  The translation of a bubble under the action of an acoustic forcing finds
applications in fields ranging from drug delivery to sonoluminesce. This
phenomenon has been widely studied for cases where the amplitude of the forcing
remains constant over time. However, in many practical applications, the
duration of the forcing is not long enough for the bubble to attain a constant
translational velocity, mainly due to the effect of the history force. Here, we
develop a formulation, valid in the limit of very viscous flow and
small-amplitude acoustic forcing, that allows us to describe the transient
dynamics of bubbles driven by acoustic pulses consisting of a finite number of
cycles. We also present an asymptotic solution to this theory for the case of a
finite-duration sinusoidal pressure pulse. This solution takes into account
both the history integral term and the transient period that the bubble needs
to achieve steady radial oscillations, being the former dominant during most of
the acceleration process. Moreover, by introducing some additional assumptions,
we derive a simplified formula that describes fairly well the time evolution of
the bubble velocity. Using this solution we show that the convergence to the
steady translational velocity, given by the so-called Bjerknes force, occurs
rather slowly, namely as $\tau^{-1/2}$, which explains the slow convergence of
the bubble velocity and stresses the importance of taking into account the
history force.
