Effects of the photospheric cut-off on the p-mode frequency stability

Sub-photospheric acoustic resonators allow for the formation of standing p-mode oscillations by reflecting acoustic waves with frequencies below the acoustic cut-off frequency. We employ the Klein-Gordon equation with a piecewise acoustic potential to study the characteristic frequencies of intermediate-degree p modes, modified by the cut-off effect. For a perfectly reflective photosphere, provided by the infinite value of the acoustic cut-off frequency, characteristic discrete frequencies of the trapped p modes are fully prescribed by the width of the acoustic potential barrier. Finite values of the acoustic cut-off frequency result in the reduction of p-mode frequencies, associated with the decrease in the sound speed by the cut-off effect. For example, for a spherical degree of l = 100, characteristic p-mode frequencies are found to decrease by up to 200 mu Hz and the effect is more pronounced for higher radial harmonics. The frequency separation between two consecutive radial harmonics is shown to behave non-asymptotically with non-uniform spacing in the radial harmonic number due to the cut-off effect. We also show how the 11-yr variability of the Sun's photospheric magnetic field can result in the p-mode frequency shifts through the link between the acoustic cut-off frequency and the plasma parameter $\beta$. Using this model, we readily reproduce the observed typical amplitudes of the p-mode frequency shift and its phase behaviour relative to other 11-yr solar cycle proxies. The use of the developed model for comparison with observations requires its generalization for 2D effects, more realistic profiles of the acoustic potential, and broad-band stochastic drivers.