  Wave propagation in magnetically structured atmospheres is a thoroughly
studied, yet practically inexhaustible well of investigations in the field of
solar magneto-seismology. A simple but powerful example is the examination of
wave behaviour in a magnetic slab. Our previous study (Zs\'amberger, Allcock
and Erd\'elyi, Astrophys. J., 853, p. 136, 2018) used an analytical approach to
derive the general dispersion relation for magneto-acoustic waves in a magnetic
slab of homogeneous plasma, which was enclosed in an asymmetric magnetic
environment. In the present study, we focus on the analysis of wave propagation
in various limiting cases applicable to solar and space plasma or astrophysics.
The thin- and wide-slab approximations, as well as the limits of low and high
plasma-beta values are considered. Utilising the fact that in a weakly
asymmetric slab, the dispersion relation can be decoupled, the behaviour of
quasi-sausage and quasi-kink modes is studied in further analytical and
numerical detail, and their avoided crossings are described. The results
highlight how the asymmetry influences the wave properties, e.g. the phase
speed of eigenmodes, depending on the ratios of external-to-internal densities
and magnetic fields on the two sides. Notably, the phase speeds of surface
modes will converge to different values for quasi-sausage and quasi-kink modes
in the wide-slab limit, and cut-off frequencies are introduced with respect to
both surface and body modes, in thin as well as wide slabs, beyond which the
solutions become leaky. These obtained properties of MHD wave behaviour could
be measured with suitable high-resolution instruments in the future.
