Abstract

A previously proposed modified form of diffusion theory [Phys. Rev. <b>100</b>, 517 (1955)] is applied to problems involving point sources and plane clouds over an absorbing ground. A general but formal solution is first presented. Some analytic results are then obtained for a point source at the edge of a semi-infinite (half-space) cloud; specific formulas are given for the photon density at the edge of the cloud and for the net photon current there; asymptotic results are obtained for large distances inside the cloud along the source axis. Finally, numerical results are presented for the photon density at the mid-plane of an infinite plane cloud with thickness equal to two (transport) mean free paths and a point source at the center. An appendix summarizes analytic results of the modified theory for: a point source in an infinite medium, a point source at the center of a finite sphere, the albedo and transmission of an infinite plane slab with parallel (";solar";) illumination at an arbitrary angle, the albedo of the combination of a plane cloud over a ground plane with arbitrary albedo, and a rough analysis of ambient light levels in the ocean.

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  1. Paul I. Richards, Phys. Rev. 100, 517 (1955).
  2. G. N. Watson, Theory of Bessel Functions (Cambridge University Press, Cambridge, 1944), second edition.
  3. A. Erdelyi, editor, Tables of Integral Transforms (McGraw-Hill Book Company, Inc., New York, 1954), Vols. I and II.
  4. W. Magnus and F. Oberhettinger, Functions of Mathematical Physics (Chelsea Publishing Company, New York, 1954).
  5. D. V. Widder, Advanced Calculus (Prentice-Hall, Inc., New York, 1947).
  6. It is understood on the right-hand side that ƒ[(t2-y2)½]=0 for t<y.
  7. In this and subsequent manipulations the following identity is frequently useful:(a2+b2)½-a=b2[(a2+b2)½+a]-1.
  8. The notation O(g) denotes a quantity whose absolute values remains less than some constant times the absolute value of g.
  9. Diffusion theory applied to the same problem gives ρ=0 unless the "extrapolated end point" is used, in which case the result corresponding to expression (31) is ρ~3/2r3, using the standard extrapolation distance (0.7104λ).
  10. G. Doetsch, Theorie und Anwendung der Laplacetransformation (Dover Publications, New York, 1945), p. 231.
  11. Diffusion theory, using the 0.7104λ extrapolated end point, gives 4.26/z2 in near agreement with Eq. (33).

Doetsch, G.

G. Doetsch, Theorie und Anwendung der Laplacetransformation (Dover Publications, New York, 1945), p. 231.

Magnus, W.

W. Magnus and F. Oberhettinger, Functions of Mathematical Physics (Chelsea Publishing Company, New York, 1954).

Oberhettinger, F.

W. Magnus and F. Oberhettinger, Functions of Mathematical Physics (Chelsea Publishing Company, New York, 1954).

Richards, Paul I.

Paul I. Richards, Phys. Rev. 100, 517 (1955).

Watson, G. N.

G. N. Watson, Theory of Bessel Functions (Cambridge University Press, Cambridge, 1944), second edition.

Widder, D. V.

D. V. Widder, Advanced Calculus (Prentice-Hall, Inc., New York, 1947).

Other (11)

Paul I. Richards, Phys. Rev. 100, 517 (1955).

G. N. Watson, Theory of Bessel Functions (Cambridge University Press, Cambridge, 1944), second edition.

A. Erdelyi, editor, Tables of Integral Transforms (McGraw-Hill Book Company, Inc., New York, 1954), Vols. I and II.

W. Magnus and F. Oberhettinger, Functions of Mathematical Physics (Chelsea Publishing Company, New York, 1954).

D. V. Widder, Advanced Calculus (Prentice-Hall, Inc., New York, 1947).

It is understood on the right-hand side that ƒ[(t2-y2)½]=0 for t<y.

In this and subsequent manipulations the following identity is frequently useful:(a2+b2)½-a=b2[(a2+b2)½+a]-1.

The notation O(g) denotes a quantity whose absolute values remains less than some constant times the absolute value of g.

Diffusion theory applied to the same problem gives ρ=0 unless the "extrapolated end point" is used, in which case the result corresponding to expression (31) is ρ~3/2r3, using the standard extrapolation distance (0.7104λ).

G. Doetsch, Theorie und Anwendung der Laplacetransformation (Dover Publications, New York, 1945), p. 231.

Diffusion theory, using the 0.7104λ extrapolated end point, gives 4.26/z2 in near agreement with Eq. (33).

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