  The linear properties of the low-frequency shear Alfv\'en waves such as those
associated with the beta-induced Alfv\'en eigenmodes (BAEs) and the
low-frequency modes observed in reversed-magnetic-shear DIII-D discharges (W.
Heidbrink, et al 2021 Nucl. Fusion 61 066031) are theoretically investigated
and delineated based on the theoretical framework of the general fishbone-like
dispersion relation (GFLDR). By adopting representative experimental
equilibrium profiles, it is found that the low-frequency modes and BAEs are,
respectively, the reactive-type and dissipative-type unstable modes with
dominant Alfv\'enic polarization, thus the former being more precisely called
low-frequency Alfv\'en modes (LFAMs). More specifically, due to different
instability mechanisms, the maximal drive of BAEs occurs, in comparison to
LFAMs, when the minimum of the safety factor ($q_{min}$) deviates from a
rational number. Meanwhile, the BAE eigenfunction peaks at the radial position
of the maximum energetic particle pressure gradient, resulting in a large
deviation from the $q_{min}$ surface. Moreover, the ascending frequency
spectrum patterns of the experimentally observed BAEs and LFAMs can be
theoretically reproduced by varying $q_{min}$ and also be well interpreted
based on the GFLDR. The present analysis illustrates the solid predictive
capability of the GFLDR and its practical usefulness in enhancing the
interpretative capability of both experimental and numerical simulation
results.
