  Complete asymptotic expansions are developed for slow, Alfv\'en and fast
magnetohydrodynamic waves at the base of an isothermal three-dimensional (3D)
plane stratified atmosphere. Together with existing convergent Frobenius series
solutions about $z=\infty$, matchings are numerically calculated that
illuminate the fates of slow and Alfv\'en waves injected from below. An
Alfv\'en wave in a two-dimensional model is 2.5D in the sense that the wave
propagates in the plane of the magnetic field but its polarization is normal to
it in an ignorable horizontal direction, and the wave remains an Alfv\'en wave
throughout. The rotation of the plane of wave propagation away from the
vertical plane of the magnetic field pushes the plasma displacement vector away
from horizontal, thereby coupling it to stratification. It is shown that potent
slow-Alfv\'en coupling occurs in such 3D models. It is found that about 50% of
direction-averaged Alfv\'en wave flux generated in the low atmosphere at
frequencies comparable to or greater than the acoustic cutoff can reach the top
as Alfv\'en flux for small magnetic field inclinations $\theta$, and this
increases to 80% or more with increasing $\theta$. On the other hand,
direction-averaged slow waves can be 40% effective in converting to Alfv\'en
waves at small inclination, but this reduces sharply with increasing $\theta$
and wave frequency. Together with previously explored fast-slow and
fast-Alfv\'en couplings, this provides valuable insights into which injected
transverse waves can reach the upper atmosphere as Alfv\'en waves, with
implications for solar and stellar coronal heating and solar/stellar wind
acceleration.

Solar and Stellar Astrophysics