  In the solar atmosphere, the acoustic cutoff frequency is a local quantity
which depends on the atmospheric height. It separates the low-frequency
evanescent waves from the high-frequency propagating waves. We measure the
cutoff frequency of slow magnetoacoustic waves at various heights of a sunspot
umbra and compare the results with the estimations from several analytical
formulae. We analyzed the oscillations in the umbra of a sunspot belonging to
active region NOAA 12662 observed in the 10830 \AA\ spectral region with the
GREGOR Infrared Spectrograph and in the Fe I 5435 \AA\ line with the GREGOR
Fabry-P\'erot Interferometer. Both instrumets are attached to the GREGOR
telescope at the Observatorio del Teide, Tenerife, Spain. We have computed the
phase and amplification spectra between the velocity measured from different
pairs of lines that sample various heights of the solar atmosphere. The cutoff
frequency and its height variation have been estimated from the inspection of
the spectra. At the deep umbral photosphere the cutoff frequency is around 5
mHz and it increases to 6 mHz at higher photospheric layers. At the
chromosphere the cutoff is $\sim 3.1$ mHz. The comparison of the
observationally determined cutoff with the theoretically predicted values
reveals an agreement in the general trend and a reasonable match at the
chromosphere, but also significant quantitative differences at the photosphere.
Our analyses show strong evidence of the variation of the cutoff frequency with
height in a sunspot umbra, which is not fully accounted for by current
analytical estimations. This result has implications for our understanding of
wave propagation, the seismology of active regions, and the evaluation of
heating mechanisms based on compressible waves.
