  From a classical analysis, it is shown that the nondiffractive accelerating
gravitational Airy wave packets are solutions of Einstein equations for their
linearized tensor modes in a Friedmann-Lema\^itre-Robertson-Walker cosmological
background filled with a perfect fluid, with equations of state $w=1/3$ and
$w=-1/3$. These solutions have finite energy, presenting accelerating behavior
due to the structured spatial form of the wavepacket. This is manifested by
curved trajectories along the wave path. Also, using spectral functions, it is
possible, with these packets, to construct more general, arbitrary wave
packets. All these new solutions bring insights on new forms for gravitational
wave propagation.
