The transport and diffusion theory of a line source in an infinite half-space with internal reflection

A closed form analytical solution is obtained for a three-dimensional transport theory problem, namely that of a line source in a half-space in one-speed transport theory with an internally reflecting surface. The theory is developed by using Fourier transforms in the transverse directions and a Laplace transform in the axial direction, together with the Wiener–Hopf technique. Both specular and diffuse internal reflection are considered and it is shown that a closed form solution is not available for the specular case but is available for the diffuse case. The surface particle scalar intensity and current are obtained and their sensitivity to absorption and the reflection coefficient are assessed. We also obtain a number of analytic asymptotic estimates for the intensity valid at large distances from the source. Diffusion theory is also employed and compared numerically with the transport solution. In general, diffusion theory gives very satisfactory results within its regions of limitation. We also offer this paper as an exercise in analytical transport theory; an aspect of reactor physics which is not often seen in recent times.