  We analyze the propagation of waves in sunspots from the photosphere to the
chromosphere using time series of co-spatial Ca II H intensity spectra
(including its line blends) and polarimetric spectra of Si I 10827 and the He I
10830 multiplet. From the Doppler shifts of these lines we retrieve the
variation of the velocity along the line-of-sight at several heights. Phase
spectra are used to obtain the relation between the oscillatory signals. Our
analysis reveals standing waves at frequencies lower than 4 mHz and a
continuous propagation of waves at higher frequencies, which steepen into
shocks in the chromosphere when approaching the formation height of the Ca II H
core. The observed non-linearities are weaker in Ca II H than in He I lines.
Our analysis suggests that the Ca II H core forms at a lower height than the He
I 10830 line: a time delay of about 20 s is measured between the Doppler signal
detected at both wavelengths. We fit a model of linear slow magnetoacoustic
wave propagation in a stratified atmosphere with radiative losses according to
Newton's cooling law to the phase spectra and derive the difference in the
formation height of the spectral lines. We show that the linear model describes
well the wave propagation up to the formation height of Ca II H, where
non-linearities start to become very important.
