  We study the worldsheet S-matrix of a string attached to a D-brane in
AdS$_5\times$S$^5$. The D-brane is either a giant graviton or a dual giant
graviton. In the gauge theory, the operators we consider belong to the
$su(2|3)$ sector of the theory. Magnon excitations of open strings can exhibit
both elastic (when magnons in the bulk of the string scatter) and inelastic
(when magnons at the endpoint of an open string participate) scattering. Both
of these $S$-matrices are determined (up to an overall phase) by the
$su(2|2)^2$ global symmetry of the theory. In this note we study the $S$-matrix
for inelastic scattering. We show that it exhibits poles corresponding to
boundstates of bulk and boundary magnons. A crossing equation is derived for
the overall phase. It reproduces the crossing equation for maximal giant
gravitons, in the appropriate limit. Finally, scattering in the $su(2)$ sector
is computed to two loops. This two loop result, which determines the overall
phase to two loops, will be useful when a unique solution to the crossing
equation is to be selected.
