Masters Theses


"Exact formulations are presented for the source function, intensity and flux for a two-dimensional, isotropically scattering medium of finite thickness. Spatially varying collimated and diffuse radiation of the following types are incident on the medium: (1) cosine varying, (2) semi-infinite step, (3) step at the origin, and (4) finite strip. The solutions for the cosine varying collimated boundary condition are used to construct solutions for the other boundary conditions. The two-dimensional integral equations are reduced to one-dimensional equations by separating variables. Through a series of transformations, the one-dimensional equations are put into a form in which the source function, intensity and radiative flux can be calculated at the boundaries of the medium without knowing the distribution within the medium. The equations are then solved numerically for a wide range of parameters.

From the two-dimensional finite thick results the range of validity for the simpler one-dimensional analysis was determined. The one-dimensional approximation is useful when the spatial frequency is less than 0.01 for the cosine varying models or when the optical distance from the origin is more than twenty for the semi-infinite step models. However, for very large optical thicknesses with albedos near unity, smaller spatial frequencies or large optical distances are required.

From the two-dimensional finite thick results, the range of validity of the simpler two-dimensional semi-infinite analysis was determined. For both the cosine varying and semi-infinite step boundary conditions, the semi-infinite analysis is a good approximation to the finite two-dimensional analysis for an optical thickness greater than ten for an albedo less than 0.90.

The effect of decreasing the albedo is to decrease all of the source functions and intensities at the boundaries and the flux leaving the boundaries. The influence of the albedo is most pronounced for the one-dimensional case (ß = 0 or Ty → ∞)"--Abstract, pages ii-iii.


Crosbie, A. L. (Alfred L.)

Committee Member(s)

Look, Dwight C., 1938-
Pagano, Sylvester J., 1924-2006


Mechanical and Aerospace Engineering

Degree Name

M.S. in Mechanical Engineering


National Science Foundation (U.S.)


Financial support from the National Science Foundation through grant GK-35859


University of Missouri--Rolla

Publication Date



xxvi, 273 pages

Note about bibliography

Includes bibliographical references (pages 140-142).


© 1974 James Warren Koewing, All rights reserved.

Document Type

Thesis - Open Access

File Type




Subject Headings

Radiative transfer -- Mathematical models
Scattering (Physics) -- Mathematical models
Finite element method

Thesis Number

T 2979

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