  We consider a series of MHD simulations in which a small amplitude,
transverse velocity perturbation is introduced into a complex magnetic field.
We analysed the deformation of the wave fronts as the perturbation propagates
through the braided magnetic structures and explore the nature of Alfv\'enic
wave phase mixing in this regime. Spatial gradients in the local Alfv\'en speed
and variations in the length of magnetic field lines ensure that small scales
form throughout the propagating wave front due to phase mixing. Additionally,
the presence of complex, intricate current sheets associated with the
background field locally modifies the polarisation of the wave front. The
combination of these two effects enhances the rate of viscous dissipation,
particularly in more complex field configurations. Unlike in classical phase
mixing configurations, the greater spatial extent of Alfv\'en speed gradients
ensures that wave energy is deposited over a larger cross-section of the
magnetic structure. Further, the complexity of the background magnetic field
ensures that small gradients in a wave driver can map to large gradients within
the coronal plasma. The phase mixing of MHD waves in a complex magnetic field
will progress throughout the braided volume. As a result, in a non-ideal regime
wave energy will be dissipated over a greater cross-section than in classical
phase mixing models. The formation rate of small spatial scales in a
propagating wave front is a function of the complexity of the background
magnetic field. As such, if the coronal field is sufficiently complex it
remains plausible that phase mixing induced wave heating can contribute to
maintaining observed temperatures. Furthermore, the weak compressibility of the
transverse wave and the observed phase mixing pattern may provide seismological
information about the nature of the background plasma.
