  We provide a rigorous justication of nonlinear geometric optics expansions
for reflecting \emph{pulses} in space dimensions $n>1$. The pulses arise as
solutions to variable coefficient semilinear first-order hyperbolic systems.
The justification applies to $N\times N$ systems with $N$ interacting pulses
which depend on phases that may be nonlinear. The \emph{coherence} assumption
made in a number of earlier works is dropped. We consider problems in which
incoming pulses are generated from pulse boundary data as well as problems in
which a single outgoing pulse reflects off a possibly curved boundary to
produce a number of incoming pulses. Although we focus here on boundary
problems, it is clear that similar results hold by similar methods for the
Cauchy problem for $N\times N$ systems in free space.
