  We present a study of diffusion enhancement of underdamped Brownian particles
in 1D symmetric space-periodic potential due to external symmetric
time-periodic forcing with zero mean. We show that the diffusivity can be
enhanced by many orders of magnitude at appropriate choice of the forcing
amplitude and frequency. The diffusivity demonstrates TAD, abnormal
(decreasing) temperature dependence at forcing amplitudes exceeding certain
value. At any fixed forcing frequency $\Omega$ normal temperature dependence of
the diffusivity is restored at low enough temperatures,
$T<T_\textrm{TAD}(\Omega)$ --- in contrast with the problem with constant
external forcing. At fixed temperature at small forcing frequency the
diffusivity either slowly decreases with $\Omega$, or (at stronger forcing)
goes through a maximum near $\Omega_2$, reciprocal superdiffusion stage
duration. At high frequencies, between $\Omega_2$ and a fraction of the
oscillation frequency at the potential minimum, the diffusivity is shown to
decrease with $\Omega$ according to a power law, with exponent related to the
transient superdiffusion exponent. This behavior is found similar for the cases
of sinusoidal in time and piecewise constant periodic ("square") forcing.
