Velocity oscillations in sunspot umbrae have been measured simultaneously
in two spectral lines: the photospheric
Silicon {\sc i} \hbox{10827 \AA} line and the chromospheric Helium
{\sc i} \hbox{10830 \AA} multiplet. From the full Stokes inversion of
temporal series of spectropolarimetric observations we retrieved,
among other parameters, the line of sight velocity
temporal variations at photospheric and chromospheric heights.
Chromospheric velocity oscillations show a three minute period with a clear
sawtooth shape typical of propagating shock wave fronts. Photospheric velocity
oscillations have basically a five minute period, although the power spectrum
also shows a secondary peak in the three minute band which has proven to be
predecessor for its chromospheric counterpart.
The derived phase spectra yield a value of the atmospheric cut-off frequency
around $4$ mHz and give evidence for the upward propagation of higher frequency
oscillation modes. The phase spectrum has been reproduced with a simple model
of linear vertical propagation of slow magneto-acoustic waves in a stratified
magnetized atmosphere
that accounts for radiative losses through Newton's cooling law. The model
explains the main features in the phase spectrum, and allows us to compute
the theoretical time delay between the photospheric and
chromospheric signals, which happens to have a strong dependence on frequency.
We find a very good agreement between this and the time delay obtained
directly from the cross-correlation of
photospheric and chromospheric velocity maps filtered around the 6 mHz band.
This allows us to infer that the 3-minute power observed at chromospheric
heights comes directly from the photosphere by means of linear wave
propagation, rather than from non-linear interaction of 5-minute (and/or
higher frequency) modes.