  Within the context of a first-order phase transition in the early Universe,
we study the collision process for vacuum bubbles expanding in a plasma. The
effects of the plasma are simulated by introducing a damping term in the
equations of motion for a $U(1)$ global field. We find that Lorentz-contracted
spherically symmetric domain walls adequately describe the overdamped motion of
the bubbles in the thin wall approximation, and study the process of collision
and phase equilibration both numerically and analytically. With an analytical
model for the phase propagation in 1+1 dimensions, we prove that the phase
waves generated in the bubble merging are reflected by the walls of the true
vacuum cavity, giving rise to a long-lived oscillating state that delays the
phase equilibration. The existence of such a state in the 3+1 dimensional model
is then confirmed by numerical simulations, and the consequences for the
formation of vortices in three-bubble collisions are considered.
