  As a model for the semiclassical analysis of quantum-mechanical systems with
both potentials and boundary conditions, we construct the WKB propagator for a
linear potential sloping away from an impenetrable boundary. First, we find all
classical paths from point $y$ to point $x$ in time $t$ and calculate the
corresponding action and amplitude functions. A large part of space-time turns
out to be classically inaccessible, and the boundary of this region is a
caustic of an unusual type, where the amplitude vanishes instead of diverging.
We show that this curve is the limit of caustics in the usual sense when the
reflecting boundary is approximated by steeply rising smooth potentials. Then,
to improve the WKB approximation we construct the propagator for initial data
in momentum space; this requires classifying the interesting variety of
classical paths with initial momentum $p$ arriving at $x$ after time $t$. The
two approximate propagators are compared by applying them to Gaussian initial
packets by numerical integration; the results show physically expected
behavior, with advantages to the momentum-based propagator in the classically
forbidden regime (large $t$).
