  Protostars interact with their surroundings through jets and winds impacting
on the envelope and creating shocks, but the nature of these shocks is still
poorly understood. Our aim is to survey far-infrared molecular line emission
from a uniform and significant sample of deeply-embedded low-mass young stellar
objects in order to characterize shocks and the possible role of ultraviolet
radiation in the immediate protostellar environment. Herschel/PACS spectral
maps of 22 objects in the Perseus molecular cloud were obtained as part of the
`William Herschel Line Legacy' survey. Line emission from H$_\mathrm{2}$O, CO,
and OH is tested against shock models from the literature.
  Observed line ratios are remarkably similar and do not show variations with
source physical parameters. Observations show good agreement with the shock
models when line ratios of the same species are compared. Ratios of various
H$_\mathrm{2}$O lines provide a particularly good diagnostic of pre-shock gas
densities, $n_\mathrm{H}\sim10^{5}$ cm$^{-3}$, in agreement with typical
densities obtained from observations of the post-shock gas. The corresponding
shock velocities, obtained from comparison with CO line ratios, are above 20
km\,s$^{-1}$. However, the observations consistently show one-to-two orders of
magnitude lower H$_\mathrm{2}$O-to-CO and H$_\mathrm{2}$O-to-OH line ratios
than predicted by the existing shock models.
  The overestimated model H$_\mathrm{2}$O fluxes are most likely caused by an
overabundance of H$_\mathrm{2}$O in the models since the excitation is
well-reproduced. Illumination of the shocked material by ultraviolet photons
produced either in the star-disk system or, more locally, in the shock, would
decrease the H$_\mathrm{2}$O abundances and reconcile the models with
observations. Detections of hot H$_\mathrm{2}$O and strong OH lines support
this scenario.
