The Earth's interior, especially at the core-mantle boundary (CMB), is believed to deviate from the state of hydrostatic equilibrium. This is mainly because the customary approaches in computing the period of free-core nutation (FCN) for an Earth model in hydrostatic equilibrium yield predicted values larger than the observed value by an average of about 30 sidereal days (sd). However, results from new computational approaches suggest that the predicted period of this mode may be closer to the observed value. In this work, I study computationally the dynamics of a compressible liquid core (LC) model, bounded by the rigid mantle and vary the \bv frequency such that the LC density profile is weakly-stably, neutrally, and weakly-unstably stratified, and compute the periods of the CW and FCN. I will show that the period of the CW is unaffected by the variations of the \bv frequency. The period of the FCN is also not significantly affected by these variations except for certain range of \bv frequency for which the core is weakly-unstably stratified. Therefore, we conclude that for most Earth models the LC maybe considered neutrally stratified, i.e., Adams-Williamson condition is met, when computing these modes. Hence, the source of the discrepancy in the computed and observed period of the FCN may be in the treatment of the dynamics of the elastic mantle (MT).