We observed the oscillations in the umbrae of two sunspots using the M
CCD imaging spectrograph at the Mees Solar Observatory (MSO) on Haleak
ala, Maui. We study velocity oscillations with 5 minute periods using
the Doppler shifts of 18 molecular lines observed in both umbrae for o
ver 50 hours each during the interval from 1991 May 11 to 16. In the u
mbrae we find only a weak correlation between the velocity measured wi
th molecular lines and the velocity measured using two iron lines. We
remove solar rotation, image drift, and interpolate all the data onto
an even time grid. We perform four spatial analyses of the umbral velo
city and find (1) there is more power traveling toward the center of t
he umbrae than leaving the center of the umbrae (this provides a direc
t measure of the absorption of p-modes by the sunspot umbrae) (2) the
umbral oscillations display power in the same spatial and temporal fre
quency band as the quiet-Sun oscillations, within the resolution provi
ded by the data, (3) Fourier-Bessel analysis of one umbra shows no obv
ious resonant frequencies which might represent natural oscillation mo
des, and (4) the centers of the umbrae have less rms velocity at high
spatial wavenumber than the edges of the umbrae. From these analyses w
e conclude: (1) the photospheric umbral oscillations are driven by a s
ource external to the sunspot, the global p-mode oscillations, (2) the
re are no resonant frequencies in the oscillations, which suggest ther
e is no acoustic cavity formed by the umbrae, and (3) the absorption o
f acoustic waves occurs inside the umbrae. We compare the observed abs
orption function for one umbra with simple wave absorption models and
find that a point absorber at the umbral center does not fit the obser
vations; rather the data equally support either a linear acoustic abso
rption across the umbra or total acoustic absorption on a ring surface
at a radius of 5 Mm from the umbral center.