  We study how tunneling through a potential barrier $V(x)$ can be enhanced by
an additional harmonically oscillating electric field ${\mathfrak
E}(t)={\mathfrak E}_0\cos(\omega t)$. To this end, we transform into the
Kramers-Henneberger frame and calculate the coupled Floquet channels
numerically. We find distinct signatures of resonances when the incident energy
$E$ equals the driving frequency $\omega=E$ which clearly shows the breakdown
of the time-averaged potential approximation. As a simple model for
experimental applications (e.g., in solid state physics), we study the
rectangular potential, which can also be benchmarked with respect to analytical
results. Finally, we consider the truncated Coulomb potential relevant for
nuclear fusion.
