Gravitational interactions between mass anomalies in the core and the
mantle cause rotational oscillations of the Earth that can be detected
as fluctuations in the length of day. A simple theoretical expression
for the frequency of these oscillations is derived using a Lagrangian
formulation. The value of the frequency depends on the aspherical dis
tribution of mass, providing a useful constraint on the structure of t
he Earth. Numerical estimates are obtained using two different models
of density, both of which are based on seismically inferred density an
omalies. The predicted period of the gravitational oscillation is 2 to
3 years, depending on the choice of density model. Longer period mode
s of oscillation can also arise from the smaller convective fluctuatio
ns in the core about the hydrostatic state. Qualitative estimates for
these low frequency oscillations depend on the size of the density flu
ctuations, but typical values yield a period of several hundred years.