We present extensive spectroscopic time series observations of the mul
tiperiodic, rapidly rotating, delta Scuti star tau Pegasi. Information
about the oscillations is contained within the patterns of line-profi
le variation of the star's blended absorption-line spectrum. We introd
uce the new technique of Doppler deconvolution with which to extract t
hese patterns by modeling the intrinsic stellar spectrum and the broad
ening functions for each spectrum in the time series. Frequencies and
modes of oscillation are identified from the variations using the tech
nique of Fourier-Doppler imaging and a two-dimensional least-squares c
leaning algorithm. We find a rich mode spectrum with degrees up to l =
20 and with frequencies below about 35 cycles day(-1). Those modes wi
th the largest amplitudes have frequencies that lie within a narrow ba
nd. We conclude that the observed spectrum can be explained if the mod
es of tau Peg propagate in the prograde direction with l similar or eq
ual to \m\ and with frequencies that are about equal in the corotating
frame of the star. We discuss the implications of these results for t
he prospect of delta Scuti seismology.