  The arched field lines forming coronal arcades are often observed to undulate
as magnetohydrodynamic (MHD) waves propagate both across and along the magnetic
field. These waves are most likely a combination of resonantly coupled fast
magnetoacoustic waves and Alfv\'en waves. The coupling results in resonant
absorption of the fast waves, converting fast wave energy into Alfv\'en waves.
The fast eigenmodes of the arcade have proven difficult to compute or derive
analytically, largely because of the mathematical complexity that the coupling
introduces. When a traditional spectral decomposition is employed, the discrete
spectrum associated with the fast eigenmodes is often subsumed into the
continuous Alfv\'en spectrum. Thus fast eigenmodes, become collective modes or
quasi-modes. Here we present a spectral decomposition that treats the
eigenmodes as having real frequencies but complex wavenumbers. Using this
procedure we derive dispersion relations, spatial damping rates, and
eigenfunctions for the resonant, fast eigenmodes of the arcade. We demonstrate
that resonant absorption introduces a fast mode that would not exist otherwise.
This new mode is heavily damped by resonant absorption, only travelling a few
wavelengths before losing most of its energy.
