  We give an explicit formula for the signature of handlebody bundles over the
circle in terms of the homological monodromy. This gives a cobounding function
of Meyer's signature cocycle on the mapping class group of a $3$-dimensional
handlebody, i.e., the handlebody group. As an application, we give a
topological interpretation for the generator of the first cohomology group of
the hyperelliptic handlebody group.
