  Solar chromospheric observations of sunspot umbrae offer an exceptional view
of magneto-hydrodynamic wave phenomena. In recent years, a wealth of wave
signatures related to propagating magneto-acoustic modes have been presented,
which demonstrate complex spatial and temporal structuring of the wave
components. Theoretical modelling has demonstrated how these ubiquitous waves
are consistent with an m=0 slow magneto-acoustic mode, which are excited by
trapped sub-photospheric acoustic (p-mode) waves. However, the spectrum of
umbral waves is broad, suggesting that the observed signatures represent the
superposition of numerous frequencies and/or modes. We apply Fourier filtering,
in both spatial and temporal domains, to extract chromospheric umbral wave
characteristics consistent with an m=1 slow magneto-acoustic mode. This
identification has not been described before. Angular frequencies of 0.037 +/-
0.007 rad/s (2.1 +/- 0.4 deg/s), corresponding to a period approximately 170 s
for the m=1 mode are uncovered for spatial wavenumbers in the range of
0.45<k<0.90 arcsec^-1 (5000-9000 km). Theoretical dispersion relations are
solved, with corresponding eigenfunctions computed, which allows the density
perturbations to be investigated and compared with our observations. Such
magnetohydrodynamic modelling confirms our interpretation that the identified
wave signatures are the first direct observations of an m=1 slow
magneto-acoustic mode in the chromospheric umbra of a sunspot.
