  This paper is the third in a series of investigations by the authors. The
nature of fast magnetoacoustic and Alfv\'en waves is investigated in a 2D
$\beta=0$ plasma in the neighbourhood of two dipoles. We use both numerical
simulations (two-step Lax-Wendroff scheme) and analytical techniques (WKB
approximation). It is found that the propagation of the linear fast wave is
dictated by the Alfv\'en speed profile and that close to the null, the wave is
attracted to the neutral point. However, it is also found that in this magnetic
configuration some of the wave can escape the refraction effect; this had not
been seen in previous investigations by the authors. The wave split occurs near
the regions of very high Alfv\'en speed (found near the loci of the two
dipoles). Also, for the set-up investigated it was found that 40% of the wave
energy accumulates at the null. Ohmic dissipation will then extract the wave
energy at this point. The Alfv\'en wave behaves in a different manner in that
part of the wave accumulates along the separatrices and part escapes. Hence,
the current density will accumulate at this part of the topology and this is
where wave heating will occur. The phenomenon of wave accumulation at a
specific place is a feature of both wave types, as is the result that a
fraction of the wave can now escape the numerical box when propagating in this
magnetic configuration.
