  The excitation of density and bending waves in Saturn's C ring by planetary
oscillation modes presents a unique opportunity to learn about gas giant
interiors and rotation. However, theoretical complications related to Saturn's
rapid and differential rotation pose a barrier to the full utilization of ring
wave detections. We calculate oscillation modes using a complete,
non-perturbative treatment of differential rotation modelled after Saturn's
zonal winds in self-consistently computed, polytropic equilibria. We find that
previous, approximate treatments of the effects of differential rotation in
Saturn overestimate shifts in the frequencies of fundamental modes (f-modes)
thought to be responsible for the majority of the waves detected in the C ring,
due to an omitted modification of the equilibrium shape and structure of the
planet by differential rotation. The bias introduced by these frequency
overestimates is small, but significant relative to the uncertainties afforded
by Cassini data. We additionally consider the non-perturbative effects of
Saturn-like differential rotation on the rotational mixing of f-modes and
internal gravity modes (g-modes), which is relevant to detections of multiple
density waves with very closely split pattern speeds. We find that higher order
rotational effects can produce orders-of-magnitude enhancements in the surface
gravitational perturbations of g-modes dominated by large spherical harmonic
degrees $\ell$, regardless of frequency separation from the sectoral f-mode.
Despite this enhancement, we find that the observed fine-splitting of density
waves is unlikely to involve g-modes dominated by $\ell\gtrsim 10$. This
restriction may aid in the inference of possible internal structures for
Saturn.
