  In this paper we show that the phase mixing of continuum Alfv\'{e}n waves
and/or continuum slow waves in magnetic structures of the solar atmosphere as,
e.g., coronal arcades, can create the illusion of wave propagation across the
magnetic field. This phenomenon could be erroneously interpreted as fast
magnetosonic waves. The cross-field propagation due to phase mixing of
continuum waves is apparent because there is no real propagation of energy
across the magnetic surfaces. We investigate the continuous Alfv\'{e}n and slow
spectra in 2D Cartesian equilibrium models with a purely poloidal magnetic
field. We show that apparent superslow propagation across the magnetic surfaces
in solar coronal structures is a consequence of the existence of continuum
Alfv\'{e}n waves and continuum slow waves that naturally live on those
structures and phase mix as time evolves. The apparent cross-field phase
velocity is related to the spatial variation of the local Alfv\'{e}n/slow
frequency across the magnetic surfaces and is slower than the Alfv\'{e}n/sound
velocities for typical coronal conditions. Understanding the nature of the
apparent cross-field propagation is important for the correct analysis of
numerical simulations and the correct interpretation of observations.
