  Measurements of oscillation frequencies of the Sun and stars can provide
important independent constraints on their internal structure and dynamics.
Seismic models of these oscillations are used to connect structure and rotation
of the star to its resonant frequencies, which are then compared with
observations, the goal being that of minimizing the difference between the two.
Even in the case of the Sun, for which structure models are highly tuned,
observed frequencies show systematic deviations from modeled frequencies, a
phenomenon referred to as the "surface term." The dominant source of this
systematic effect is thought to be vigorous near-surface convection, which is
not well accounted for in both stellar modeling and mode-oscillation physics.
Here we bring to bear the method of homogenization, applicable in the
asymptotic limit of large wavelengths (in comparison to the correlation scale
of convection), to characterize the effect of small-scale surface convection on
resonant-mode frequencies in the Sun. We show that the full oscillation
equations, in the presence of temporally stationary 3-D flows, can be reduced
to an effective "quiet-Sun" wave equation with altered sound speed,
Br\"{u}nt--V\"{a}is\"{a}la frequency and Lamb frequency. We derive the modified
equation and relations for the appropriate averaging of three dimensional flows
and thermal quantities to obtain the properties of this effective medium. Using
flows obtained from three dimensional numerical simulations of near-surface
convection, we quantify their effect on solar oscillation frequencies, and find
that they are shifted systematically and substantially. We argue therefore that
consistent interpretations of resonant frequencies must include modifications
to the wave equation that effectively capture the impact of vigorous
hydrodynamic convection.
