Water in star-forming regions with Herschel (WISH): II. Evolution of 557 GHz 110-101 emission in low-mass protostars

(Abridged) Water is a key tracer of dynamics and chemistry in low-mass protostars, but spectrally resolved observations have so far been limited in sensitivity and angular resolution. In this first systematic survey of spectrally resolved water emission in low-mass protostellar objects, H2O was observed in the ground-state transition at 557 GHz with HIFI on Herschel in 29 embedded Class 0 and I protostars. Complementary far-IR and sub-mm continuum data (including PACS data from our program) are used to constrain the spectral energy distribution of each source. H2O intensities are compared to inferred envelope and outflow properties and CO 3-2 emission. H2O emission is detected in all objects except one. The line profiles are complex and consist of several kinematic components. The profiles are typically dominated by a broad Gaussian emission feature, indicating that the bulk of the water emission arises in outflows, not the quiescent envelope. Several sources show multiple shock components in either emission or absorption, thus constraining the internal geometry of the system. Furthermore, the components include inverse P-Cygni profiles in 7 sources (6 Class 0, 1 Class I) indicative of infalling envelopes, and regular P-Cygni profiles in 4 sources (3 Class I, 1 Class 0) indicative of expanding envelopes. "Bullets" moving at >50 km/s are seen in 4 Class 0 sources; 3 of these are new detections. In the outflow, the H2O/CO abundance ratio as a function of velocity is nearly the same for all sources, increasing from 10^-3 at 10^-1 at >10 km/s. The H2O abundance in the outer envelope is low, ~10^-10. The different H2O profile components show a clear evolutionary trend: in the Class 0 sources, emission is dominated by outflow components originating inside an infalling envelope. When the infall diminishes during the Class I phase, the outflow weakens and H2O emission disappears