  Characterizing the gap eigenmode of shear Alfv\'{e}n waves (SAW) and its
interaction with energetic ions is important to the success of magnetically
confined fusion. Previous studies have reported an experimental observation of
the spectral gap of SAW on LAPD (Zhang et al 2008 Phys. Plasmas 15 012103), a
linear large plasma device (Gekelman et al 1991 Rev. Sci. Instrum. 62 2875)
possessing easier diagnostic access and lower cost compared with traditional
fusion devices, and analytical theory and numerical gap eigenmode using ideal
conditions (Chang 2014 PhD Thesis at Australian National University). To guide
experimental implementation, the present work models the gap eigenmode of SAW
using exact LAPD parameters. A full picture of the wave field for previous
experiment reveals that the previously observed spectral gap is not global but
an axially local result. To form a global spectral gap, the number of magnetic
mirrors has to be increased and stronger static magnetic field makes it more
clear. Such a spectral gap is obtained for the magnetic field of
$B_0(z)=1.2+0.6\cos[2\pi (z-33.68)/3.63]$ with $7.74$~m magnetic beach. By
introducing two types of local defects (corresponding to $E_\theta(z_0)=0$ and
$E_\theta'(z_0)=0$ respectively), odd-parity and even-parity discrete
eigenmodes are formed clearly inside the gap. The strength of these gap
eigenmodes decreases significantly with collision frequency, which is
consistent with previous studies. Parameter scans show that these gap
eigenmodes can be even formed successfully for the field strength of
$B_0(z)=0.2+0.1\cos[2\pi (z-33.68)/3.63]$ and with only $4$ magnetic mirrors,
which are achievable by LAPD at its present status. This work can serve as a
strong motivation and direct reference for the experimental implementation of
the gap eigenmode of SAW on LAPD and other linear plasma devices.
