Y. Fan et al., SCATTERING OF P-MODES BY SUNSPOTS .2. CALCULATIONS OF PHASE-SHIFTS FROM A PHENOMENOLOGICAL MODEL, The Astrophysical journal, 451(2), 1995, pp. 877-888
We model the scattering of p-mode waves in a polytropic atmosphere by
localized inhomogeneities in wave speed, pressure, and density of the
medium. The effect of the inhomogeneities is attributed to a source te
rm in the pressure wave equation. The inhomogeneous wave equation for
the scattered waves is solved under the simplification of the Born app
roximation. From the solution for the scattered waves, we compute the
phase shifts between the incoming and outgoing waves of individual mod
es. We find that the variations of the computed phase shifts with degr
ee I and radial order n of the modes show different behavior for inhom
ogeneities with different characteristic depths. Depths significantly
shallower than the depth of the modes seem to show a phase shift depen
dence on I and n that is similar to the qualitative behavior of the ob
served phase shifts produced by sunspots. Direct quantitative comparis
on of the computed phase shifts with observations is limited to modes
with lower degree (I less than or equal to 200) in which the observed
phase shifts are reasonably small so that the Born approximation is ap
plicable. We find that for inhomogeneities with a wave speed contrast
reasonable for sunspots, occupying a volume described by a characteris
tic depth D approximate to 10(8) cm and horizontal radius R approximat
e to 2.5 x 10(9) cm, the computed phase shifts at lower l range are in
agreement with the observed phase shifts from sunspots in both their
magnitudes as well as their variation with I and frequency (or n).