  The molecular richness of fast protostellar jets within 20-100 au of their
source, despite strong ultraviolet irradiation, remains a challenge for the
models investigated so far. We aim to investigate the effect of interaction
between a time-variable jet and a surrounding steady disk wind, to assess the
possibility of jet chemical enrichement by the wind, and the characteristic
signatures of such a configuration. We have constructed an analytic model of a
jet bow shock driven into a surrounding slower disk wind in the thin shell
approximation. The refilling of the post bow shock cavity from below by the
disk wind is also studied. An extension of the model to the case of two or more
successive internal working surfaces (IWS) is made. We then compared this
analytic model with numerical simulations with and without a surrounding disk
wind. We find that at early times (of order the variability period), jet bow
shocks travel in refilled pristine disk-wind material, before interacting with
the cocoon of older bow shocks. This opens the possibility of bow shock
chemical enrichment (if the disk wind is molecular and dusty) and of probing
the unperturbed disk wind structure near the jet base. Several distinctive
signatures of the presence of a surrounding disk wind are identified, in the
bow shock morphology and kinematics. Numerical simulations validate our
analytical approach and further show that at large scale, the passage of many
jet IWS inside a disk wind produces a stationary V-shaped cavity, closing down
onto the axis at a finite distance from the source.
