  The magnetic geometry of the solar atmosphere, combined with projection
effects, makes it difficult to accurately map the propagation of ubiquitous
waves in fibrillar structures. These waves are of interest due to their ability
to carry energy into the chromosphere and deposit it through damping and
dissipation mechanisms. To this end, the Interferometric Bidimensional
Spectrometer (IBIS) at the Dunn Solar Telescope was employed to capture high
resolution H$\alpha$ spectral scans of a sunspot, with the transverse
oscillations of a prominent super-penumbral fibril examined in depth. The
oscillations are re-projected from the helioprojective-cartesian frame to a new
frame of reference oriented along the average fibril axis through non-linear
force-free field extrapolations. The fibril was found to be carrying an
elliptically polarised, propagating kink oscillation with a period of $430$ s
and a phase velocity of $69\pm4$ km s$^{-1}$. The oscillation is damped as it
propagates away from the sunspot with a damping length of approximately $9.2$
Mm, resulting in the energy flux decreasing at a rate on the order of $460$ W
m$^{-2}$/Mm. The H$\alpha$ line width is examined and found to increase with
distance from the sunspot; a potential sign of a temperature increase.
Different linear and non-linear mechanisms are investigated for the damping of
the wave energy flux, but a first-order approximation of their combined effects
is insufficient to recreate the observed damping length by a factor of at least
$3$. It is anticipated that the re-projection methodology demonstrated in this
study will aid with future studies of transverse waves within fibrillar
structures.
