Context. Understanding the physical phenomena involved in the earlierst stages of protostellar evolution requires knowledge of the heating and cooling processes that occur in the surroundings of a young stellar object. Spatially resolved information from its constituent gas and dust provides the necessary constraints to distinguish between different theories of accretion energy dissipation into the envelope. Aims. Our aims are to quantify the far-infrared line emission from low-mass protostars and the contribution of different atomic and molecular species to the gas cooling budget, to determine the spatial extent of the emission, and to investigate the underlying excitation conditions. Analysis of the line cooling will help us characterize the evolution of the relevant physical processes as the protostar ages. Methods. Far-infrared Herschel-PACS spectra of 18 low-mass protostars of various luminosities and evolutionary stages are studied in the context of the WISH key program. For most targets, the spectra include many wavelength intervals selected to cover specific CO, H 2 O, OH, and atomic lines. For four targets the spectra span the entire 55-200 μm region. The PACS field-of-view covers ∼47 with the resolution of 9.4 . Results. Most of the protostars in our sample show strong atomic and molecular far-infrared emission. Water is detected in 17 out of 18 objects (except TMC1A), including 5 Class I sources. The high-excitation H 2 O 8 18 -7 07 63.3 μm line (E u /k B = 1071 K) is detected in 7 sources. CO transitions from J = 14−13 up to J = 49−48 are found and show two distinct temperature components on Boltzmann diagrams with rotational temperatures of ∼350 K and ∼700 K. H 2 O has typical excitation temperatures of ∼150 K. Emission from both Class 0 and I sources is usually spatially extended along the outflow direction but with a pattern that depends on the species and the transition. In the extended sources, emission is stronger off source and extended on ≥10 000 AU scales; in the compact sample, more than half of the flux originates within 1000 AU of the protostar. The Conclusions. The PACS data probe at least two physical components. The H 2 O and CO emission very likely arises in non-dissociative (irradiated) shocks along the outflow walls with a range of pre-shock densities. Some OH is also associated with this component, most likely resulting from H 2 O photodissociation. UV-heated gas contributes only a minor fraction to the CO emission observed by PACS, based on the strong correlation between the shock-dominated CO 24-23 line and the CO 14-13 line. [O i] and some of the OH emission probe dissociative shocks in the inner envelope. The total far-infrared cooling is dominated by H 2 O and CO, with the fraction contributed by [O i] increasing for Class I sources. Consistent with previous studies, the ratio of total far-infrared line emission over bolometric luminosity decreases with the evolutionary state.

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We consider the dynamical response of a finite, simply supported Timoshenko beam loaded by a force moving with a constant velocity. The classical solution for the transverse displacement and the rotation of the cross section of a Timoshenko beam has a form of a sum of two infinite series, one of which represents the force vibrations (aperiodic vibrations) and the other one free vibrations of the beam. We show that one of the series, which represents aperiodic (force) vibrations of the beam, can be presented in a closed form. The closed form solutions take different forms depending if the velocity of the moving force is smaller or larger than the velocities of certain shear and bar velocities.

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