We consider the two-region Milne problem, defined as the steady-state
monoenergetic linear transport problem for two adjoining homogeneous s
ource-free half-spaces, with a particle source coming from infinity in
one of the half-spaces. We demonstrate that the asymptotic (Case disc
rete mode) component of the solution for the scalar flux is easily and
explicitly written in terms of Chandrasekhar's H-function for each me
dium. This asymptotic solution is shown to exhibit a discontinuity in
both the scalar flux and current at the interface between the two half
-spaces. Numerical benchmark results for the linear extrapolation dist
ance and the discontinuities are given for various combinations of the
mean number of secondaries (c) characterizing the two media. Contact
is also made with a variational treatment. In particular, the variatio
nal formalism is shown to predict the linear extrapolation distance an
d these asymptotic discontinuities correct to first order in the diffe
rence between the values of c characterizing the two half-spaces.