  Alfven waves are considered as potential transporters of energy heating the
solar corona. We seek spectroscopic signatures of the Alfven waves in the
chromosphere occupied by a bright network element, investigating temporal
variations of the spectral width, intensity, Dopplershift, and the asymmetry of
the core of the Halpha spectral line observed by the tunable Lyot filter
installed on the Dutch Open Telescope. The spectral characteristics are derived
through the fitting of five intensity samples, separated from each other by
0.35 A, by a 4th-order polynomial. The bright network element displays the most
pronounced variations of the Dopplershift varying from 0 to 4 km/s about the
average of 1.5 km/s. This fact implies a persistent redshift of the Halpha core
with a redward asymmetry of about 0.5 km/s, suggesting an inverse-C bisector.
The variations of the core intensity up to +-10% and the core width up to +-5%
about the respective averages are much less pronounced, but still detectable.
The core intensity variations lag behind the Dopplershift variations about 2.1
min. The Halpha core width tends to correlate with the Dopplershift and
anticorrelate with the asymmetry, suggesting that more redshifted Halpha
profiles are wider and the broadening of the Halpha core is accompanied with a
change of the core asymmetry from redward to blueward. We also found a striking
anticorrelation between the core asymmetry and the Dopplershift, suggesting a
change of the core asymmetry from redward to blueward with an increasing
redshift of the Halpha core. The data and the applied analysis do not show
meaningful tracks of Alfven waves in the selected network element.
