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Thermal-Wave Fields in One Dimension

  • Andreas Mandelis

Abstract

Much of the Green function formalism of Chapter 1 will be used in this chapter to address several “case studies” in the form of detailed solutions to one-dimensional thermal-wave problems most frequently encountered in applications. The Green-function approach results in the use of very few constitutive formulas, which are, nevertheless, capable of covering a large spectrum of boundary-value problems. A thermal-wave field generated by a spatially arbitrary, harmonic source function Q(r,w) is the solution to the inhomogeneous equation (1.9):
$$ {\nabla ^2}T(r,\omega ) - {\sigma ^2}(\omega )T(r,\omega ) = - \frac{1}{k}Q(r,\omega ) $$
(2.1)
where σ is the thermal wavenumber, defined by Eq. (1.5), and k is the thermal conductivity of the medium. In the Introduction chapter, it was shown that a straightforward algebraic combination of Eq. (2.1) and the equation for the Green function, Eq. (1.24′), yields the general solution for the thermal-wave field, Eq. (1.30):
$$ T(r,\omega ) = \left( {\frac{\alpha }{k}} \right)G(r\left| {{r_0}} \right.;\omega )d{V_0} + \alpha \oint_{{S_0}} {\left[ {G(r\left| {r_o^s;\omega ){\nabla _0}({r^s},\omega ) - T({r^2},\omega )} \right.{\nabla _0}G(r\left| {r_o^s;\omega } \right.} \right]} \bullet d{S_0} $$
(2.2)
Here, S 0 is the surface surrounding the domain volume V 0, which includes the harmonic source.\( Q({r_0},\omega ).r_o^s \) is a coordinate point on S 0.

Keywords

Heat Transfer Coefficient Green Function Optical Absorption Coefficient Thermal Effusivity Dimensionless Thickness 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. L. C. Aamodt, J. C. Murphy and J. G. Parker, J. Appl. Phys. 48, 927 (1977).ADSCrossRefGoogle Scholar
  2. D. P. Almond and P. M. Patel, Photothermal Science and Techniques (Chapman & Hall, London, 1996).Google Scholar
  3. C. A. Bennett and R. R. Patty, Appl. Opt. 21, 49 (1982).ADSCrossRefGoogle Scholar
  4. M. Beyfuss, J. Baumann, and R. Tilgner, in Photoacoustic and Photothermal Phenomena II, Springer Ser. Opt. Sci. 62 (J. C. Murphy, J. W. Maclachlan-Spicer, L. C. Aamodt, and B. S. H. Royce, eds.), (Springer-Verlag New York, 1990), p. 17.Google Scholar
  5. B. Bonno, J. L. Laporte, and Y. Rousset, J. Appl. Phys. 67, 2253 (1990).ADSCrossRefGoogle Scholar
  6. G. Busse, Appl. Phys. Lett. 35, 759 (1979).ADSCrossRefGoogle Scholar
  7. G. Busse and A. Ograbek, J. Appl. Phys. 51, 3576 (1980).ADSCrossRefGoogle Scholar
  8. G. Busse and H. G. Walther, in Principles and Perspectives of Photothermal and Photoacoustic Phenomena, Progress in Photothermal and Photoacoustic Science and Technology, Vol. I (A. Mandelis, ed.), (Elsevier, New York, 1992), Chap. 5.Google Scholar
  9. H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids (Oxford University Press, Oxford, 1959).Google Scholar
  10. P. Dennery and A. Krzywicki, Mathematics for Physicists (Dover, Mineola, NY, 1995), Sect. 25.2.Google Scholar
  11. J. M. C. Duhamel, Journal Ecole Polytechnique Paris 14, Cah. 22, 20 (1833).Google Scholar
  12. J. D. Jackson, Classical Electrodynamics, 2nd Ed. (Wiley, New York, 1975).zbMATHGoogle Scholar
  13. P. K. John, L. C. M. Miranda and A. C. Rastogi, Phys. Rev. B34, 4342 (1986).ADSGoogle Scholar
  14. L. E. Kinsler and A. R. Frey, Fundamentals of Acoustics, 2nd Ed. (Wiley, New York, 1962), Sect. 5.4.Google Scholar
  15. A. Lachaine and P. Poulet, Appl. Phys. Lett. 45, 953 (1984).ADSCrossRefGoogle Scholar
  16. A. Lehto, J. Jaarinen, T. Tiusanen, M. Jokinen, and M. Luukkala, Electron. Lett. 17, 364(1981).Google Scholar
  17. N. F. Leite, N. Cella, H. Vargas, and L. C. M. Miranda, J. Appl. Phys. 61, 3025 (1987).ADSCrossRefGoogle Scholar
  18. A. Mandelis, J. Appl. Phys. 78, 647 (1995).ADSCrossRefGoogle Scholar
  19. J. Mathews and R. L. Walker, Mathematical Methods of Physics, 2nd. ed. (Benjamin/Cummings, Reading, MA, 1970).Google Scholar
  20. I. Morris, P. M. Patel, D. P. Almond, and H. Reiter, Surf. Coat. Technol. 34, 51 (1988).CrossRefGoogle Scholar
  21. P. M. Morse and H. Feshbach, Methods of Theoretical Physics, Vol. I (McGraw-Hill, New York, 1953).zbMATHGoogle Scholar
  22. P. M. Morse and K. U. Ingard, Theoretical Acoustics (McGraw-Hill, New York, 1968).Google Scholar
  23. K. No and J. F. McClelland, J. Appl. Phys. 64, 1730 (1988).ADSCrossRefGoogle Scholar
  24. J. Opsal, Review of Progress in Quantitative NDE, Vol. 6A (D. O. Thompson and D. E. Chimenti, eds.), (Plenum, New York, 1987), p. 217.Google Scholar
  25. J. Opsal and A. Rosencwaig, J. Appl. Phys. 53, 4240 (1982).ADSCrossRefGoogle Scholar
  26. J. G. Parker, Appl. Opt. 12, 2974 (1973).ADSCrossRefGoogle Scholar
  27. J.-L. Parpal, J.-P. Monchalin, L. Bertrand and J.-M. Gagne, J. Appl. Phys. 52, 6879(1981).ADSCrossRefGoogle Scholar
  28. O. Pessoa Jr., C. L. Cesar, N. A. Patel, H. Vargas, C.C. Guizoni, and L. C. M. Miranda, J. Appl. Phys. 59, 1316 (1986).ADSCrossRefGoogle Scholar
  29. A. Rosencwaig and A. Gersho, J. Appl. Phys. 47, 64 (1976).ADSCrossRefGoogle Scholar
  30. J. Shen and A. Mandelis, Rev. Sci. Instrum. 66, 4999 (1995).ADSCrossRefGoogle Scholar
  31. I. N. Sneddon, Fourier Transforms (McGraw-Hill, New York, 1951), p. 164.Google Scholar
  32. N. Teramae and S. Tanaka, Anal. Chem. 57, 95 (1985).CrossRefGoogle Scholar
  33. C. H. Wang and A. Mandelis, J. Appl. Phys. 85, 8366 (1999).ADSCrossRefGoogle Scholar

Copyright information

© 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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