SCATTERING OF P-MODES BY SUNSPOTS .2. CALCULATIONS OF PHASE-SHIFTS FROM A PHENOMENOLOGICAL MODEL

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 term in the pressure wave equation. The inhomogeneous wave equation for the scattered waves is solved under the simplification of the Born approximation. From the solution for the scattered waves, we compute the phase shifts between the incoming and outgoing waves of individual modes. We find that the variations of the computed phase shifts with degree I and radial order n of the modes show different behavior for inhomogeneities with different characteristic depths. Depths significantly shallower than the depth of the modes seem to show a phase shift dependence on I and n that is similar to the qualitative behavior of the observed phase shifts produced by sunspots. Direct quantitative comparison 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 applicable. We find that for inhomogeneities with a wave speed contrast reasonable for sunspots, occupying a volume described by a characteristic depth D approximate to 10(8) cm and horizontal radius R approximate 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).