  Previous studies analyzing the evanescent nature of acoustic waves in the
lower solar atmosphere, up to 300\,km above the photosphere, have shown an
unexpected phase shift of an order of 1\,s between different heights. Those
studies investigated the spectral line \ion{Fe}{1} 6173.3\,\AA, commonly used
for helioseismic measurements. Such phase-shifts can contribute to a
misinterpretation of the measured travel times in local helioseismology,
complicating inferences of, e.g., the deep meridional flow. In this study, we
carry out phase-shift computations using a simulated, fully radiative, and
convective atmosphere from which the \ion{Fe}{1} 6173.3\,\AA\ line is
synthesized. The resulting phase-shifts as functions of frequency across
multiple heights show non-zero values in evanescent waves, similar to what was
found in observational data. Comparing the Doppler-velocities estimated from
the synthesized absorption line with the true velocities directly obtained from
the simulated plasma motions, we find substantial differences in phase-shifts
between the two. This leads us to hypothesize that the non-adiabaticity of the
solar atmosphere yields extra phase-shift contributions to Doppler velocities.
Finally, computing phase-differences for different viewing angles reveals a
systematic center-to-limb variation, similar to what is present in
observations. Overall, this study helps to improve our understanding of the
physical cause of the helioseismic center-to-limb effect.
