  Fluid flows in nature and applications are frequently subject to periodic
velocity modulations. Surprisingly, even for the generic case of flow through a
straight pipe, there is little consensus regarding the influence of pulsation
on the transition threshold to turbulence: while most studies predict a
monotonically increasing threshold with pulsation frequency (i.e. Womersley
number, $\alpha$), others observe a decreasing threshold for identical
parameters and only observe an increasing threshold at low $\alpha$. In the
present study we apply recent advances in the understanding of transition in
steady shear flows to pulsating pipe flow. For moderate pulsation amplitudes we
find that the first instability encountered is subcritical (i.e. requiring
finite amplitude disturbances) and gives rise to localized patches of
turbulence ("puffs") analogous to steady pipe flow. By monitoring the impact of
pulsation on the lifetime of turbulence we map the onset of turbulence in
parameter space. Transition in pulsatile flow can be separated into three
regimes. At small Womersley numbers the dynamics are dominated by the decay
turbulence suffers during the slower part of the cycle and hence transition is
delayed significantly. As shown in this regime thresholds closely agree with
estimates based on a quasi steady flow assumption only taking puff decay rates
into account. The transition point predicted in the zero $\alpha$ limit equals
to the critical point for steady pipe flow offset by the oscillation Reynolds
number. In the high frequency limit puff lifetimes are identical to those in
steady pipe flow and hence the transition threshold appears to be unaffected by
flow pulsation. In the intermediate frequency regime the transition threshold
sharply drops (with increasing $\alpha$) from the decay dominated (quasi
steady) threshold to the steady pipe flow level.
