  We consider recent observations of the chromospheric network, and argue that
the bright network grains observed in the Ca II H & K lines are heated by an as
yet unidentified quasi-steady process. We propose that the heating is caused by
dissipation of short-period magnetoacoustic waves in magnetic flux tubes
(periods less than 100 s). Magnetohydrodynamic (MHD) models of such waves are
presented. We consider wave generation in the network due to two separate
processes: (a) by transverse motions at the base of the flux tube; and (b) by
the absorption of acoustic waves generated in the ambient medium. We find that
the former mechanism leads to an efficient heating of the chromosphere by slow
magnetoacoustic waves propagating along magnetic field lines. This heating is
produced by shock waves with a horizontal size of a few hundred kilometers. In
contrast, acoustic waves excited in the ambient medium are converted into
transverse fast modes that travel rapidly through the flux tube and do not form
shocks, unless the acoustic sources are located within 100 km from the tube
axis. We conclude that the magnetic network may be heated by magnetoacoustic
waves that are generated in or near the flux tubes.
