  Abridged abstract: For stars to form, material must fall inwards from core
scales through the envelope towards the central protostar. The velocity profile
around protostars is poorly constrained. 6 Class 0 protostars and one Class I
protostars observed with HIFI on board Herschel as part of the "Water in
star-forming regions with Herschel" (WISH) survey show infall signatures in
water line observations. We use 1-D non-LTE RATRAN radiative transfer models of
the observed water lines to constrain the infall velocity and chemistry in the
protostellar envelopes of these sources. We assume a free-fall velocity profile
and, having found the best fit, vary the radii over which infall takes place.
In the well-studied Class 0 protostar NGC1333-IRAS4A we find that infall takes
place over the whole envelope to which our observations are sensitive (r>~1000
AU). For 4 sources infall takes place on core to envelope scales (i.e.
~10000-3000 AU). In 2 sources the inverse P-Cygni profiles seen in the
ground-state lines are more likely due to larger-scale motions or foreground
clouds. Models including a simple consideration of the chemistry are consistent
with the observations, while using step abundance profiles are not. The
non-detection of excited water in the inner envelope in 6/7 protostars is
further evidence that water must be heavily depleted from the gas-phase at
these radii. Infall in four of the sources is supersonic and infall in all
sources must take place at the outer edge of the envelope, which may be
evidence that collapse is global or outside-in rather than inside-out. The mass
infall rate in IRAS4A is large (>~10^-4 msol\yr), higher than the mass outflow
rate and expected mass accretion rates onto the star, suggesting that any
flattened disk-like structure on small scales will be gravitationally unstable,
potentially leading to rotational fragmentation and/or episodic accretion.
