Shockingly low water abundances in

Context. Protostars interact with their surroundings through jets and winds impinging on the envelope and creating shocks, but the nature of these shocks is still poorly understood. Aims. Our aim is to survey far-infrared molecular line emission from a uniform and significant sample of deeply-embedded low-mass young stellar objects (YSOs) in order to characterize shocks and the possible role of ultraviolet radiation in the immediate protostellar environment. Methods. Herschel/PACS spectral maps of 22 objects in the Perseus molecular cloud were obtained as part of the William Herschel Line Legacy (WILL) survey. Line emission from H2O, CO, and OH is tested against shock models from the literature. Results. Observed line ratios are remarkably similar and do not show variations with physical parameters of the sources (luminosity, envelope mass). Most ratios are also comparable to those found at off-source outflow positions. Observations show good agreement with the shock models when line ratios of the same species are compared. Ratios of various H2O lines provide a particularly good diagnostic of pre-shock gas densities, nH ~ 105 cm-3, in agreement with typical densities obtained from observations of the post-shock gas when a compression factor on the order of 10 is applied (for non-dissociative shocks). The corresponding shock velocities, obtained from comparison with CO line ratios, are above 20 km s-1. However, the observations consistently show H2O-to-CO and H2O-to-OH line ratios that are one to two orders of magnitude lower than predicted by the existing shock models. Conclusions. The overestimated model H2O fluxes are most likely caused by an overabundance of H2O 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 H2O abundances and reconcile the models with observations. Detections of hot H2O and strong OH lines support this scenario