For weakly damped non-autonomous hyperbolic equations,
we introduce a new concept Condition (C*), denote the set of all
functions satisfying Condition (C*) by L2C* $(R;X)$ which
are translation bounded but not translation compact in
$L^2$ loc$(R;X)$, and show that there are many functions
satisfying Condition (C*); then we study the uniform attractors for
weakly damped non-autonomous hyperbolic equations with this new
class of time dependent external forces $g(x,t)\in $ L2C* $(R;X)$ and prove the existence of the uniform attractors for the
family of processes corresponding to the equation in $H^1_0\times
L^2$ and $D(A)\times H^1_0$.