  An outstanding issue in braneworld theory concerns the setting up of proper
boundary conditions for the brane-bulk system. Boundary conditions (BC's)
employing regulatory branes or demanding that the bulk metric be nonsingular
have yet to be implemented in full generality. In this paper, we take a
different route and specify boundary conditions directly on the brane thereby
arriving at a local and closed system of equations (on the brane). We consider
a one-parameter family of boundary conditions involving the anisotropic stress
of the projection of the bulk Weyl tensor on the brane and derive an exact
system of equations describing scalar cosmological perturbations on a generic
braneworld with induced gravity. Depending upon our choice of boundary
conditions, perturbations on the brane either grow moderately (region of
stability) or rapidly (instability). In the instability region, the evolution
of perturbations usually depends upon the scale: small scale perturbations grow
much more rapidly than those on larger scales. This instability is caused by a
peculiar gravitational interaction between dark radiation and matter on the
brane. Generalizing the boundary conditions obtained by Koyama and Maartens, we
find for the Dvali-Gabadadze-Porrati model an instability, which leads to a
dramatic scale-dependence of the evolution of density perturbations in matter
and dark radiation. A different set of BC's, however, leads to a more moderate
and scale-independent growth of perturbations. For the mimicry braneworld,
which expands like LCDM, this class of BC's can lead to an earlier epoch of
structure formation.
