### 2.2 Properties of solar oscillations

The five-minute solar oscillations were first discovered by Leighton et al. (1962) and interpreted as
standing acoustic waves by Ulrich (1970) and Leibacher and Stein (1971). Deubner (1975) then confirmed
that the power in the oscillations is concentrated at discrete frequencies for any given horizontal
wavenumber, as predicted by Ulrich’s theory. The driving mechanism of solar oscillations is
believed to be near-surface turbulent convection (Goldreich and Keeley, 1977). Solar and stellar
oscillations are discussed in details by Cox (1980), Gough (1993), Unno et al. (1989), and
Christensen-Dalsgaard (2002). Particularly useful are the lecture notes of J. Christensen-Dalsgaard
(Christensen-Dalsgaard, 2003).
The small oscillations of a sphere can be represented by a linear superposition of eigenmodes, each
characterized by a set of three indices: the radial order the spherical harmonic degree and the
azimuthal order . For instance, the radial displacement of a fluid element can be written as

where is the radius, and are spherical-polar coordinates (colatitude and longitude), and is
time. The are spherical harmonics, is a complex mode amplitude, and is the radial
eigenfunction of the mode with frequency . By convention, corresponds to the number of
nodes of the radial eigenfunction, indicates the total number of nodal lines on spheres,
and tells how many of these nodal lines cross the equator. A spherically symmetric star
would give rise to a spectrum of azimuthally degenerate frequencies. However, rotation and
other perturbations lift the -fold -degeneracy of the frequency of nonradial mode
().
Figure 1 shows -averaged power spectra of solar oscillations derived from series of tracked Doppler
images of size 30° × 30° (near disk center) observed simultaneously by the MDI and TON instruments.
Each ridge in the power spectrum corresponds to a different radial order . The lowest frequency ridge
() is for the fundamental (f) modes. The f modes are identified as surface gravity waves,
with nearly the dispersion relation for deep water waves, , where is the angular
temporal frequency, is the gravitational acceleration at the Sun’s surface,
is the horizontal wavenumber, and is the solar radius. The f modes
propagate horizontally. All other ridges, denoted , correspond to acoustic modes, or p modes.
The restoring force for p modes is pressure. The ridge immediately above the f mode ridge is
, the next one , and so forth. Low- and high- modes penetrate deeper inside the
Sun. For frequencies above the acoustic cutoff frequency (5.3 mHz), acoustic waves are not
trapped inside the Sun. Acoustic modes with similar values of propagate to similar
depths inside the Sun. For degrees larger than about 150, wave damping becomes significant and
modes are not resolved anymore (continuous ridges). Figure 2 is another beautiful example of a
power spectrum of solar oscillations, obtained from data collected during the 1994 south pole
campaign.