  In the three-dimensional pure Einstein gravity, the geometries of the vacuum
space-times are always trivial, and gravitational waves (gravitons) are
strictly forbidden. For the first time, we find a vacuum circularly symmetric
black hole with nontrivial geometries in $f(R)$ gravity theory, in which a true
singularity appears. In this frame with nontrivial geometry, a perturbative
gravitational wave does exist. Beyond the perturbative level, we make a
constructive proof of the existence of a gravitational wave in $f(R)$ gravity,
where the Birkhoff-like theorem becomes invalid. We find two classes of exact
solutions of circularly symmetric pure gravitational wave radiation and
absorption.
