  Solar prominences are subject to both field-aligned (longitudinal) and
transverse oscillatory motions, as evidenced by an increasing number of
observations. Large-amplitude longitudinal motions provide valuable information
on the geometry of the filament-channel magnetic structure that supports the
cool prominence plasma against gravity. Our pendulum model, in which the
restoring force is the gravity projected along the dipped field lines of the
magnetic structure, best explains these oscillations. However, several factors
can influence the longitudinal oscillations, potentially invalidating the
pendulum model. The aim of this work is to study the influence of large-scale
variations in the magnetic field strength along the field lines, i.e.,
variations of the cross-sectional area along the flux tubes supporting
prominence threads. We studied the normal modes of several flux tube
configurations, using linear perturbation analysis, to assess the influence of
different geometrical parameters on the oscillation properties. We found that
the influence of the symmetric and asymmetric expansion factors on longitudinal
oscillations is small.}{We conclude that the longitudinal oscillations are not
significantly influenced by variations of the cross-section of the flux tubes,
validating the pendulum model in this context.
