  Along the lines of the Einstein-Rosen wave equation of General Relativity
(GR), we derive a gravitational wave equation with cylindrical symmetry in the
Einstein-aether (EA) theory. We show that the gravitational wave in the EA is
periodic in time for both the metric functions $\Psi(r,t)$ and $H(r,t)$.
However, in GR, $\Psi(r,t)$ is periodic in time, but $H(r,t)$ is semi-periodic
in time, having a secular drifting in the wave frequency. The evolution of wave
pulses of a given width is entirely different in both theories in the $H(r,t)$
metric function due to this frequency drifting. Another fundamental difference
between the two theories is the gravitational wave velocity. While in GR, the
waves propagate with the speed of light, in EA, there is no upper limit to the
wave velocity, reaching infinity if $c_{13} \rightarrow 1$ and zero if $c_{13}
\rightarrow -\infty$. We also show that energy-momentum pseudotensor and
superpotential get contributions from aether in addition to the usual
gravitational field part. All these characteristics are observational
signatures that differentiate GR and EA.
