  We consider brane-world models embedded in a five-dimensional bulk spacetime
with a large extra dimension and a cosmological constant. The cosmology in $5D$
possesses "wave-like" character in the sense that the metric coefficients in
the bulk are assumed to have the form of plane waves propagating in the fifth
dimension. We model the brane as the "plane" of collision of waves propagating
in opposite directions along the extra dimension. This plane is a jump
discontinuity which presents the usual ${\bf Z}_2$ symmetry of brane models.
The model reproduces the {\em generalized} Friedmann equation for the evolution
on the brane, regardless of the specific details in $5D$. Model solutions with
spacelike extra coordinate show the usual {\em big-bang} behavior, while those
with timelike extra dimension present a {\em big bounce}. This bounce is an
genuine effect of a timelike extra dimension. We argue that, based on our
current knowledge, models having a large timelike extra dimension cannot be
dismissed as mathematical curiosities in non-physical solutions. The size of
the extra dimension is small today, but it is {\em increasing} if the universe
is expanding with acceleration. Also, the expansion rate of the fifth dimension
can be expressed in a simple way through the four-dimensional "deceleration"
and Hubble parameters as $- q H$. These predictions could have important
observational implications, notably for the time variation of rest mass,
electric charge and the gravitational "constant". They hold for the three $(k =
0, + 1, - 1)$ models with arbitrary cosmological constant, and are independent
of the signature of the extra dimension.
