  The wealth of asteroseismic data for red-giant stars and the precision with
which these data have been observed over the last decade calls for
investigations to further understand the internal structures of these stars.
The aim of this work is to validate a method to measure the underlying period
spacing, coupling term and mode offset of pure gravity modes that are present
in the deep interiors of red-giant stars. We subsequently investigate the
physical conditions of the evanescent zone between the gravity mode cavity and
the pressure mode cavity. We implement an alternative mathematical description
to analyse observational data and to extract the underlying physical parameters
that determine the frequencies of mixed modes. This description takes the
radial order of the modes explicitly into account, which reduces its
sensitivity to aliases. Additionally, and for the first time, this method
allows us to constrain the gravity mode offset for red-giant stars. We
determine the period spacing and the coupling term for the dipole modes within
a few percent of literature values. Additionally, we find that the gravity mode
offset varies on a star by star basis and should not be kept fixed in the
analysis. Furthermore, we find that the coupling factor is logarithmically
related to the physical width of the evanescent region normalised by the radius
at which the evanescent zone is located. Finally, the local density contrast at
the edge of the core of red giant branch models shows a tentative correlation
with the offset. (abstract abriged)
