Radio multifrequency observations of galaxy clusters. The Abell
  399$-$401 pair
  Galaxy clusters are assembled via merging of smaller structures, in a process
that generates shocks and turbulence in the intra cluster medium and produces
radio emission in the form of halos and relics. The cluster pair A 399-A 401
represents a special case: both clusters host a radio halo and recent LOFAR
observations at 140~MHz revealed the presence of a radio bridge connecting the
two clusters and two candidate relics, one South of A 399 and the other in
between the two clusters in proximity of a shock front detected in X-ray
observations. In this paper we present Westerbork observations at 1.7, 1.4 and
1.2~GHz and 346~MHz of the A 399-A 401 cluster pair. We detected the radio halo
in the A 399 cluster at 346~MHz, extending up to $\sim 650$~kpc and with a $125
\pm 6$~mJy flux density. Its spectral index between 1.4~GHz and 346~MHz and
between 140~MHz and 346~MHz is $\alpha = 1.47 \pm 0.05$, and $\alpha = 1.75 \pm
0.14$ respectively. The two candidate relics are also seen at 346~MHz and we
determined their spectral index to be $\alpha = 1.10 \pm 0.14$ and $\alpha =
1.46 \pm 0.14$. The low surface brightness bridge connecting the two clusters
is below the noise level at 346~MHz, therefore we constrained the bridge
average spectral to be steep, i.e. $\alpha > 1.5$ at $2\sigma$ confidence
level. This result favours the scenario where dynamically-induced turbulence is
a viable mechanism to reaccelerate a population of mildly relativistic
particles and amplify magnetic fields even in cluster bridges, i.e. on scales
of a few Mpcs.
