We study the propagation of waves from the photosphere to the chromosphere of sunspots. From time series of cospatial Ca II H (including its line blends) intensity spectra and polarimetric spectra of Si I λ 1082.7 nm and He I λ 1083.0 nm we retrieve the line-of-sight velocity at several heights. The analysis of the phase difference and amplification spectra shows standing waves for frequencies below 4 mHz and propagating waves for higher frequencies, and allows us to infer the temperature and height where the lines are formed. Using these observational data, we have constructed a model of sunspot, and we have introduced the velocity measured with the photospheric Si I λ 1082.7 nm line as a driver. The numerically propagated wave pattern fits reasonably well with the observed using the lines formed at higher layers, and the simulations reproduce many of the observed features. The observed waves are slow MHD waves propagating longitudinally along field lines.