang, Int. J. Thermophys. 18, 221 (1997).ADSCrossRefGoogle Scholar
  12. A. Mandelis, J. Opt. Soc. Am. A 6, 298 (1989)ADSCrossRefGoogle Scholar
  13. A. Mandelis, J. Phys. Math. Gen. 24, 2485 (1991).ADSzbMATHCrossRefGoogle Scholar
  14. A. Mandelis, J. Appl. Phys. 78, 647 (1995).ADSCrossRefGoogle Scholar
  15. A. Mandelis and J. F. Power, Appl. Opt. 27, 3397 (1988).ADSCrossRefGoogle Scholar
  16. P. M. Morse and H. Feshbach, Methods of Theoretical Physics, Vol. I (McGraw-Hill, New York, 1953).zbMATHGoogle Scholar
  17. P. M. Morse and K. U. Ingard, Theoretical Acoustics (McGraw-Hill, New York, 1968).Google Scholar
  18. B. Ploss and C. Albrecht, Ferroelectrics 165, 171 (1995).CrossRefGoogle Scholar
  19. 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
  20. A. Rosencwaig and J. Opsal, IEEE Trans. Ultrason. Ferroelectr. Freq. Control UFFC-33, 516 (1986).ADSCrossRefGoogle Scholar
  21. A. Sommerfeld, Arm. Phys. 28, 665 (1909).zbMATHGoogle Scholar
  22. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), p. 573.zbMATHGoogle Scholar
  23. R. L. Thomas, J. J. Pouch, Y. H. Wong, L. D. Favro, P. K. Kuo, and A. Rosencwaig, J. Appl. Phys. 51, 1152 (1980).ADSCrossRefGoogle Scholar
  24. P. R. Wallace, Mathematical Analysis of Physical Problems (Dover, New York, 1984).zbMATHGoogle Scholar

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© Springer Science+Business Media New York 2001

Authors and Affiliations

  • Andreas Mandelis
    • 1
  1. 1.Department of Mechanical and Industrial Engineering, Photothermal and Optoelectronic Diagnostics LaboratoryUniversity of TorontoTorontoCanada

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