Abstract

The diffuse reflection and transmission of light by a plane-parallel suspension of randomly oriented particles, scattering independently, are treated using radiative-transfer theory. Exact allowance is made for external and internal reflection at the plane surfaces of the matrix. General equations are derived in a form suitable for automatic computation. Some comparisons made between theoretical predictions and experimental measurements of the total transmittance of an optically thick diffusing suspension show satisfactory agreement with radiative-transfer theory and also with the Eddington approximation.

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  1. P. Debye, in Second Interdisciplinary Conference on Electromagnetic Scattering, R. L. Rowell and R. S. Stein, Eds. (Gordon and Breach, New York, 1967).
  2. S. Chandrasekhar, Radiative Transfer (Dover Publications, Inc., New York, 1960).
  3. V. V. Sobolev, A Treatise on Radiative Transfer (D. van Nostrand Co., Inc., Princeton, N. J., 1963).
  4. G. W. Kattawar and G. N. Plass, Appl. Opt. 7, 1519 (1968).
  5. S. E. Orchard, J. Oil Col. Chem. Assoc. 51, 44 (1968).
  6. P. Kubelka and F. Munk, Z. Phys. 12, 593 (1931).
  7. E. Pitts, Proc. Phys. Soc. B67, 105 (1954).
  8. H. C. van de Hulst, Astrophys. J. 107, 220 (1948).
  9. C. M. Chu and S. W. Churchill, IRE Trans. AP-4, 142 (1956).
  10. G. N. Plass and G. W. Kattawar, Appl. Opt. 7, 361 (1968).
  11. M. de Belder, J. de Kerf, J. Jespers, and R. Verbrugghe, J. Opt. Soc. Am. 55, 1261 (1965).
  12. E. M. Feigel'son, Light and Heat Conduction in Stratus Clouds (Oidbourne Press, London, 1966).
  13. Another possibility is to rewrite the equation of transfer as an integral equation, and express the solution as a Neumann series [W. M. Irvine, Astrophys. J. 142, 1563 (1965)]. For thick diffusing layers the convergence is very slow, but can be speeded up by a thickness-doubling method due to Van de Hulst [H. C. van de Hulst, in The Atmospheres of Venus and Mars, J. C. Brandt and M. B. McElroy, Eds. (Gordon and Breach, New York, 1968)]; this method has been applied by Irvine31 since the present work was done.
  14. I. W. Busbridge, The Mathematics of Radiative Transfer (Cambridge University Press, New York, 1960).
  15. T. W. Mullikin, Trans. Am. Math. Soc. 113, 316 (1964).
  16. I. W. Busbridge and S. E. Orchard, Astrophys. J. 149, 655 (1967)
  17. D. B. Judd and G. Wyszecki, Color in Business, Science, and Industry (John Wiley & Sons, Inc., New York, 1963).
  18. R. E. Bellman, R. E. Kalaba, and M. C. Prestrud, Invariant Imbedding and Radiative Transfer in Slabs of Finite Thickness (American Elsevier Publishing Co., New York, 1963).
  19. T. W. Mullikin, in Interdisciplinary Conference on Electromagnetic Scattering, M. Kerker, Ed. (Pergamon Press, Ltd., Oxford, 1963).
  20. L. B. Evans, C. M. Chu, and S. W. Churchill, J. Heat Transfer 87C, 381 (1965).
  21. J. W. Ryde and B. S. Cooper, Proc. Roy. Soc. (London) A131, 464 (1931).
  22. A. Brockes, Farbe 9, 53 (1960).
  23. R. G. Giovanelli, Opt. Acta 2, 153 (1955).
  24. L. Fox, Numerical Solution of Ordinary and Partial Differential Equations (Pergamon Press, Inc., New York, 1962).
  25. J. L. Carlstedt and T. W. Mullikin, Astrophys J. Suppl. 12, 449 (1966).
  26. R. Bellman, H. Kagiwada, R. Kalaba, and S. Ueno, J. Quant. Spectry. Radiative Transfer 6, 479 (1966).
  27. S. E. Orchard, Astrophys. J. 149, 665 (1967).
  28. I. W. Busbridge and S. E. Orchard, Astrophys. J. 154, 729 (1968).
  29. H. H. Kagiwada and R. E. Kalaba, J. Quant. Spectry. Radiative Transfer 7, 295 (1967).
  30. J. W. Ryde, Proc. Roy. Soc. (London) A131, 451 (1931).
  31. W. M. Irvine, Astrophys. J. 152, 823 (1968).

Bellman, R.

R. Bellman, H. Kagiwada, R. Kalaba, and S. Ueno, J. Quant. Spectry. Radiative Transfer 6, 479 (1966).

Bellman, R. E.

R. E. Bellman, R. E. Kalaba, and M. C. Prestrud, Invariant Imbedding and Radiative Transfer in Slabs of Finite Thickness (American Elsevier Publishing Co., New York, 1963).

Brockes, A.

A. Brockes, Farbe 9, 53 (1960).

Busbridge, I. W.

I. W. Busbridge and S. E. Orchard, Astrophys. J. 149, 655 (1967)

I. W. Busbridge, The Mathematics of Radiative Transfer (Cambridge University Press, New York, 1960).

I. W. Busbridge and S. E. Orchard, Astrophys. J. 154, 729 (1968).

Carlstedt, J. L.

J. L. Carlstedt and T. W. Mullikin, Astrophys J. Suppl. 12, 449 (1966).

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Dover Publications, Inc., New York, 1960).

Chu, C. M.

C. M. Chu and S. W. Churchill, IRE Trans. AP-4, 142 (1956).

L. B. Evans, C. M. Chu, and S. W. Churchill, J. Heat Transfer 87C, 381 (1965).

Churchill, S. W.

L. B. Evans, C. M. Chu, and S. W. Churchill, J. Heat Transfer 87C, 381 (1965).

C. M. Chu and S. W. Churchill, IRE Trans. AP-4, 142 (1956).

Cooper, B. S.

J. W. Ryde and B. S. Cooper, Proc. Roy. Soc. (London) A131, 464 (1931).

de Belder, M.

M. de Belder, J. de Kerf, J. Jespers, and R. Verbrugghe, J. Opt. Soc. Am. 55, 1261 (1965).

de Kerf, J.

M. de Belder, J. de Kerf, J. Jespers, and R. Verbrugghe, J. Opt. Soc. Am. 55, 1261 (1965).

Debye, P.

P. Debye, in Second Interdisciplinary Conference on Electromagnetic Scattering, R. L. Rowell and R. S. Stein, Eds. (Gordon and Breach, New York, 1967).

Evans, L. B.

L. B. Evans, C. M. Chu, and S. W. Churchill, J. Heat Transfer 87C, 381 (1965).

Feigel'son, E. M.

E. M. Feigel'son, Light and Heat Conduction in Stratus Clouds (Oidbourne Press, London, 1966).

Fox, L.

L. Fox, Numerical Solution of Ordinary and Partial Differential Equations (Pergamon Press, Inc., New York, 1962).

Giovanelli, R. G.

R. G. Giovanelli, Opt. Acta 2, 153 (1955).

Irvine, W. M.

Another possibility is to rewrite the equation of transfer as an integral equation, and express the solution as a Neumann series [W. M. Irvine, Astrophys. J. 142, 1563 (1965)]. For thick diffusing layers the convergence is very slow, but can be speeded up by a thickness-doubling method due to Van de Hulst [H. C. van de Hulst, in The Atmospheres of Venus and Mars, J. C. Brandt and M. B. McElroy, Eds. (Gordon and Breach, New York, 1968)]; this method has been applied by Irvine31 since the present work was done.

W. M. Irvine, Astrophys. J. 152, 823 (1968).

Jespers, J.

M. de Belder, J. de Kerf, J. Jespers, and R. Verbrugghe, J. Opt. Soc. Am. 55, 1261 (1965).

Judd, D. B.

D. B. Judd and G. Wyszecki, Color in Business, Science, and Industry (John Wiley & Sons, Inc., New York, 1963).

Kagiwada, H.

R. Bellman, H. Kagiwada, R. Kalaba, and S. Ueno, J. Quant. Spectry. Radiative Transfer 6, 479 (1966).

Kagiwada, H. H.

H. H. Kagiwada and R. E. Kalaba, J. Quant. Spectry. Radiative Transfer 7, 295 (1967).

Kalaba, R.

R. Bellman, H. Kagiwada, R. Kalaba, and S. Ueno, J. Quant. Spectry. Radiative Transfer 6, 479 (1966).

Kalaba, R. E.

H. H. Kagiwada and R. E. Kalaba, J. Quant. Spectry. Radiative Transfer 7, 295 (1967).

R. E. Bellman, R. E. Kalaba, and M. C. Prestrud, Invariant Imbedding and Radiative Transfer in Slabs of Finite Thickness (American Elsevier Publishing Co., New York, 1963).

Kattawar, G. W.

G. W. Kattawar and G. N. Plass, Appl. Opt. 7, 1519 (1968).

G. N. Plass and G. W. Kattawar, Appl. Opt. 7, 361 (1968).

Kubelka, P.

P. Kubelka and F. Munk, Z. Phys. 12, 593 (1931).

Mullikin, T. W.

T. W. Mullikin, in Interdisciplinary Conference on Electromagnetic Scattering, M. Kerker, Ed. (Pergamon Press, Ltd., Oxford, 1963).

T. W. Mullikin, Trans. Am. Math. Soc. 113, 316 (1964).

J. L. Carlstedt and T. W. Mullikin, Astrophys J. Suppl. 12, 449 (1966).

Munk, F.

P. Kubelka and F. Munk, Z. Phys. 12, 593 (1931).

Orchard, S. E.

S. E. Orchard, J. Oil Col. Chem. Assoc. 51, 44 (1968).

S. E. Orchard, Astrophys. J. 149, 665 (1967).

I. W. Busbridge and S. E. Orchard, Astrophys. J. 154, 729 (1968).

I. W. Busbridge and S. E. Orchard, Astrophys. J. 149, 655 (1967)

Pitts, E.

E. Pitts, Proc. Phys. Soc. B67, 105 (1954).

Plass, G. N.

G. W. Kattawar and G. N. Plass, Appl. Opt. 7, 1519 (1968).

G. N. Plass and G. W. Kattawar, Appl. Opt. 7, 361 (1968).

Prestrud, M. C.

R. E. Bellman, R. E. Kalaba, and M. C. Prestrud, Invariant Imbedding and Radiative Transfer in Slabs of Finite Thickness (American Elsevier Publishing Co., New York, 1963).

Ryde, J. W.

J. W. Ryde and B. S. Cooper, Proc. Roy. Soc. (London) A131, 464 (1931).

J. W. Ryde, Proc. Roy. Soc. (London) A131, 451 (1931).

Sobolev, V. V.

V. V. Sobolev, A Treatise on Radiative Transfer (D. van Nostrand Co., Inc., Princeton, N. J., 1963).

Ueno, S.

R. Bellman, H. Kagiwada, R. Kalaba, and S. Ueno, J. Quant. Spectry. Radiative Transfer 6, 479 (1966).

van de Hulst, H. C.

H. C. van de Hulst, Astrophys. J. 107, 220 (1948).

Verbrugghe, R.

M. de Belder, J. de Kerf, J. Jespers, and R. Verbrugghe, J. Opt. Soc. Am. 55, 1261 (1965).

Wyszecki, G.

D. B. Judd and G. Wyszecki, Color in Business, Science, and Industry (John Wiley & Sons, Inc., New York, 1963).

Other (31)

P. Debye, in Second Interdisciplinary Conference on Electromagnetic Scattering, R. L. Rowell and R. S. Stein, Eds. (Gordon and Breach, New York, 1967).

S. Chandrasekhar, Radiative Transfer (Dover Publications, Inc., New York, 1960).

V. V. Sobolev, A Treatise on Radiative Transfer (D. van Nostrand Co., Inc., Princeton, N. J., 1963).

G. W. Kattawar and G. N. Plass, Appl. Opt. 7, 1519 (1968).

S. E. Orchard, J. Oil Col. Chem. Assoc. 51, 44 (1968).

P. Kubelka and F. Munk, Z. Phys. 12, 593 (1931).

E. Pitts, Proc. Phys. Soc. B67, 105 (1954).

H. C. van de Hulst, Astrophys. J. 107, 220 (1948).

C. M. Chu and S. W. Churchill, IRE Trans. AP-4, 142 (1956).

G. N. Plass and G. W. Kattawar, Appl. Opt. 7, 361 (1968).

M. de Belder, J. de Kerf, J. Jespers, and R. Verbrugghe, J. Opt. Soc. Am. 55, 1261 (1965).

E. M. Feigel'son, Light and Heat Conduction in Stratus Clouds (Oidbourne Press, London, 1966).

Another possibility is to rewrite the equation of transfer as an integral equation, and express the solution as a Neumann series [W. M. Irvine, Astrophys. J. 142, 1563 (1965)]. For thick diffusing layers the convergence is very slow, but can be speeded up by a thickness-doubling method due to Van de Hulst [H. C. van de Hulst, in The Atmospheres of Venus and Mars, J. C. Brandt and M. B. McElroy, Eds. (Gordon and Breach, New York, 1968)]; this method has been applied by Irvine31 since the present work was done.

I. W. Busbridge, The Mathematics of Radiative Transfer (Cambridge University Press, New York, 1960).

T. W. Mullikin, Trans. Am. Math. Soc. 113, 316 (1964).

I. W. Busbridge and S. E. Orchard, Astrophys. J. 149, 655 (1967)

D. B. Judd and G. Wyszecki, Color in Business, Science, and Industry (John Wiley & Sons, Inc., New York, 1963).

R. E. Bellman, R. E. Kalaba, and M. C. Prestrud, Invariant Imbedding and Radiative Transfer in Slabs of Finite Thickness (American Elsevier Publishing Co., New York, 1963).

T. W. Mullikin, in Interdisciplinary Conference on Electromagnetic Scattering, M. Kerker, Ed. (Pergamon Press, Ltd., Oxford, 1963).

L. B. Evans, C. M. Chu, and S. W. Churchill, J. Heat Transfer 87C, 381 (1965).

J. W. Ryde and B. S. Cooper, Proc. Roy. Soc. (London) A131, 464 (1931).

A. Brockes, Farbe 9, 53 (1960).

R. G. Giovanelli, Opt. Acta 2, 153 (1955).

L. Fox, Numerical Solution of Ordinary and Partial Differential Equations (Pergamon Press, Inc., New York, 1962).

J. L. Carlstedt and T. W. Mullikin, Astrophys J. Suppl. 12, 449 (1966).

R. Bellman, H. Kagiwada, R. Kalaba, and S. Ueno, J. Quant. Spectry. Radiative Transfer 6, 479 (1966).

S. E. Orchard, Astrophys. J. 149, 665 (1967).

I. W. Busbridge and S. E. Orchard, Astrophys. J. 154, 729 (1968).

H. H. Kagiwada and R. E. Kalaba, J. Quant. Spectry. Radiative Transfer 7, 295 (1967).

J. W. Ryde, Proc. Roy. Soc. (London) A131, 451 (1931).

W. M. Irvine, Astrophys. J. 152, 823 (1968).

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