  We describe the relaxation dynamics of a coherently split one-dimensional
(1D) Bose gas in the harmonic approximation. A dephased, prethermalized state
emerges in a light-cone-like evolution which is connected to the spreading of
correlations with a characteristic velocity. In our description we put special
emphasis on the influence of the longitudinal trapping potential and the finite
size of the system, both of which are highly relevant in experiments. In
particular, we quantify their influence on the phase correlation properties and
the characteristic velocity with which the prethermalized state is established.
Finally, we show that the trapping potential has an important effect on the
recurrences of coherence which are expected to appear in a finite size system.
